Einstein Model of a Graph to Characterize Protein Folded/Unfolded States.

The folded structures of proteins can be accurately predicted by deep learning algorithms from their amino-acid sequences. By contrast, in spite of decades of research studies, the prediction of folding pathways and the unfolded and misfolded states of proteins, which are intimately related to diseases, remains challenging. A two-state (folded/unfolded) description of protein folding dynamics hides the complexity of the unfolded and misfolded microstates. Here, we focus on the development of simplified order parameters to decipher the complexity of disordered protein structures. First, we show that any connected, undirected, and simple graph can be associated with a linear chain of atoms in thermal equilibrium. This analogy provides an interpretation of the usual topological descriptors of a graph, namely the Kirchhoff index and Randic resistance, in terms of effective force constants of a linear chain. We derive an exact relation between the Kirchhoff index and the average shortest path length for a linear graph and define the free energies of a graph using an Einstein model. Second, we represent the three-dimensional protein structures by connected, undirected, and simple graphs. As a proof of concept, we compute the topological descriptors and the graph free energies for an all-atom molecular dynamics trajectory of folding/unfolding events of the proteins Trp-cage and HP-36 and for the ensemble of experimental NMR models of Trp-cage. The present work shows that the local, nonlocal, and global force constants and free energies of a graph are promising tools to quantify unfolded/disordered protein states and folding/unfolding dynamics. In particular, they allow the detection of transient misfolded rigid states.

Exploring Structural Flexibility and Stability of α-Synuclein by the Landau-Ginzburg-Wilson Approach.

α-Synuclein (αS) is the principal protein component of the Lewy body and Lewy neurite deposits that are found in the brains of the victims of one of the most prevalent neurodegenerative disorders, Parkinson's disease. αS can be qualified as a chameleon protein because of the large number of different conformations that it is able to adopt: it is disordered under physiological conditions in solution, in equilibrium with a minor α-helical tetrameric form in the cytoplasm, and is α-helical when bound to a cell membrane. Also, in vitro, αS forms polymorphic amyloid fibrils with unique arrangements of cross-ß-sheet motifs. Therefore, it is of interest to elucidate the origins of the structural flexibility of αS and what makes αS stable in different conformations. We address these questions here by analyzing the experimental structures of the micelle-bound, tetrameric, and fibrillar αS in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger equation. It is illustrated that without molecular dynamics simulations the kinks are capable of identifying the key residues causing structural flexibility of αS. Also, the stability of the experimental structures of αS is investigated by simulating heating/cooling trajectories using the Glauber algorithm. The findings are consistent with experiments.

Wild-Type α-Synuclein and Variants Occur in Different Disordered Dimers and Pre-Fibrillar Conformations in Early Stage of Aggregation.

α-Synuclein is a 140 amino-acid intrinsically disordered protein mainly found in the brain. Toxic α-synuclein aggregates are the molecular hallmarks of Parkinson's disease. In vitro studies showed that α-synuclein aggregates in oligomeric structures of several 10th of monomers and into cylindrical structures (fibrils), comprising hundred to thousands of proteins, with polymorphic cross-ß-sheet conformations. Oligomeric species, formed at the early stage of aggregation remain, however, poorly understood and are hypothezised to be the most toxic aggregates. Here, we studied the formation of wild-type (WT) and mutant (A30P, A53T, and E46K) dimers of α-synuclein using coarse-grained molecular dynamics. We identified two principal segments of the sequence with a higher propensity to aggregate in the early stage of dimerization: residues 36-55 and residues 66-95. The transient α-helices (residues 53-65 and 73-82) of α-synuclein monomers are destabilized by A53T and E46K mutations, which favors the formation of fibril native contacts in the N-terminal region, whereas the helix 53-65 prevents the propagation of fibril native contacts along the sequence for the WT in the early stages of dimerization. The present results indicate that dimers do not adopt the Greek key motif of the monomer fold in fibrils but form a majority of disordered aggregates and a minority (9-15%) of pre-fibrillar dimers both with intra-molecular and intermolecular ß-sheets. The percentage of residues in parallel ß-sheets is by increasing order monomer < disordered dimers < pre-fibrillar dimers. Native fibril contacts between the two monomers are present in the NAC domain for WT, A30P, and A53T and in the N-domain for A53T and E46K. Structural properties of pre-fibrillar dimers agree with rupture-force atomic force microscopy and single-molecule Förster resonance energy transfer available data. This suggests that the pre-fibrillar dimers might correspond to the smallest type B toxic oligomers. The probability density of the dimer gyration radius is multi-peaks with an average radius that is 10 Å larger than the one of the monomers for all proteins. The present results indicate that even the elementary α-synuclein aggregation step, the dimerization, is a complicated phenomenon that does not only involve the NAC region.

