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The consistency of energy landscape theory predictions with available experimental data, as well as direct evidence from molecular simulations, have shown that protein folding mechanisms are largely determined by the contacts present in the native structure. As expected, native contacts are generally energetically favorable. However, there are usually at least as many energetically favorable nonnative pairs owing to the greater number of possible nonnative interactions. This apparent frustration must therefore be reduced by the greater cooperativity of native interactions. In this work, we analyze the statistics of contacts in the unbiased all-atom folding trajectories obtained by Shaw and coworkers, focusing on the unfolded state. By computing mutual cooperativities between contacts formed in the unfolded state, we show that native contacts form the most cooperative pairs, while cooperativities among nonnative or between native and nonnative contacts are typically much less favorable or even anticooperative. Furthermore, we show that the largest network of cooperative interactions observed in the unfolded state consists mainly of native contacts, suggesting that this set of mutually reinforcing interactions has evolved to stabilize the native state. [ABSTRACT FROM AUTHOR]

The ability of protein chains to spontaneously form their three-dimensional structures is a long-standing mystery in molecular biology. The most conceptual aspect of this mystery is how the protein chain can find its native, "working" spatial structure (which, for not too big protein chains, corresponds to the global free energy minimum) in a biologically reasonable time, without exhaustive enumeration of all possible conformations, which would take billions of years. This is the so-called "Levinthal's paradox." In this review, we discuss the key ideas and discoveries leading to the current understanding of protein folding kinetics, including folding landscapes and funnels, free energy barriers at the folding/unfolding pathways, and the solution of Levinthal's paradox. A special role here is played by the "all-or-none" phase transition occurring at protein folding and unfolding and by the point of thermodynamic (and kinetic) equilibrium between the "native" and the "unfolded" phases of the protein chain (where the theory obtains the simplest form). The modern theory provides an understanding of key features of protein folding and, in good agreement with experiments, it (i) outlines the chain length-dependent range of protein folding times, (ii) predicts the observed maximal size of "foldable" proteins and domains. Besides, it predicts the maximal size of proteins and domains that fold under solely thermodynamic (rather than kinetic) control. Complementarily, a theoretical analysis of the number of possible protein folding patterns, performed at the level of formation and assembly of secondary structures, correctly outlines the upper limit of protein folding times. [ABSTRACT FROM AUTHOR]

Ankyrin (ANK) repeat proteins are coded by tandem occurrences of patterns with around 33 amino acids. They often mediate protein–protein interactions in a diversity of biological systems. These proteins have an elongated non-globular shape and often display complex folding mechanisms. This work investigates the energy landscape of representative proteins of this class made up of 3, 4 and 6 ANK repeats using the energy-landscape visualisation method (ELViM). By combining biased and unbiased coarse-grained molecular dynamics AWSEM simulations that sample conformations along the folding trajectories with the ELViM structure-based phase space, one finds a three-dimensional representation of the globally funnelled energy surface. In this representation, it is possible to delineate distinct folding pathways. We show that ELViMs can project, in a natural way, the intricacies of the highly dimensional energy landscapes encoded by the highly symmetric ankyrin repeat proteins into useful low-dimensional representations. These projections can discriminate between multiplicities of specific parallel folding mechanisms that otherwise can be hidden in oversimplified depictions.

Ankyrin (ANK) repeat proteins are coded by tandem occurrences of patterns with around 33 amino acids. They often mediate protein-protein interactions in a diversity of biological systems. These proteins have an elongated non-globular shape and often display complex folding mechanisms. This work investigates the energy landscape of representative proteins of this class made up of 3, 4 and 6 ANK repeats using the energy-landscape visualisation method (ELViM). By combining biased and unbiased coarse-grained molecular dynamics AWSEM simulations that sample conformations along the folding trajectories with the ELViM structurebased phase space, one finds a three-dimensional representation of the globally funnelled energy surface. In this representation, it is possible to delineate distinct folding pathways. We show that ELViMs can project, in a natural way, the intricacies of the highly dimensional energy landscapes encoded by the highly symmetric ankyrin repeat proteins into useful low-dimensional representations. These projections can discriminate between multiplicities of specific parallel folding mechanisms that otherwise can be hidden in oversimplified depictions. [ABSTRACT FROM AUTHOR]

