Effect of short- and long-range interactions on protein folding

The relative importance of short- and long-range interactions is examined using a Monte Carlo simulation of protein folding on bovine pancreatic trypsin inhibitor. The model of the protein and the interaction energies were parametrized using X-ray structures of 30 native proteins. A nearest neighbor Ising model is used to determine the conformational state at each stage of the Monte Carlo procedure. Long-range interactions are simulated by contact free energies which become effective as two residues, separated by four or more residues along the chain, approach each other, and by disulfide-bond energies. Short-range interactions for residues separated by one, two, or three residues along the chain are also modeled by contact free energies and by -helical hydrogen bonds. A hard-sphere model is used to represent repulsive interactions. The ratios of short- to long-range interactions studied are 1:1, 2:1, 1:2, 0:1, and 1:0; e.g., for the 2:1 ratio, short-range interactions are weighted twice as much as long-range interactions, and for the 1:0 ratio, long-range interactions are omitted. For each ratio of short- to long-range interactions, a native conformation is found by a Monte Carlo procedure, a segment of 11 residues (residue numbers 1–11) is then rotated away from the rest of the molecule [breaking the 5–55 native disulfide bond, and moving this segment so that the distance between the sulfur atoms of the 5 and 55 cystine side chains (averaged for all native conformations) increases from 3.9 to 7.3 Å], and the Monte Carlo simulation is carried out (allowing the conformation of the whole molecule to change) until equilibrium is attained. For each ratio, the refolded conformation is compared to the native one using triangular distance maps and differential geometry distance criteria. With ratios of short- to long-range interaction energies of 1:1 and 0:1, the native disulfide bond could be re-formed; with ratios of 2:1 and 1:2 it did not; and with the 1:0 ratio, even a stable native conformation was not achieved. Therefore, long-range interactions (in addition to short-range ones) are required to bring remote parts of the protein together and to stabilize its native conformation.NIH Postdoctoral Fellow, 1977–1978.     

Low-throughput model design of protein folding inhibitors

The stabilization energy of proteins in their native conformation is not distributed uniformly among all the amino acids, but is concentrated in few (short) fragments, fragments which play a key role in the folding process and in the stability of the protein. Peptides displaying the same sequence as these key fragments can compete with the formation of the most important native contacts, destabilizing the protein and thus inhibiting its biological activity. We present an essentially automatic method to individuate such peptidic inhibitors based on a low-throughput screening of the fragments which build the target protein. The efficiency and generality of the method is tested on proteins Src-SH3, G, CI2, and HIV-1-PR with the help of a simplified computational model. In each of the cases studied, we find few peptides displaying strong inhibitory properties, properties which are quite robust with respect to point mutations. The possibility of implementing the method through low-throughput experimental screening of the target protein is discussed.     

Testing a new Monte Carlo algorithm for protein folding

We demonstrate that the recently proposed pruned-enriched Rosenbluth method (PERM) (Grassberger, Phys. Rev. E 56:3682, 1997) leads to extremely efficient algorithms for the folding of simple model proteins. We test it on several models for lattice heteropolymers, and compare it to published Monte Carlo studies of the properties of particular sequences. In all cases our method is faster than the previous ones, and in several cases we find new minimal energy states. In addition to producing more reliable candidates for ground states, our method gives detailed information about the thermal spectrum and thus allows one to analyze thermodynamic aspects of the folding behavior of arbitrary sequences. Proteins 32:52–66, 1998. © 1998 Wiley-Liss, Inc.     

