Principle Of Minimal Frustration Research Articles

Frustraevo: a web server to localize and quantify the conservation of local energetic frustration in protein families.

According to the Principle of Minimal Frustration, folded proteins canonly have a minimal number of strong energetic conflicts in their native states. However, not all interactions are energetically optimized for folding but some remain in energetic conflict, i.e. they are highly frustrated. This remaining local energetic frustration has been shown to be statistically correlated with distinct functional aspects such as protein-protein interaction sites, allosterism and catalysis. Fuelled by the recent breakthroughs in efficient protein structure prediction that have made available good quality models for most proteins, we have developed a strategy to calculate local energetic frustration within large protein families and quantify its conservation over evolutionary time. Based on this evolutionary information we can identify how stability and functional constraints have appeared at the common ancestor of the family and have been maintained over the course of evolution. Here, we present FrustraEvo, a web server tool to calculate and quantify the conservation of local energetic frustration in protein families.

Exploring the structural acrobatics of fold-switching proteins using simplified structure-based models.

Metamorphic proteins are a paradigm of the protein folding process, by encoding two or more native states, highly dissimilar in terms of their secondary, tertiary, and even quaternary structure, on a single amino acid sequence. Moreover, these proteins structurally interconvert between these native states in a reversible manner at biologically relevant timescales as a result of different environmental cues. The large-scale rearrangements experienced by these proteins, and their sometimes high mass interacting partners that trigger their metamorphosis, makes the computational and experimental study of their structural interconversion challenging. Here, we present our efforts in studying the refolding landscapes of two quintessential metamorphic proteins, RfaH and KaiB, using simplified dual-basin structure-based models (SBMs), rigorously footed on the energy landscape theory of protein folding and the principle of minimal frustration. By using coarse-grained models in which the native contacts and bonded interactions extracted from the available experimental structures of the two native states of RfaH and KaiB are merged into a single Hamiltonian, dual-basin SBM models can be generated and savvily calibrated to explore their fold-switch in a reversible manner in molecular dynamics simulations. We also describe how some of the insights offered by these simulations have driven the design of experiments and the validation of the conformational ensembles and refolding routes observed using this simple and computationally efficient models.

Funneled energy landscape unifies principles of protein binding and evolution

Most proteins have evolved to spontaneously fold into native structure and specifically bind with their partners for the purpose of fulfilling biological functions. According to Darwin, protein sequences evolve through random mutations, and only the fittest survives. The understanding of how the evolutionary selection sculpts the interaction patterns for both biomolecular folding and binding is still challenging. In this study, we incorporated the constraint of functional binding into the selection fitness based on the principle of minimal frustration for the underlying biomolecular interactions. Thermodynamic stability and kinetic accessibility were derived and quantified from a global funneled energy landscape that satisfies the requirements of both the folding into the stable structure and binding with the specific partner. The evolution proceeds via a bowl-like evolution energy landscape in the sequence space with a closed-ring attractor at the bottom. The sequence space is increasingly reduced until this ring attractor is reached. The molecular-interaction patterns responsible for folding and binding are identified from the evolved sequences, respectively. The residual positions participating in the interactions responsible for folding are highly conserved and maintain the hydrophobic core under additional evolutionary constraints of functional binding. The positions responsible for binding constitute a distributed network via coupling conservations that determine the specificity of binding with the partner. This work unifies the principles of protein binding and evolution under minimal frustration and sheds light on the evolutionary design of proteins for functions.

Frustration and inhomogeneous environments in relaxation of open chains with Ising-type interactions.