“How do proteins fold?” Researchers have been studying different aspects of this question for more than 50 years. The most conceptual aspect of the problem is how protein can find the global free energy minimum in a biologically reasonable time, without exhaustive enumeration of all possible conformations, the so-called “Levinthal’s paradox.” Less conceptual but still critical are aspects about factors defining folding times of particular proteins and about perspectives of machine learning for their prediction. We will discuss in this review the key ideas and discoveries leading to the current understanding of folding kinetics, including the solution of Levinthal’s paradox, as well as the current state of the art in the prediction of protein folding times.

The ability of proteins to spontaneously form their spatial structures is a long standing puzzle in molecular biol ogy. Experimentally measured rates of spontaneous folding of singledomain globular proteins range from microseconds to hours: the difference - 1011 orders of magnitude - is the same as between the lifespan of a mosquito and the age of the Universe. This review (based on the literature and some personal recollections) describes a winding road to understanding spontaneous folding of protein structure. The main attention is given to the freeenergy landscape of conformations of a protein chain - especially to the barrier separating its unfolded (U) and the natively folded (N) states - and to physical the ories of rates of crossing this barrier in both directions: from U to N, and from N to U. It is shown that theories of both these processes come to essentially the same result and outline the observed range of folding and unfolding rates for singledomain globular proteins. In addition, they predict the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control, and explain the observed maximal size of "foldable" protein domains. [ABSTRACT FROM AUTHOR]

The ability of protein chains to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. Experimentally measured folding times of single-domain globular proteins range from microseconds to hours: the difference (10–11 orders of magnitude) is the same as that between the life span of a mosquito and the age of the universe. This review describes physical theories of rates of overcoming the free-energy barrier separating the natively folded (N) and unfolded (U) states of protein chains in both directions: “U-to-N” and “N-to-U”. In the theory of protein folding rates a special role is played by the point of thermodynamic (and kinetic) equilibrium between the native and unfolded state of the chain; here, the theory obtains the simplest form. Paradoxically, a theoretical estimate of the folding time is easier to get from consideration of protein unfolding (the “N-to-U” transition) rather than folding, because it is easier to outline a good unfolding pathway of any structure than a good folding pathway that leads to the stable fold, which is yet unknown to the folding protein chain. And since the rates of direct and reverse reactions are equal at the equilibrium point (as follows from the physical “detailed balance” principle), the estimated folding time can be derived from the estimated unfolding time. Theoretical analysis of the “N-to-U” transition outlines the range of protein folding rates in a good agreement with experiment. Theoretical analysis of folding (the “U-to-N” transition), performed at the level of formation and assembly of protein secondary structures, outlines the upper limit of protein folding times (i.e., of the time of search for the most stable fold). Both theories come to essentially the same results; this is not a surprise, because they describe overcoming one and the same free-energy barrier, although the way to the top of this barrier from the side of the unfolded state is very different from the way from the side of the native state; and both theories agree with experiment. In addition, they predict the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control and explain the observed maximal size of the “foldable” protein domains. [ABSTRACT FROM AUTHOR]

To find the native conformation (fold), proteins sample a subspace that is typically hundreds of orders of magnitude smaller than their full conformational space. Whether such fast folding is intrinsic or the result of natural selection, and what is the longest foldable protein, are open questions. Here, we derive the average conformational degeneracy of a lattice polypeptide chain in water and quantitatively show that the constraints associated with hydrophobic forces are themselves sufficient to reduce the effective conformational space to a size compatible with the folding of proteins up to approximately 200 amino acids long within a biologically reasonable amount of time. This size range is in general agreement with the experimental protein domain length distribution obtained from approximately 1,200 proteins. Molecular dynamics simulations of the Trp-cage protein confirm this picture on the free energy landscape. Our analytical and computational results are consistent with a model in which the length and time scales of protein folding, as well as the modular nature of large proteins, are dictated primarily by inherent physical forces, whereas natural selection determines the native state. [ABSTRACT FROM AUTHOR]