Mechanistic Kinetic Model Reveals How Amyloidogenic Hydrophobic Patches Facilitate the Amyloid-ß Fibril Elongation.

Abnormal aggregation of amyloid ß (Aß) peptides into fibrils plays a critical role in the development of Alzheimer's disease. A two-stage "dock-lock" model has been proposed for the Aß fibril elongation process. However, the mechanisms of the Aß monomer-fibril binding process have not been elucidated with the necessary molecular-level precision, so it remains unclear how the lock phase dynamics leads to the overall in-register binding of the Aß monomer onto the fibril. To gain mechanistic insights into this critical step during the fibril elongation process, we used molecular dynamics (MD) simulations with a physics-based coarse-grained UNited-RESidue (UNRES) force field and sampled extensively the dynamics of the lock phase process, in which a fibril-bound Aß(9-40) peptide rearranged to establish the native docking conformation. Analysis of the MD trajectories with Markov state models was used to quantify the kinetics of the rearrangement process and the most probable pathways leading to the overall native docking conformation of the incoming peptide. These revealed a key intermediate state in which an intra-monomer hairpin is formed between the central core amyloidogenic patch 18VFFA21 and the C-terminal hydrophobic patch 34LMVG37. This hairpin structure is highly favored as a transition state during the lock phase of the fibril elongation. We propose a molecular mechanism for facilitation of the Aß fibril elongation by amyloidogenic hydrophobic patches.

Probing Protein Aggregation Using the Coarse-Grained UNRES Force Field.

Protein aggregation is the cause of many, often lethal, diseases, including the Alzheimer's, Parkinson's, and Huntington's diseases, and familial amyloidosis. Theoretical investigation of the mechanism of this process, including the structures of the oligomeric intermediates which are the most toxic, is difficult because of long time scale of aggregation. Coarse-grained models, which enable us to extend the simulation time scale by three or more orders of magnitude, are, therefore, of great advantage in such studies. In this chapter, we describe the application of the physics-based UNited RESidue (UNRES) force field developed in our laboratory to study protein aggregation, in both free simulations and simulations of aggregation propagation from an existing template (seed), and illustrate it with the examples of Aß-peptide aggregation and Aß-peptide-assisted aggregation of the peptides derived from the repeat domains of tau (TauRD).

Missense Mutations Modify the Conformational Ensemble of the α-Synuclein Monomer Which Exhibits a Two-Phase Characteristic.