Frustration can contribute to very slow relaxation times in large open chains, as in spin glasses and in biopolymers. However, frustration may not be sufficient to produce broken ergodicity in finite systems. Here we employ a system-plus-reservoir approach to investigate how strongly inhomogeneous environments and frustration compete in the relaxation of finite open chains. We find a sufficient condition for our inhomogeneous environments to break ergodicity. We use the microscopic model to derive a Markovian quantum master equation for a generic chain with ultrastrong intrachain couplings. We show that this microscopic model avoids a spurious broken ergodicity we find in the phenomenological model. We work out an explicit example of broken ergodicity due to the inhomogeneous environment of an unfrustrated spin chain as far as simulating a recent experiment on protein denaturation (where environment inhomogeneity is especially relevant). We finally show that an inhomogeneous environment can mitigate the effects of frustration-induced degeneracies.

Go-Kit: A Tool To Enable Energy Landscape Exploration of Proteins.

Coarse-grained Go̅-like models, based on the principle of minimal frustration, provide valuable insight into fundamental questions in the field of protein folding and dynamics. In conjunction with commonly used molecular dynamics (MD) simulations, energy landscape exploration methods like discrete path sampling (DPS) with Go̅-like models can provide quantitative details of the thermodynamics and kinetics of proteins. Here we present Go-kit, a software that facilitates the setup of MD and DPS simulations of several flavors of Go̅-like models. Go-kit is designed for use with MD (GROMACS) and DPS (PATHSAMPLE) simulation engines that are open source. The Go-kit code is written in python2.7 and is also open source. A case study for the ribosomal protein S6 is discussed to illustrate the utility of the software, which is available at https://github.com/gokit1/gokit .

Local frustration around enzyme active sites

Conflicting biological goals often meet in the specification of protein sequences for structure and function. Overall, strong energetic conflicts are minimized in folded native states according to the principle of minimal frustration, so that a sequence can spontaneously fold, but local violations of this principle open up the possibility to encode the complex energy landscapes that are required for active biological functions. We survey the local energetic frustration patterns of all protein enzymes with known structures and experimentally annotated catalytic residues. In agreement with previous hypotheses, the catalytic sites themselves are often highly frustrated regardless of the protein oligomeric state, overall topology, and enzymatic class. At the same time a secondary shell of more weakly frustrated interactions surrounds the catalytic site itself. We evaluate the conservation of these energetic signatures in various family members of major enzyme classes, showing that local frustration is evolutionarily more conserved than the primary structure itself.

Features of Energy Landscape Topography in the Space of Torsion Angles for Macromolecules that Form Unique 3D Structures

The topography of the potential energy surface (PES) for macromolecules in the configuration space of torsion angles was considered. The effects that interatomic repulsion exerts on the surface topography are discussed. Singularities in interatomic potentials were not found to critically affect the energy landscapes in the space of torsion angles because singularity points are inaccessible at thermal energies. Classically forbidden regions for motions in the vicinity of singularity points lead to an exponential decrease in the coefficients of the Fourier series expansion of the potential energy function as harmonic numbers increase. The minimal frustration principle, which underlies the current ideas that energy funnels exist to ensure the folding of biopolymers into unique 3D structures, was extended to include maxima of the energy landscape, yielding PESs whose topography is characterized not only by a global minimum, but also by a global maximum of the energy landscape (atomic spatial conflicts are excluded in this context). In this case with each of the angular variables, the coordinates of the extreme points differ strongly by π. The resulting surfaces are antisymmetric with respect to the reversal of the directions of the acting interatomic forces (namely, the regions of the global minimum and global maximum interchange when the sign of the interaction energy changes). The topography of such surfaces ideally provides the folding of a macromolecular chain into a unique 3D structure. Entropy effects that arise during the motion of a representative point along the PES form a smooth barrier around the entrance to the energy funnel at lower temperatures and destroy the energy funnel at higher temperatures. The effects agree with data on protein-folding kinetics.