Coarse-grained, self-contained polymer models are powerful tools in the study of protein folding. They are also essential to assess predictions from less rigorous theoretical approaches that lack an explicit-chain representation. Here we review advances in coarse-grained modeling of cooperative protein folding, noting in particular that the Levinthal paradox was raised in response to the experimental discovery of two-state-like folding in the late 1960s, rather than to the problem of conformational search per se. Comparisons between theory and experiment indicate a prominent role of desolvation barriers in cooperative folding, which likely emerges generally from a coupling between local conformational preferences and nonlocal packing interactions. Many of these principles have been elucidated by native-centric models, wherein nonnative interactions may be treated perturbatively. We discuss these developments as well as recent applications of coarse-grained chain modeling to knotted proteins and to intrinsically disordered proteins. [ABSTRACT FROM AUTHOR]

Coarse-grained, self-contained polymer models are powerful tools in the study of protein folding. They are also essential to assess predictions from less rigorous theoretical approaches that lack an explicit-chain representation. Here we review advances in coarse-grained modeling of cooperative protein folding, noting in particular that the Levinthal paradox was raised in response to the experimental discovery of two-state-like folding in the late 1960s, rather than to the problem of conformational search per se. Comparisons between theory and experiment indicate a prominent role of desolvation barriers in cooperative folding, which likely emerges generally from a coupling between local conformational preferences and nonlocal packing interactions. Many of these principles have been elucidated by native-centric models, wherein nonnative interactions may be treated perturbatively. We discuss these developments as well as recent applications of coarse-grained chain modeling to knotted proteins and to intrinsically disordered proteins. [ABSTRACT FROM AUTHOR]

Abstract: We present a new computationally efficient method for large-scale polypeptide folding using coarse-grained elastic networks and gradient-based continuous optimization techniques. The folding is governed by minimization of energy based on Miyazawa–Jernigan contact potentials. Using this method we are able to substantially reduce the computation time on ordinary desktop computers for simulation of polypeptide folding starting from a fully unfolded state. We compare our results with available native state structures from Protein Data Bank (PDB) for a few de-novo proteins and two natural proteins, Ubiquitin and Lysozyme. Based on our simulations we are able to draw the energy landscape for a small de-novo protein, Chignolin. We also use two well known protein structure prediction software, MODELLER and GROMACS to compare our results. In the end, we show how a modification of normal elastic network model can lead to higher accuracy and lower time required for simulation. [Copyright &y& Elsevier]

Abstract: In recent years, intrinsically disordered proteins (IDPs) have attracted a lot of attention given the functional importance inherent to their flexible nature. One of the most intriguing features of IDPs is their ability to undergo disorder-to-order transitions upon binding in order to perform their function. Although the importance of intermolecular interactions involving IDPs has been widely recognized, there are divergent views on their binding mechanisms. Among the existing mechanistic models, two of them have gained popularity in the IDP field: the ‘conformational selection’ and the ‘coupled folding and binding.’ The first mechanism suggests that folding of IDPs precedes binding, while the second mechanism argues that folding may only take place upon binding. It has been suggested that both models are valid, although they work independently. However, reinterpretation of recent experimental and theoretical data indicates that both models have much more in common that it has been thought. In this manuscript, it is proposed that both mechanistic models should be merged into a single one: the synergistic model. In this model, both ‘conformational selection’ and ‘coupled folding and binding’ will synergistically participate in the binding of IDPs. To what extent each model will contribute to the full binding mechanism will depend on the required rate of binding, IDPs concentration, the native local plasticity of IDPs, the degree of binding degeneracy and the type of disorder-to-order transition. Furthermore, it is proposed that combination of the two mechanisms would bring tremendous advantages to IDP binding. For example, synergy may effectively modulate binding kinetics, balance the delicate interplay between enthalpy and entropy by using the funneled energy landscape more efficiently, thus yielding high specificity with carefully balanced free energy of binding. Given the advantages of the synergistic model, it is proposed that it will provide the basis to fully understand the complex nature of IDP binding. [Copyright &y& Elsevier]