α-Synuclein is an intrinsically disordered protein occurring in different conformations and prone to aggregate in ß-sheet structures, which are the hallmark of the Parkinson disease. Missense mutations are associated with familial forms of this neuropathy. How these single amino-acid substitutions modify the conformations of wild-type α-synuclein is unclear. Here, using coarse-grained molecular dynamics simulations, we sampled the conformational space of the wild type and mutants (A30P, A53P, and E46K) of α-synuclein monomers for an effective time scale of 29.7 ms. To characterize the structures, we developed an algorithm, CUTABI (CUrvature and Torsion based of Alpha-helix and Beta-sheet Identification), to identify residues in the α-helix and ß-sheet from Cα -coordinates. CUTABI was built from the results of the analysis of 14,652 selected protein structures using the Dictionary of Secondary Structure of Proteins (DSSP) algorithm. DSSP results are reproduced with 93% of success for 10 times lower computational cost. A two-dimensional probability density map of α-synuclein as a function of the number of residues in the α-helix and ß-sheet is computed for wild-type and mutated proteins from molecular dynamics trajectories. The density of conformational states reveals a two-phase characteristic with a homogeneous phase (state B, ß-sheets) and a heterogeneous phase (state HB, mixture of α-helices and ß-sheets). The B state represents 40% of the conformations for the wild-type, A30P, and E46K and only 25% for A53T. The density of conformational states of the B state for A53T and A30P mutants differs from the wild-type one. In addition, the mutant A53T has a larger propensity to form helices than the others. These findings indicate that the equilibrium between the different conformations of the α-synuclein monomer is modified by the missense mutations in a subtle way. The α-helix and ß-sheet contents are promising order parameters for intrinsically disordered proteins, whereas other structural properties such as average gyration radius, R g , or probability distribution of R g cannot discriminate significantly the conformational ensembles of the wild type and mutants. When separated in states B and HB, the distributions of R g are more significantly different, indicating that global structural parameters alone are insufficient to characterize the conformational ensembles of the α-synuclein monomer.

Investigation of Phosphorylation-Induced Folding of an Intrinsically Disordered Protein by Coarse-Grained Molecular Dynamics.

Apart from being the most common mechanism of regulating protein function and transmitting signals throughout the cell, phosphorylation has an ability to induce disorder-to-order transition in an intrinsically disordered protein. In particular, it was shown that folding of the intrinsically disordered protein, eIF4E-binding protein isoform 2 (4E-BP2), can be induced by multisite phosphorylation. Here, the principles that govern the folding of phosphorylated 4E-BP2 (pT37pT46 4E-BP218-62) are investigated by analyzing canonical and replica exchange molecular dynamics trajectories, generated with the coarse-grained united-residue force field, in terms of local and global motions and the time dependence of formation of contacts between Cαs of selected pairs of residues. The key residues involved in the folding of the pT37pT46 4E-BP218-62 are elucidated by this analysis. The correlations between local and global motions are identified. Moreover, for a better understanding of the physics of the formation of the folded state, the experimental structure of the pT37pT46 4E-BP218-62 is analyzed in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger equation. It is shown that without molecular dynamics simulations the kinks are able to identify not only the phosphorylated sites of protein, the key players in folding, but also the reasons for the weak stability of the pT37pT46 4E-BP218-62.

How Useful can the Voigt Profile be in Protein Folding Processes?

The analytical expression for the Voigt profile, along with its simplified forms for the Gaussian and Lorentzian dominance, is presented. The applicability of the Voigt profile in the description of anomalous diffusion phenomena, ubiquitous in different fields of science including protein folding, is discussed. It is shown that the Voigt profile is a good descriptor of the processes occurring in protein folding and in the native state. The usefulness of the Voigt profile in deriving important information of the diffusive motions in proteins from a quasielastic incoherent neutron scattering experiments is illustrated.

Curvature and Torsion of Protein Main Chain as Local Order Parameters of Protein Unfolding.

Thermal protein unfolding resembles a global (two-state) phase transition. At the local scale, protein unfolding is, however, heterogeneous and probe dependent. Here, we consider local order parameters defined by the local curvature and torsion of the protein main chain. Because chemical shifts (CS's) measured by NMR spectroscopy are extremely sensitive to the local atomic environment, CS has served as a local probe of thermal unfolding of proteins by varying the position of the atomic isotope along the amino acid sequence. The variation of the CS of each Cα atom along the sequence as a function of the temperature defines a local heat-induced denaturation curve. We demonstrate that these local heat-induced denaturation curves mirror the local protein nativeness defined by the free energy landscape of the local curvature and torsion of the protein main chain described by the Cα-Cα virtual bonds. Comparison between molecular dynamics simulations and CS data of the gpW protein demonstrates that some local native states defined by the local curvature and torsion of the main chain, mainly located in secondary structures, are coupled to each other whereas others, mainly located in flexible protein segments, are not. Consequently, CS's of some residues are faithful reporters of global protein unfolding, with heat-induced denaturation curves similar to the average global one, whereas other residues remain silent about the protein unfolded state. For the latter, the local deformation of the protein main chain, characterized by its local curvature and torsion, is not cooperatively coupled to global unfolding.