Evidence for the principle of minimal frustration in the evolution of protein folding landscapes

Theoretical and experimental studies have firmly established that protein folding can be described by a funneled energy landscape. This funneled energy landscape is the result of foldable protein sequences evolving following the principle of minimal frustration, which allows proteins to rapidly fold to their native biologically functional conformations. For a protein family with a given functional fold, the principle of minimal frustration suggests that, independent of sequence, all proteins within this family should fold with similar rates. However, depending on the optimal living temperature of the organism, proteins also need to modulate their thermodynamic stability. Consequently, the difference in thermodynamic stability should be primarily caused by differences in the unfolding rates. To test this hypothesis experimentally, we performed comprehensive thermodynamic and kinetic analyses of 15 different proteins from the thioredoxin family. Eight of these thioredoxins were extant proteins from psychrophilic, mesophilic, or thermophilic organisms. The other seven protein sequences were obtained using ancestral sequence reconstruction and can be dated back over 4 billion years. We found that all studied proteins fold with very similar rates but unfold with rates that differ up to three orders of magnitude. The unfolding rates correlate well with the thermodynamic stability of the proteins. Moreover, proteins that unfold slower are more resistant to proteolysis. These results provide direct experimental support to the principle of minimal frustration hypothesis.

Globular-disorder transition in proteins: a compromise between hydrophobic and electrostatic interactions?

The charge-hydrophobicity correlation of globular and disordered proteins is explored using a generalized self-consistent field theoretical method combined with Monte Carlo simulations. Globular and disordered protein sequences with varied mean net charge and mean hydrophobicity are designed by theory, while Metropolis Monte Carlo generates a suitable ensemble of conformations. Results imply a transition of the dominant interactions between globular and disordered proteins across the charge-hydrophobicity boundary. It is observed that the charge-hydrophobicity boundary actually represents a trade-off between the repulsive and attractive interactions in a protein sequence. The attractive interactions predominate on the globular side of the boundary, while the repulsive interactions prevail on the disordered side. For globular proteins, core forming hydrophobic interactions are dominant leading to a minimally frustrated native conformation. For disordered proteins, the repulsive electrostatic interactions prevail yielding a minimally frustrated region comprising of an expanded, dynamic conformational ensemble. Thus, protein disorder, like protein folding, satisfies the principle of minimal frustration. All results are compared to real globular and disordered proteins. Thus this algorithm may be useful to probe the conformational characteristics of disordered proteins.

Constructing a folding model for protein S6 guided by native fluctuations deduced from NMR structures.

The diversity in a set of protein nuclear magnetic resonance (NMR) structures provides an estimate of native state fluctuations that can be used to refine and enrich structure-based protein models (SBMs). Dynamics are an essential part of a protein's functional native state. The dynamics in the native state are controlled by the same funneled energy landscape that guides the entire folding process. SBMs apply the principle of minimal frustration, drawn from energy landscape theory, to construct a funneled folding landscape for a given protein using only information from the native structure. On an energy landscape smoothed by evolution towards minimal frustration, geometrical constraints, imposed by the native structure, control the folding mechanism and shape the native dynamics revealed by the model. Native-state fluctuations can alternatively be estimated directly from the diversity in the set of NMR structures for a protein. Based on this information, we identify a highly flexible loop in the ribosomal protein S6 and modify the contact map in a SBM to accommodate the inferred dynamics. By taking into account the probable native state dynamics, the experimental transition state is recovered in the model, and the correct order of folding events is restored. Our study highlights how the shared energy landscape connects folding and function by showing that a better description of the native basin improves the prediction of the folding mechanism.

Predictive energy landscapes for folding membrane protein assemblies.

We study the energy landscapes for membrane protein oligomerization using the Associative memory, Water mediated, Structure and Energy Model with an implicit membrane potential (AWSEM-membrane), a coarse-grained molecular dynamics model previously optimized under the assumption that the energy landscapes for folding α-helical membrane protein monomers are funneled once their native topology within the membrane is established. In this study we show that the AWSEM-membrane force field is able to sample near native binding interfaces of several oligomeric systems. By predicting candidate structures using simulated annealing, we further show that degeneracies in predicting structures of membrane protein monomers are generally resolved in the folding of the higher order assemblies as is the case in the assemblies of both nicotinic acetylcholine receptor and V-type Na(+)-ATPase dimers. The physics of the phenomenon resembles domain swapping, which is consistent with the landscape following the principle of minimal frustration. We revisit also the classic Khorana study of the reconstitution of bacteriorhodopsin from its fragments, which is the close analogue of the early Anfinsen experiment on globular proteins. Here, we show the retinal cofactor likely plays a major role in selecting the final functional assembly.

