Exploring the Free Energy Landscape: From Dynamics to Networks and Back

The knowledge of the Free Energy Landscape topology is the essential key to understand many biochemical processes. The determination of the conformers of a protein and their basins of attraction takes a central role for studying molecular isomerization reactions. In this work, we present a novel framework to unveil the features of a Free Energy Landscape answering questions such as how many meta-stable conformers are, how the hierarchical relationship among them is, or what the structure and kinetics of the transition paths are. Exploring the landscape by molecular dynamics simulations, the microscopic data of the trajectory are encoded into a Conformational Markov Network. The structure of this graph reveals the regions of the conformational space corresponding to the basins of attraction. In addition, handling the Conformational Markov Network, relevant kinetic magnitudes as dwell times or rate constants, and the hierarchical relationship among basins, complete the global picture of the landscape. We show the power of the analysis studying a toy model of a funnel-like potential and computing efficiently the conformers of a short peptide, the dialanine, paving the way to a systematic study of the Free Energy Landscape in large peptides.Comment: PLoS Computational Biology (in press

Orientation-dependent backbone-only residue pair scoring functions for fixed backbone protein design

<p>Abstract</p> <p>Background</p> <p>Empirical scoring functions have proven useful in protein structure modeling. Most such scoring functions depend on protein side chain conformations. However, backbone-only scoring functions do not require computationally intensive structure optimization and so are well suited to protein design, which requires fast score evaluation. Furthermore, scoring functions that account for the distinctive relative position and orientation preferences of residue pairs are expected to be more accurate than those that depend only on the separation distance.</p> <p>Results</p> <p>Residue pair scoring functions for fixed backbone protein design were derived using only backbone geometry. Unlike previous studies that used spherical harmonics to fit 2D angular distributions, Gaussian Mixture Models were used to fit the full 3D (position only) and 6D (position and orientation) distributions of residue pairs. The performance of the 1D (residue separation only), 3D, and 6D scoring functions were compared by their ability to identify correct threading solutions for a non-redundant benchmark set of protein backbone structures. The threading accuracy was found to steadily increase with increasing dimension, with the 6D scoring function achieving the highest accuracy. Furthermore, the 3D and 6D scoring functions were shown to outperform side chain-dependent empirical potentials from three other studies. Next, two computational methods that take advantage of the speed and pairwise form of these new backbone-only scoring functions were investigated. The first is a procedure that exploits available sequence data by averaging scores over threading solutions for homologs. This was evaluated by applying it to the challenging problem of identifying interacting transmembrane alpha-helices and found to further improve prediction accuracy. The second is a protein design method for determining the optimal sequence for a backbone structure by applying Belief Propagation optimization using the 6D scoring functions. The sensitivity of this method to backbone structure perturbations was compared with that of fixed-backbone all-atom modeling by determining the similarities between optimal sequences for two different backbone structures within the same protein family. The results showed that the design method using 6D scoring functions was more robust to small variations in backbone structure than the all-atom design method.</p> <p>Conclusions</p> <p>Backbone-only residue pair scoring functions that account for all six relative degrees of freedom are the most accurate and including the scores of homologs further improves the accuracy in threading applications. The 6D scoring function outperformed several side chain-dependent potentials while avoiding time-consuming and error prone side chain structure prediction. These scoring functions are particularly useful as an initial filter in protein design problems before applying all-atom modeling.</p

Molecular Structures of Quiescently Grown and Brain-Derived Polymorphic Fibrils of the Alzheimer Amyloid Aβ9-40 Peptide: A Comparison to Agitated Fibrils

The presence of amyloid deposits consisting primarily of Amyloid-β (Aβ) fibril in the brain is a hallmark of Alzheimer's disease (AD). The morphologies of these fibrils are exquisitely sensitive to environmental conditions. Using molecular dynamics simulations combined with data from previously published solid-state NMR experiments, we propose the first atomically detailed structures of two asymmetric polymorphs of the Aβ9-40 peptide fibril. The first corresponds to synthetic fibrils grown under quiescent conditions and the second to fibrils derived from AD patients' brain-extracts. Our core structure in both fibril structures consists of a layered structure in which three cross-β subunits are arranged in six tightly stacked β-sheet layers with an antiparallel hydrophobic-hydrophobic and an antiparallel polar-polar interface. The synthetic and brain-derived structures differ primarily in the side-chain orientation of one β-strand. The presence of a large and continually exposed hydrophobic surface (buried in the symmetric agitated Aβ fibrils) may account for the higher toxicity of the asymmetric fibrils. Our model explains the effects of external perturbations on the fibril lateral architecture as well as the fibrillogenesis inhibiting action of amphiphilic molecules

Potentials of Mean Force for Protein Structure Prediction Vindicated, Formalized and Generalized

Understanding protein structure is of crucial importance in science, medicine and biotechnology. For about two decades, knowledge based potentials based on pairwise distances -- so-called "potentials of mean force" (PMFs) -- have been center stage in the prediction and design of protein structure and the simulation of protein folding. However, the validity, scope and limitations of these potentials are still vigorously debated and disputed, and the optimal choice of the reference state -- a necessary component of these potentials -- is an unsolved problem. PMFs are loosely justified by analogy to the reversible work theorem in statistical physics, or by a statistical argument based on a likelihood function. Both justifications are insightful but leave many questions unanswered. Here, we show for the first time that PMFs can be seen as approximations to quantities that do have a rigorous probabilistic justification: they naturally arise when probability distributions over different features of proteins need to be combined. We call these quantities reference ratio distributions deriving from the application of the reference ratio method. This new view is not only of theoretical relevance, but leads to many insights that are of direct practical use: the reference state is uniquely defined and does not require external physical insights; the approach can be generalized beyond pairwise distances to arbitrary features of protein structure; and it becomes clear for which purposes the use of these quantities is justified. We illustrate these insights with two applications, involving the radius of gyration and hydrogen bonding. In the latter case, we also show how the reference ratio method can be iteratively applied to sculpt an energy funnel. Our results considerably increase the understanding and scope of energy functions derived from known biomolecular structures