Recent experimental studies suggest that the mature GFP has an unconventional landscape composed of an early folding event with a typical funneled landscape, followed by a very slow search and rearrangement step into the locked, active chromophore-containing structure. As we have shown previously, the substantial difference in time scales is what generates the observed hysteresis in thermodynamic folding. The interconversion between locked and the soft folding structures at intermediate denaturant concentrations is so slow that it is not observed under the typical experimental observation time. Simulations of a coarse-grained model were used to describe the fast folding event as well as identify native-like intermediates on energy landscapes enroute to the fluorescent native fold. Interestingly, these simulations reveal structural features of the slow dynamic transition to chromophore activation. Experimental evidence presented here shows that the trapped, native-like intermediate has structural heterogeneity in residues previously linked to chromophore formation. We propose that the final step of GFP folding is a "locking" mechanism leading to chromophore formation and high stability. The combination of previous experimental work and current simulation work is explained in the context of a dual-basin folding mechanism described above. [ABSTRACT FROM AUTHOR]

The homochirality, or isotacticity, of the natural amino acids facilitates the formation of regular secondary structures such as α-helices and β-sheets. However, many examples exist in nature where novel polypeptide topologies use both l- and d-amino acids. In this study, we explore how stereochemistry of the polypeptide backbone influences basic properties such as compactness and the size of fold space by simulating both lattice and all-atom polypeptide chains. We formulate a rectangular lattice chain model in both two and three dimensions, where monomers are chiral, having the effect of restricting local conformation. Syndiotactic chains with alternating chirality of adjacent monomers have a very large ensemble of accessible conformations characterized predominantly by extended structures. Isotactic chains on the other hand, have far fewer possible conformations and a significant fraction of these are compact. Syndiotactic chains are often unable to access maximally compact states available to their isotactic counterparts of the same length. Similar features are observed in all-atom models of isotactic versus syndiotactic polyalanine. Our results suggest that protein isotacticity has evolved to increase the enthalpy of chain collapse by facilitating compact helical states and to reduce the entropic cost of folding by restricting the size of the unfolded ensemble of competing states. [ABSTRACT FROM AUTHOR]

Water is essential for life in many ways, and without it biomolecules might no longer truly be biomolecules. In particular, water is important to the structure, stability, dynamics, and function of biological macromolecules. In protein folding, water mediates the collapse of the chain and the search for the native topology through a funneled energy landscape. Water actively participates in molecular recognition by mediating the interactions between binding partners and contributes to either enthalpic or entropic stabilization. Accordingly, water must be included in recognition and structure prediction codes to capture specificity. Thus water should not be treated as an inert environment, but rather as an integral and active component of biomolecular systems, where it has both dynamic and structural roles. Focusing on water sheds light on the physics and function of biological machinery and self-assembly and may advance our understanding of the natural design of proteins and nucleic acids. [ABSTRACT FROM AUTHOR]

Molecular machines order and disorder polypeptides as they form and dissolve large intermolecular interfaces, but the biological significance of coupled ordering and binding has been established in few, if any, macromolecular systems. The ordering and binding of GroES co-chaperonin mobile loops accompany an ATP-dependent conformational change in the GroEL chaperonin that promotes client protein folding. Following ATP hydrolysis, disordering of the mobile loops accompanies co-chaperonin dissociation, reversal of the GroEL conformational change, and release of the client protein. 'High-affinity' GroEL mutants were identified by their compatibility with 'low-affinity' co-chaperonin mutants and incompatibility with high-affinity co-chaperonin mutants. Analysis of binding kinetics using the intrinsic fluorescence of tryptophan-containing co-chaperonin variants revealed that excessive affinity causes the chaperonin to stall in a conformation that forms in the presence of ATP. Destabilizing the β-hairpins formed by the mobile loops restores the normal rate of dissociation. Thus, the free energy of mobile-loop ordering and disordering acts like the inertia of an engine's flywheel by modulating the speed of chaperonin conformational changes. [ABSTRACT FROM AUTHOR]