New Insights into Folding, Misfolding, and Nonfolding Dynamics of a WW Domain.

Intermediate states in protein folding are associated with formation of amyloid fibrils, which are responsible for a number of neurodegenerative diseases. Therefore, prevention of the aggregation of folding intermediates is one of the most important problems to overcome. Recently, we studied the origins and prevention of formation of intermediate states with the example of the Formin binding protein 28 (FBP28) WW domain. We demonstrated that the replacement of Leu26 by Asp26 or Trp26 (in ∼15% of the folding trajectories) can alter the folding scenario from three-state folding, a major folding scenario for the FBP28 WW domain (WT) and its mutants, toward two-state or downhill folding at temperatures below the melting point. Here, for a better understanding of the physics of the formation/elimination of intermediates, (i) the dynamics and energetics of formation of ß-strands in folding, misfolding, and nonfolding trajectories of these mutants (L26D and L26W) is investigated; (ii) the experimental structures of WT, L26D, and L26W are analyzed in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger equation. We show that the formation of each ß-strand in folding trajectories is accompanied by the emergence of kinks in internal coordinate space as well as a decrease in local free energy. In particular, the decrease in downhill folding trajectory is ∼7 kcal/mol, while it varies between 31 and 48 kcal/mol for the three-state folding trajectory. The kink analyses of the experimental structures give new insights into formation of intermediates, which may become a useful tool for preventing aggregation.

Dependence of the Formation of Tau and Aß Peptide Mixed Aggregates on the Secondary Structure of the N-Terminal Region of Aß.

One of the hallmarks of Alzheimer's disease is the formation of aggregates of the tau protein, a process that can be facilitated by the presence of fibrils formed by the amyloid ß peptide (Aß). However, the mechanism that triggers tau aggregation is still a matter of debate. The effect of Aß40 fibrils on the aggregation of the repeat domain of tau (TauRD) is investigated here by employing coarse-grained molecular dynamics simulations. The results indicate that the repeat domain of tau has a high affinity for Aß40 fibrils, with the 261GSTENLK267 fragment of tau driving TauRD toward the 16KLVFFA21 fragment in Aß40. Monomeric Aß40, in which the 16KLVFFA21 fragment is rarely found in an extended conformation (as in the fibril), has a low affinity for the TauRD, indicating that the ability of Aß40 fibrils to bind to the TauRD depends on the 16KLVFFA21 fragment of Aß adopting an extended conformation.

Statistical Model To Decipher Protein Folding/Unfolding at a Local Scale.

Protein folding/unfolding can be analyzed experimentally at a local scale by monitoring the physical properties of local probes as a function of the temperature, for example, the distance between fluorophores or the values of chemical shifts of backbone atoms. Here, the analytical Lifson-Roig model for the helix-coil transition is modified to analyze local thermal unfolding of the fast-folder W protein of bacteriophage lambda (gpW) simulated by all-atom molecular dynamics (MD) simulations in explicit solvent at 15 different temperatures. The protein structure is described by the coarse-grained dihedral angles (γ) and bond angles (θ) built between successive Cα-Cα virtual bonds. Each (γ,θ) pair serves as a local probe of protein unfolding. Local native/non-native states are defined for each pair of (γ,θ) angles by analyzing the free-energy landscapes Δ G(γ,θ) computed from MD trajectories. The three local elementary equilibrium constants of the model are extracted for each (γ,θ) pair along the sequence from MD simulations, and the model predictions are compared to the MD data. Using only the local equilibrium constants as an input, we show that the local denaturation curves computed from the model partition function fit their MD simulated counterparts in a satisfying manner without any adjustment. In the model and MD simulations, gpW unfolds gradually between 320 and 340 K, with an average native percentage decreasing from 0.8 (320 K) to 0.2 (340 K). In the prism of the model, there is no stable structure at the local scale in this 20 K unfolding temperature range. The enthalpy change upon local unfolding computed from the model and from MD trajectories suggests that the unfolded state between 320 and 340 K corresponds to a dynamical equilibrium between a large ensemble of constantly changing structures. The present results confirm the downhill unfolding of gpW, which does not obey a two-state global folding/unfolding model, and shed light on the interpretation of local denaturation curves.