Evolution, energy landscapes and the paradoxes of protein folding.

Protein folding has been viewed as a difficult problem of molecular self-organization. The search problem involved in folding however has been simplified through the evolution of folding energy landscapes that are funneled. The funnel hypothesis can be quantified using energy landscape theory based on the minimal frustration principle. Strong quantitative predictions that follow from energy landscape theory have been widely confirmed both through laboratory folding experiments and from detailed simulations. Energy landscape ideas also have allowed successful protein structure prediction algorithms to be developed. The selection constraint of having funneled folding landscapes has left its imprint on the sequences of existing protein structural families. Quantitative analysis of co-evolution patterns allows us to infer the statistical characteristics of the folding landscape. These turn out to be consistent with what has been obtained from laboratory physicochemical folding experiments signaling a beautiful confluence of genomics and chemical physics.

Native structure-based modeling and simulation of biomolecular systems per mouse click.

BackgroundMolecular dynamics (MD) simulations provide valuable insight into biomolecular systems at the atomic level. Notwithstanding the ever-increasing power of high performance computers current MD simulations face several challenges: the fastest atomic movements require time steps of a few femtoseconds which are small compared to biomolecular relevant timescales of milliseconds or even seconds for large conformational motions. At the same time, scalability to a large number of cores is limited mostly due to long-range interactions. An appealing alternative to atomic-level simulations is coarse-graining the resolution of the system or reducing the complexity of the Hamiltonian to improve sampling while decreasing computational costs. Native structure-based models, also called Gō-type models, are based on energy landscape theory and the principle of minimal frustration. They have been tremendously successful in explaining fundamental questions of, e.g., protein folding, RNA folding or protein function. At the same time, they are computationally sufficiently inexpensive to run complex simulations on smaller computing systems or even commodity hardware. Still, their setup and evaluation is quite complex even though sophisticated software packages support their realization.ResultsHere, we establish an efficient infrastructure for native structure-based models to support the community and enable high-throughput simulations on remote computing resources via GridBeans and UNICORE middleware. This infrastructure organizes the setup of such simulations resulting in increased comparability of simulation results. At the same time, complete workflows for advanced simulation protocols can be established and managed on remote resources by a graphical interface which increases reusability of protocols and additionally lowers the entry barrier into such simulations for, e.g., experimental scientists who want to compare their results against simulations. We demonstrate the power of this approach by illustrating it for protein folding simulations for a range of proteins.ConclusionsWe present software enhancing the entire workflow for native structure-based simulations including exception-handling and evaluations. Extending the capability and improving the accessibility of existing simulation packages the software goes beyond the state of the art in the domain of biomolecular simulations. Thus we expect that it will stimulate more individuals from the community to employ more confidently modeling in their research.Electronic supplementary materialThe online version of this article (doi:10.1186/1471-2105-15-292) contains supplementary material, which is available to authorized users.

Simulating Biomolecular Folding and Function by Native‐Structure‐Based/Go‐Type Models

AbstractThe 2013 Nobel Prize in Chemistry highlights how crucial computer simulations have become for many scientific and engineering fields. Nowadays, scientific progress is not only driven by the interplay of new experimental measurements and increasingly sophisticated theoretical frameworks, but also by an incredible toolbox of complex computational models meeting ubiquitously available computing power and data storage facilities. Quantum mechanical (QM) calculations can be condensed into molecular mechanics (MM) force fields and coupled QM/MM calculations can derive atomic and molecular properties of biomolecular or materials science systems with high accuracy. Pure MM simulations driven by Monte Carlo or molecular dynamics algorithms are widely applied in biological chemistry/physics and can investigate large biomolecular systems, such as proteins, DNA, or RNA. One coarse‐grained class of these models, native‐structure‐based or Go models, are based on energy landscape theory and the principle of minimal frustration. Herein, an ensemble of converging pathways guide protein folding on a funnel‐like shape of the entire energy landscape towards the native state. Simulations based on these ideas have been tremendously successful in explaining protein folding and function. Their history and recent application highlights are reviewed.