We present a simple formalism for the dynamics of proteins on a potential energy landscape, using connectedness of configurational domains as an order parameter. This formalism clearly shows that the energy bias required to form a unit correct contact toward the native configuration of a two-state folder, to overcome Levinthal's paradox, is Ebias ≅ RT ln 2. This result agrees well with earlier studies and indicates that the bias is mainly due to hydrophobic interaction. Further investigations have shown that the landscape funnel could be experimentally mapped onto a two-dimensional space formed by denaturant concentration and the connectedness of configurational domains. The theoretical value of the depth-of-folding funnel in terms of denaturant concentration has been calculated for a model protein (P450cam), which agrees well with the experimental value. Using our model, it is also possible to explain the turnover nature of heat-capacity change upon unfolding of proteins and the existence of enthalpy and entropy convergence temperatures during unfolding without any strict assumptions as proposed in earlier studies. [ABSTRACT FROM AUTHOR]

Molecular conformation problems arising in computational chemistry require the global minimization of a non-convex potential energy function representing the interactions of, for example, the component atoms in a molecular system. Typically the number of local minima on the potential energy surface grows exponentially with system size, and often becomes enormous even for relatively modestly sized systems. Thus the simple multistart strategy of randomly sampling local minima becomes impractical. However, for many molecular conformation potential energy surfaces the local minima can be organized by a simple adjacency relation into a single or at most a small number of funnels. A distinguished local minimum lies at the bottom of each funnel and a monotonically descending sequence of adjacent local minima connects every local minimum in the funnel with the funnel bottom. Thus the global minimum can be found among the comparatively small number of funnel bottoms, and a multistart strategy based on sampling funnel bottoms becomes viable. In this paper we present such an algorithm of the basin-hopping type and apply it to the Lennard–Jones cluster problem, an intensely studied molecular conformation problem which has become a benchmark for global optimization algorithms. Results of numerical experiments are presented which confirm both the multifunneling character of the Lennard–Jones potential surface as well as the efficiency of the algorithm. The algorithm has found all of the current putative global minima in the literature up to 110 atoms, as well as discovered a new global minimum for the 98-atom cluster of a novel geometrical class. [ABSTRACT FROM AUTHOR]

In this review, the main concepts of protein folding, as deduced from both theoretical and experimental in vitro studies, are presented. The thermodynamic aspects from Anfinsen's postulate, Levinthal's paradox to the concept of folding funnel as proposed by Wolynes and coworkers are described. Concerning the folding pathway(s), particular attention is brought to bear on the early steps that initiate the process in the light of the results of the fast and even ultrafast techniques presently being used. The role of structural domains as folding units is discussed. Last, from the recent studies, it can be concluded that the main rules deduced from the in vitro folding studies are valid for the folding of a nascent polypeptide chain in vivo. [ABSTRACT FROM AUTHOR]

"How do proteins fold?" Researchers have been studying different aspects of this question for more than 50 years. The most conceptual aspect of the problem is how protein can find the global free energy minimum in a biologically reasonable time, without exhaustive enumeration of all possible conformations, the so-called "Levinthal's paradox." Less conceptual but still critical are aspects about factors defining folding times of particular proteins and about perspectives of machine learning for their prediction. We will discuss in this review the key ideas and discoveries leading to the current understanding of folding kinetics, including the solution of Levinthal's paradox, as well as the current state of the art in the prediction of protein folding times. [ABSTRACT FROM AUTHOR]