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. © 2013 Wiley Periodicals, Inc.

Frustration and hydrophobicity interplay in protein folding and protein evolution

A lattice model is used to study mutations and compacting effects on protein folding rates and folding temperature. In the context of protein evolution, we address the question regarding the best scenario for a polypeptide chain to fold: either a fast nonspecific collapse followed by a slow rearrangement to form the native structure or a specific collapse from the unfolded state with the simultaneous formation of the native state. This question is investigated for optimized sequences, whose native state has no frustrated contacts between monomers, and also for mutated sequences, whose native state has some degree of frustration. It is found that the best scenario for folding may depend on the amount of frustration of the native structure. The implication of this result on protein evolution is discussed. (c) 2006 American Institute of Physics.

Exploring the origins of topological frustration: Design of a minimally frustrated model of fragment B of protein A

Topological frustration in an energetically unfrustrated off-lattice model of the helical protein fragment B of protein A from Staphylococcus aureus was investigated. This Gō-type model exhibited thermodynamic and kinetic signatures of a well-designed two-state folder with concurrent collapse and folding transitions and single exponential kinetics at the transition temperature. Topological frustration is determined in the absence of energetic frustration by the distribution of Fersht φ values. Topologically unfrustrated systems present a unimodal distribution sharply peaked at intermediate φ, whereas highly frustrated systems display a bimodal distribution peaked at low and high φ values. The distribution of φ values in protein A was determined both thermodynamically and kinetically. Both methods yielded a unimodal distribution centered at φ = 0.3 with tails extending to low and high φ values, indicating the presence of a small amount of topological frustration. The contacts with high φ values were located in the turn regions between helices I and II and II and III, intimating that these hairpins are in large part required in the transition state. Our results are in good agreement with all-atom simulations of protein A, as well as lattice simulations of a three- letter code 27-mer (which can be compared with a 60-residue helical protein). The relatively broad unimodal distribution of φ values obtained from the all-atom simulations and that from the minimalist model for the same native fold suggest that the structure of the transition state ensemble is determined mostly by the protein topology and not energetic frustration.

Folding funnels and frustration in off-lattice minimalist protein landscapes

A full quantitative understanding of the protein folding problem is now becoming possible with the help of the energy landscape theory and the protein folding funnel concept. Good folding sequences have a landscape that resembles a rough funnel where the energy bias towards the native state is larger than its ruggedness. Such a landscape leads not only to fast folding and stable native conformations but, more importantly, to sequences that are robust to variations in the protein environment and to sequence mutations. In this paper, an off-lattice model of sequences that fold into a β-barrel native structure is used to describe a framework that can quantitatively distinguish good and bad folders. The two sequences analyzed have the same native structure, but one of them is minimally frustrated whereas the other one exhibits a high degree of frustration.

Dynamical frustration of protein's environment at the nanoseconds time scale

A 21-residue peptide in explicit water has been simulated using classical molecular dynamics. The system's trajectory has been analysed with a novel approach that quantifies the process of how atom's environment trajectories are explored. The approach is based on the measure of Statistical Complexity that extracts complete dynamical information from the signal. The introduced characteristic quantifies the system's dynamics at the nanoseconds time scale. It has been found that the peptide exhibits nanoseconds long periods that significantly differ in the rates of the exploration of the dynamically allowed configurations of the environment. During these periods the rates remain the same but different from other periods and from the rate for water. Periods of dynamical frustration are detected when only limited routes in the space of possible trajectories of the surrounding atoms are realised.

Lattice model for rapidly folding protein-like heteropolymers

Protein folding is a relatively fast process considering the astronomical number of conformations in which a protein could find itself. Within the framework of a lattice model, we show that one can design rapidly folding sequences by assigning the strongest attractive couplings to the contacts present in a target native state, Our protein design can be extended to situations with both attractive and repulsive contacts. Frustration is minimized by ensuring that all the native contacts are again strongly attractive. Strikingly, this ensures the inevitability of folding and accelerates the folding process by an order of magnitude, The evolutionary implications of our findings are discussed.