Effects of Energetic Heterogeneity on Protein Folding Dynamics Across Many Non-Homologous Proteins

Native structure based models (SBMs, “Go-models”) are based on energy landscape theory and the principle of minimal frustration [1,2]. Using this framework strongly reduces computational demands and allows us to investigate the influence of the contact-energy weight onto the folding process for many different protein families. A large set (approx. 200) of non-homologous monomeric proteins sized from 50-150 amino acids are simulated on a coarse-grained level, representing each amino acid by a single bead. A fully automatized workflow implemented with the help of eSBMTools [3] allows to quantify typical folding properties like phi-values, folding free energy landscapes and transition state ensembles (TSEs) for the simulated proteins.Conventional SBMs use sequence independent homogeneous energy weights for the native contact energy. We compare these “vanilla” models with sequence dependent heterogeneous “flavored” energy weights as introduced by Miyazawa and Jernigan [4]. Subsequently, the dynamics of forming TSEs are compared by the sequential formation of tertiary interfaces between secondary structure elements in vanilla and flavored simulations. We find that, despite the energetic dissimilarity, the sequential ordering is similar between both types of simulations and appears as a robust property. This suggests that protein folding dynamics are strongly influenced by the native protein topology.1. Onuchic, J.N. and P.G. Wolynes, Theory of protein folding. Curr. Opin. Struct. Bio., 2004. 14(1): p. 70-75.2. Schug, A. and J.N. Onuchic, Current opinion in pharmacology, 2010. 10(6): p. 709-14.3. Lutz, B., C. Sinner, G. Heuermann, A. Verma, and A. Schug, eSBMTools1.0: enhanced native structure-based modeling tools. Bioinformatics, 2013. doi:10.1093/bioinformatics/btt478.4. Miyazawa, S. and R.L. Jernigan, Residue-residue potentials with a favorable contact pair term and an unfavorable high packing density term, for simulation and threading. Journal of Molecular Biology, 1996. 256(3): p. 623-44.

Exploring the Minimally Frustrated Energy Landscape of Unfolded ACBP

The unfolded state of globular proteins is not well described by a simple statistical coil due to residual structural features, such as secondary structure or transiently formed long-range contacts. The principle of minimal frustration predicts that the unfolded ensemble is biased toward productive regions in the conformational space determined by the native structure. Transient long-range contacts, both native-like and non-native-like, have previously been shown to be present in the unfolded state of the four-helix-bundle protein acyl co-enzyme binding protein (ACBP) as seen from both perturbations in nuclear magnetic resonance (NMR) chemical shifts and structural ensembles generated from NMR paramagnetic relaxation data. To study the nature of the contacts in detail, we used paramagnetic NMR relaxation enhancements, in combination with single-point mutations, to obtain distance constraints for the acid-unfolded ensemble of ACBP. We show that, even in the acid-unfolded state, long-range contacts are specific in nature and single-point mutations affect the free-energy landscape of the unfolded protein. Using this approach, we were able to map out concerted, interconnected, and productive long-range contacts. The correlation between the native-state stability and compactness of the denatured state provides further evidence for native-like contact formation in the denatured state. Overall, these results imply that, even in the earliest stages of folding, ACBP dynamics are governed by native-like contacts on a minimally frustrated energy landscape.