From a Highly Disordered to a Metastable State: Uncovering Insights of α-Synuclein.

α-Synuclein (αS) is a major constituent of Lewy bodies, the insoluble aggregates that are the hallmark of one of the most prevalent neurodegenerative disorders, Parkinson's disease (PD). The vast majority of experiments in vitro and in vivo provide extensive evidence that a disordered monomeric form is the predominant state of αS in water solution, and it undergoes a large-scale disorder-to-helix transition upon binding to vesicles of different types. Recently, another form, tetrameric, of αS with a stable helical structure was identified experimentally. It has been shown that a dynamic intracellular population of metastable αS tetramers and monomers coexists normally; and the tetramer plays an essential role in maintaining αS homeostasis. Therefore, it is of interest to know whether the tetramer can serve as a means of preventing or delaying the start of PD. Before answering this very important question, it is, first, necessary to find out, on an atomistic level, a correlation between tetramers and monomers; what mediates tetramer formation and what makes a tetramer stable. We address these questions here by investigating both monomeric and tetrameric forms of αS. In particular, by examining correlations between the motions of the side chains and the main chain, steric parameters along the amino-acid sequence, and one- and two-dimensional free-energy landscapes along the coarse-grained dihedral angles γ and δ and principal components, respectively, in monomeric and tetrameric αS, we were able to shed light on a fundamental relationship between monomers and tetramers, and the key residues involved in mediating formation of a tetramer. Also, the reasons for the stability of tetrameric αS and inability of monomeric αS to fold are elucidated here.

Lysosomal enzyme tripeptidyl peptidase 1 destabilizes fibrillar Aß by multiple endoproteolytic cleavages within the ß-sheet domain.

Accumulation of amyloid-beta (Aß), which is associated with Alzheimer's disease, can be caused by excess production or insufficient clearance. Because of its ß-sheet structure, fibrillar Aß is resistant to proteolysis, which would contribute to slow degradation of Aß plaques in vivo. Fibrillar Aß can be internalized by microglia, which are the scavenger cells of the brain, but the fibrils are degraded only slowly in microglial lysosomes. Cathepsin B is a lysosomal protease that has been shown to proteolyze fibrillar Aß. Tripeptidyl peptidase 1 (TPP1), a lysosomal serine protease, possesses endopeptidase activity and has been shown to cleave peptides between hydrophobic residues. Herein, we demonstrate that TPP1 is able to proteolyze fibrillar Aß efficiently. Mass spectrometry analysis of peptides released from fibrillar Aß digested with TPP1 reveals several endoproteolytic cleavages including some within ß-sheet regions that are important for fibril formation. Using molecular dynamics simulations, we demonstrate that these cleavages destabilize fibrillar ß-sheet structure. The demonstration that TPP1 can degrade fibrillar forms of Aß provides insight into the turnover of fibrillar Aß and may lead to new therapeutic methods to increase degradation of Aß plaques.

Sequence-, structure-, and dynamics-based comparisons of structurally homologous CheY-like proteins.

We recently introduced a physically based approach to sequence comparison, the property factor method (PFM). In the present work, we apply the PFM approach to the study of a challenging set of sequences-the bacterial chemotaxis protein CheY, the N-terminal receiver domain of the nitrogen regulation protein NT-NtrC, and the sporulation response regulator Spo0F. These are all response regulators involved in signal transduction. Despite functional similarity and structural homology, they exhibit low sequence identity. PFM sequence comparison demonstrates a statistically significant qualitative difference between the sequence of CheY and those of the other two proteins that is not found using conventional alignment methods. This difference is shown to be consonant with structural characteristics, using distance matrix comparisons. We also demonstrate that residues participating strongly in native contacts during unfolding are distributed differently in CheY than in the other two proteins. The PFM result is also in accord with dynamic simulation results of several types. Molecular dynamics simulations of all three proteins were carried out at several temperatures, and it is shown that the dynamics of CheY are predicted to differ from those of NT-NtrC and Spo0F. The predicted dynamic properties of the three proteins are in good agreement with experimentally determined B factors and with fluctuations predicted by the Gaussian network model. We pinpoint the differences between the PFM and traditional sequence comparisons and discuss the informatic basis for the ability of the PFM approach to detect physical differences between these sequences that are not apparent from traditional alignment-based comparison.