Local Frustration Determines Molecular and Macroscopic Helix Structures

The effect of displaying cytochromes from an amyloid fibre is modelled as perturbation of -strands in a bilayer of helical -sheets, thereby explaining the spiral morphology of decorated amyloid and the dynamic response of morphology to cytochrome conformation. The morphology of the modelled fibre, which consists of minimal energy assemblies of rigid building blocks containing two anisotropic interacting units, depends primarily on the rigid constraints between units rather than the soft interactions between them. The framework is a discrete version of the bilayered frustration principle that drives morphology in Bauhinia seedpods. We show that self-assembly of frustrated long range structures can occur if the building blocks themselves are internally frustrated, e.g. amyloid morphology is governed by the conformation of the misfolded protein nucleating the fibre. Our model supports the idea that any peptide sequence can form amyloid if bilayers can form first, albeit stabilised by additional material such as chaperones or cytochromes. Analysis of experimentally derived amyloid structures supports our conclusions and suggests a range of frustration effects, which natural amyloid fibres may exploit. From this viewpoint, amyloid appears as a molecular example of a more general universal bilayered frustration principle, which may have profound implications for materials design using fibrous systems. Our model provides quantitative guidance for such applications. The relevance to longer length scales was proved by designing the morphology of a series of macroscopic magnetic stacks. Finally, this work leads to the idea of mixing controlled morphologically defined species to generate higher-order assembly and complex functional behaviour. The systematic kinking of decorated fibres and the nested frustration of the Bauhinia seed pod are two outstanding examples.

The Origin of Nonmonotonic Complex Behavior and the Effects of Nonnative Interactions on the Diffusive Properties of Protein Folding

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Foldons, protein structural modules, and exons.

Foldons, which are kinetically competent, quasi-independently folding units of a protein, may be defined using energy landscape analysis. Foldons can be identified by maxima in a scan of the ratio of a contiguous segment's energetic stability gap to the energy variance of that segment's molten globule states, reflecting the requirement of minimal frustration. The predicted foldons are compared with the exons and structural modules for 16 of the 30 proteins studied. Statistical analysis indicates a strong correlation between the energetically determined foldons and Go's geometrically defined structural modules, but there are marked sequence-dependent effects. There is only a weak correlation of foldons to exons. For gammaII-crystallin, myoglobin, barnase, alpha-lactalbumin, and cytochrome c the foldons and some noncontiguous clusters of foldons compare well with intermediates observed in experiment.

A mean field model of ligand-protein interactions: implications for the structural assessment of human immunodeficiency virus type 1 protease complexes and receptor-specific binding.

We propose a general mean field model of ligand-protein interactions to determine the thermodynamic equilibrium of a system at finite temperature. The method is employed in structural assessments of two human immuno-deficiency virus type 1 protease complexes where the gross effects of protein flexibility are incorporated by utilizing a data base of crystal structures. Analysis of the energy spectra for these complexes has revealed that structural and thermo-dynamic aspects of molecular recognition can be rationalized on the basis of the extent of frustration in the binding energy landscape. In particular, the relationship between receptor-specific binding of these ligands to human immunodeficiency virus type 1 protease and a minimal frustration principle is analyzed.

Lattice model for rapidly folding protein-like heteropolymers.

Protein folding is a relatively fast process considering the astronomical number of conformations in which a protein could find itself. Within the framework of a lattice model, we show that one can design rapidly folding sequences by assigning the strongest attractive couplings to the contacts present in a target native state. Our protein design can be extended to situations with both attractive and repulsive contacts. Frustration is minimized by ensuring that all the native contacts are again strongly attractive. Strikingly, this ensures the inevitability of folding and accelerates the folding process by an order of magnitude. The evolutionary implications of our findings are discussed.