ESBMTools 1.0: enhanced native structure-based modeling tools

Molecular dynamics simulations provide detailed insights into the structure and function of biomolecular systems. Thus, they complement experimental measurements by giving access to experimentally inaccessible regimes. Among the different molecular dynamics techniques, native structure-based models (SBMs) are based on energy landscape theory and the principle of minimal frustration. Typically used in protein and RNA folding simulations, they coarse-grain the biomolecular system and/or simplify the Hamiltonian resulting in modest computational requirements while achieving high agreement with experimental data. eSBMTools streamlines running and evaluating SBM in a comprehensive package and offers high flexibility in adding experimental- or bioinformatics-derived restraints. We present a software package that allows setting up, modifying and evaluating SBM for both RNA and proteins. The implemented workflows include predicting protein complexes based on bioinformatics-derived inter-protein contact information, a standardized setup of protein folding simulations based on the common PDB format, calculating reaction coordinates and evaluating the simulation by free-energy calculations with weighted histogram analysis method or by phi-values. The modules interface with the molecular dynamics simulation program GROMACS. The package is open source and written in architecture-independent Python2. http://sourceforge.net/projects/esbmtools/. alexander.schug@kit.edu. Supplementary data are available at Bioinformatics online.

Analyzing the effect of homogeneous frustration in protein folding

The energy landscape theory has been an invaluable theoretical framework in the understanding of biological processes such as protein folding, oligomerization, and functional transitions. According to the theory, the energy landscape of protein folding is funneled toward the native state, a conformational state that is consistent with the principle of minimal frustration. It has been accepted that real proteins are selected through natural evolution, satisfying the minimum frustration criterion. However, there is evidence that a low degree of frustration accelerates folding. We examined the interplay between topological and energetic protein frustration. We employed a Cα structure-based model for simulations with a controlled nonspecific energetic frustration added to the potential energy function. Thermodynamics and kinetics of a group of 19 proteins are completely characterized as a function of increasing level of energetic frustration. We observed two well-separated groups of proteins: one group where a little frustration enhances folding rates to an optimal value and another where any energetic frustration slows down folding. Protein energetic frustration regimes and their mechanisms are explained by the role of non-native contact interactions in different folding scenarios. These findings strongly correlate with the protein free-energy folding barrier and the absolute contact order parameters. These computational results are corroborated by principal component analysis and partial least square techniques. One simple theoretical model is proposed as a useful tool for experimentalists to predict the limits of improvements in real proteins.

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.

Analyzing Protein Folding by High-throughput Simulations

Molecular Dynamics allows investigating the dynamical properties of biomolecules. Protein folding simulations are computationally challenging when simulating all involved (solvent) atoms considering timescales of ms or slower. Native structure based models (SBM,‘Go-models') reduce computational complexity and have been proven to be a robust and efficient way for exploring the protein folding process (Schug and Onuchic 2010; Thirumalai, O'Brien et al. 2010). They are based on energy landscape theory and the principle of minimal frustration. Using this framework, we simulate protein folding for a large set (∼ 200) of non-homologous monomeric proteins sized from 50-150 amino acids in coarse-grained simulations, representing each amino acid by a single bead. A fully automatized workflow implemented with the help of eSBM Tools (Lutz et al.) guides these simulations. From the simulations, we extract typical folding properties like phi-values, folding free energy landscape and transition state ensembles. We repeat the simulations for a variant SBM with flavored contact strengths pending on amino acids composition. The resulting database estimates the robustness of folding parameters, quantifies the folding behavior, compares the behavior to existing experimental data and can serve as a baseline for comparison to future experiments or simulations of protein folding.Schug, A. and J. N. Onuchic (2010). “From protein folding to protein function and biomolecular binding by energy landscape theory.” Curr Opin Pharmacol 10(6): 709-714.Thirumalai, D., E. P. O'Brien, et al. (2010). “Theoretical perspectives on protein folding.” Annual review of biophysics 39: 159-183.Lutz, B., et al. (2012), in Proceedings of the 11th International Conference on Modeling and Applied Simulation, Wien, 2012, edited by M. Affenzeller, et al., pp. 237.