Elucidating Important Sites and the Mechanism for Amyloid Fibril Formation by Coarse-Grained Molecular Dynamics.

Fibrils formed by the ß-amyloid (Aß) peptide play a central role in the development of Alzheimer's disease. In this study, the principles governing their growth and stability are investigated by analyzing canonical and replica-exchange molecular dynamics trajectories of Aß(9-40) fibrils. In particular, an unstructured monomer was allowed to interact freely with an Aß fibril template. Trajectories were generated with the coarse-grained united-residue force field, and one- and two-dimensional free-energy landscapes (FELs) along the backbone virtual-bond angle θ and backbone virtual-bond-dihedral angle γ of each residue and principal components, respectively, were analyzed. Also, thermal unbinding (unfolding) of an Aß peptide from the fibril template was investigated. These analyses enable us to illustrate the entire process of Aß fibril elongation and to elucidate the key residues involved in it. Several different pathways were identified during the search for the fibril conformation by the monomer, which finally follows a dock-lock mechanism with two distinct locking stages. However, it was found that the correct binding, with native hydrogen bonds, of the free monomer to the fibril template at both stages is crucial for fibril elongation. In other words, if the monomer is incorrectly bound (with nonnative hydrogen bonds) to the fibril template during the first "docking" stage, it can remain attached to it for a long time before it dissociates and either attempts a different binding or allows another monomer to bind. This finding is consistent with an experimentally observed "stop-and-go" mechanism of fibril growth.

Eliminating a Protein Folding Intermediate by Tuning a Local Hydrophobic Contact.

Intermediate states in protein folding may slow folding, and sometimes can provide a starting point for aggregation. Recently, the FBP28 WW domain of the formin-binding protein was used as a model for a computational study of the origin and prevention of intermediate-state formation, and local hydrophobic interactions of Leu26 were implicated. Here, we combine new simulations over a broad temperature range with experimental temperature-jump data to study this site in more detail. We replace Leu26 by Asp26 or Trp26 to alter the folding scenario from three-state folding toward two-state or downhill folding at temperatures below the melting point, whereas the wild type shows two-state behavior only near its melting temperature. We offer an explanation of this behavior mainly in terms of principles of hydrophobic interactions.

Preventing fibril formation of a protein by selective mutation.

The origins of formation of an intermediate state involved in amyloid formation and ways to prevent it are illustrated with the example of the Formin binding protein 28 (FBP28) WW domain, which folds with biphasic kinetics. Molecular dynamics of protein folding trajectories are used to examine local and global motions and the time dependence of formation of contacts between C(α)s and C(ß)s of selected pairs of residues. Focus is placed on the WT FBP28 WW domain and its six mutants (L26D, L26E, L26W, E27Y, T29D, and T29Y), which have structures that are determined by high-resolution NMR spectroscopy. The origins of formation of an intermediate state are elucidated, viz. as formation of hairpin 1 by a hydrophobic collapse mechanism causing significant delay of formation of both hairpins, especially hairpin 2, which facilitates the emergence of an intermediate state. It seems that three-state folding is a major folding scenario for all six mutants and WT. Additionally, two-state and downhill folding scenarios were identified in ∼ 15% of the folding trajectories for L26D and L26W, in which both hairpins are formed by the Matheson-Scheraga mechanism much faster than in three-state folding. These results indicate that formation of hairpins connecting two antiparallel ß-strands determines overall folding. The correlations between the local and global motions identified for all folding trajectories lead to the identification of the residues making the main contributions in the formation of the intermediate state. The presented findings may provide an understanding of protein folding intermediates in general and lead to a procedure for their prevention.