New statistical potential for quality assessment of protein models and a survey of energy functions

Abstract Background Scoring functions, such as molecular mechanic forcefields and statistical potentials are fundamentally important tools in protein structure modeling and quality assessment. Results The performances of a number of publicly available scoring functions are compared with a statistical rigor, with an emphasis on knowledge-based potentials. We explored the effect on accuracy of alternative choices for representing interaction center types and other features of scoring functions, such as using information on solvent accessibility, on torsion angles, accounting for secondary structure preferences and side chain orientation. Partially based on the observations made, we present a novel residue based statistical potential, which employs a shuffled reference state definition and takes into account the mutual orientation of residue side chains. Atom- and residue-level statistical potentials and Linux executables to calculate the energy of a given protein proposed in this work can be downloaded from http://www.fiserlab.org/potentials. Conclusions Among the most influential terms we observed a critical role of a proper reference state definition and the benefits of including information about the microenvironment of interaction centers. Molecular mechanical potentials were also tested and found to be over-sensitive to small local imperfections in a structure, requiring unfeasible long energy relaxation before energy scores started to correlate with model quality.</p

Finite Size Effects in Simulations of Protein Aggregation

It is becoming increasingly clear that the soluble protofibrillar species that proceed amyloid fibril formation are associated with a range of neurodegenerative disorders such as Alzheimer's and Parkinson diseases. Computer simulations of the processes that lead to the formation of these oligomeric species are starting to make significant contributions to our understanding of the determinants of protein aggregation. We simulate different systems at constant concentration but with a different number of peptides and we study the how the finite number of proteins affects the underlying free energy of the system and therefore the relative stability of the species involved in the process. If not taken into account, this finite size effect can undermine the validity of theoretical predictions regarding the relative stability of the species involved and the rates of conversion from one to the other. We discuss the reasons that give rise to this finite size effect form both a probabilistic and energy fluctuations point of view and also how this problem can be dealt by a finite size scaling analysis

Discrete Kinetic Models from Funneled Energy Landscape Simulations

A general method for facilitating the interpretation of computer simulations of protein folding with minimally frustrated energy landscapes is detailed and applied to a designed ankyrin repeat protein (4ANK). In the method, groups of residues are assigned to foldons and these foldons are used to map the conformational space of the protein onto a set of discrete macrobasins. The free energies of the individual macrobasins are then calculated, informing practical kinetic analysis. Two simple assumptions about the universality of the rate for downhill transitions between macrobasins and the natural local connectivity between macrobasins lead to a scheme for predicting overall folding and unfolding rates, generating chevron plots under varying thermodynamic conditions, and inferring dominant kinetic folding pathways. To illustrate the approach, free energies of macrobasins were calculated from biased simulations of a non-additive structure-based model using two structurally motivated foldon definitions at the full and half ankyrin repeat resolutions. The calculated chevrons have features consistent with those measured in stopped flow chemical denaturation experiments. The dominant inferred folding pathway has an “inside-out”, nucleation-propagation like character

Four Distances between Pairs of Amino Acids Provide a Precise Description of their Interaction

The three-dimensional structures of proteins are stabilized by the interactions between amino acid residues. Here we report a method where four distances are calculated between any two side chains to provide an exact spatial definition of their bonds. The data were binned into a four-dimensional grid and compared to a random model, from which the preference for specific four-distances was calculated. A clear relation between the quality of the experimental data and the tightness of the distance distribution was observed, with crystal structure data providing far tighter distance distributions than NMR data. Since the four-distance data have higher information content than classical bond descriptions, we were able to identify many unique inter-residue features not found previously in proteins. For example, we found that the side chains of Arg, Glu, Val and Leu are not symmetrical in respect to the interactions of their head groups. The described method may be developed into a function, which computationally models accurately protein structures

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

A Kinetic Model of Trp-Cage Folding from Multiple Biased Molecular Dynamics Simulations

Trp-cage is a designed 20-residue polypeptide that, in spite of its size, shares several features with larger globular proteins. Although the system has been intensively investigated experimentally and theoretically, its folding mechanism is not yet fully understood. Indeed, some experiments suggest a two-state behavior, while others point to the presence of intermediates. In this work we show that the results of a bias-exchange metadynamics simulation can be used for constructing a detailed thermodynamic and kinetic model of the system. The model, although constructed from a biased simulation, has a quality similar to those extracted from the analysis of long unbiased molecular dynamics trajectories. This is demonstrated by a careful benchmark of the approach on a smaller system, the solvated Ace-Ala3-Nme peptide. For the Trp-cage folding, the model predicts that the relaxation time of 3100 ns observed experimentally is due to the presence of a compact molten globule-like conformation. This state has an occupancy of only 3% at 300 K, but acts as a kinetic trap. Instead, non-compact structures relax to the folded state on the sub-microsecond timescale. The model also predicts the presence of a state at of 4.4 Å from the NMR structure in which the Trp strongly interacts with Pro12. This state can explain the abnormal temperature dependence of the and chemical shifts. The structures of the two most stable misfolded intermediates are in agreement with NMR experiments on the unfolded protein. Our work shows that, using biased molecular dynamics trajectories, it is possible to construct a model describing in detail the Trp-cage folding kinetics and thermodynamics in agreement with experimental data