Protein folding mechanisms and the multidimensional folding funnel

An important idea that emerges from the energy landscape theory of protein folding is that subtle global features of the protein landscape can profoundly affect the apparent mechanism of folding. The relationship between various characteristic temperatures in the phase diagrams and landmarks in the folding funnel at fixed temperatures can be used to classify different folding behaviors. The one-dimensional picture of a folding funnel classifies folding kinetics into four basic scenarios, depending on the relative location of the thermodynamic barrier and the glass transition as a function of a single-order parameter. However, the folding mechanism may not always be quantitatively described by a single-order parameter. Several other order parameters, such as degree of secondary structure formation, collapse and topological order, are needed to establish the connection between minimalist models and proteins in the laboratory. In this article we describe a simple multidimensional funnel based on two-order parameters that measure the degree of collapse and topological order. The appearance of several different “mechanisms” is illustrated by analyzing lattice models with different potentials and sequences with different degrees of design. In most cases, the two-dimensional analysis leads to a classification of mechanisms totally in keeping with the one-dimensional scheme, but a topologically distinct scenario of fast folding with traps also emerges. The nature of traps depends on the relative location of the glass transition surface and the thermodynamic barrier in the multidimensional funnel. Proteins 32:136–158, 1998. © 1998 Wiley-Liss, Inc.     

A simple lattice model that exhibits a protein-like cooperative all-or-none folding transition

In a recent paper (D. Gront et al., Journal of Chemical Physics, Vol. 115, pp. 1569, 2001) we applied a simple combination of the Replica Exchange Monte Carlo and the Histogram methods in the computational studies of a simplified protein lattice model containing hydrophobic and polar units and sequence-dependent local stiffness. A well-defined, relatively complex Greek-key topology, ground (native) conformations was found; however, the cooperativity of the folding transition was very low. Here we describe a modified minimal model of the same Greek-key motif for which the folding transition is very cooperative and has all the features of the "all-or-none" transition typical of real globular proteins. It is demonstrated that the all-or-none transition arises from the interplay between local stiffness and properly defined tertiary interactions. The tertiary interactions are directional, mimicking the packing preferences seen in proteins. The model properties are compared with other minimal protein-like models, and we argue that the model presented here captures essential physics of protein folding (structurally well-defined protein-like native conformation and cooperative all-or-none folding transition).     

Domains in folding of model proteins.

By means of Monte Carlo simulation, we investigated the equilibrium between folded and unfolded states of lattice model proteins. The amino acid sequences were designed to have pronounced energy minimum target conformations of different length and shape. For short fully compact (36-mer) proteins, the all-or-none transition from the unfolded state to the native state was observed. This was not always the case for longer proteins. Among 12 designed sequences with the native structure of a fully compact 48-mer, a simple all-or-none transition was observed in only three cases. For the other nine sequences, three states of behavior-the native, denatured, and intermediate states-were found. The contiguous part of the native structure (domain) was conserved in the intermediate state, whereas the remaining part was completely unfolded and structureless. These parts melted separately from each other.     

Monte Carlo simulation of protein folding in the presence of residue-specific binding sites

Ensemble growth Monte Carlo (EGMC) and dynamic Monte Carlo (DMC) simulations are used to study sequential folding and thermodynamic stability of hydrophobic-polar (HP) chains that fold to a compact structure. Molecularly imprinted cavities are modeled as hard walls having sites that are attractive to specific polar residues on the chain. Using EGMC simulation, we find that the folded conformation can be stabilized using a small number of carefully selected residue-specific sites while a random selection of surface-bound residues may only slightly contribute toward stabilizing the folded conformation, and in some cases may hinder the folding of the chain. DMC simulations of the surface-bound chain confirm increased stability of the folded conformation over a free chain. However, a different trend of the equilibrium population of folded chains as a function of residue-external site interactions is predicted with the two simulation methods.     

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 C_α 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.Proteins 2013; 81:1727–1737. © 2013 Wiley Periodicals, Inc.     

Non-native interactions play an effective role in protein folding dynamics

Systematic Monte Carlo simulations of simple lattice models show that the final stage of protein folding is an ordered process where native contacts get locked (i.e., the residues come into contact and remain in contact for the duration of the folding process) in a well‐defined order. The detailed study of the folding dynamics of protein‐like sequences designed as to exhibit different contact energy distributions, as well as different degrees of sequence optimization (i.e., participation of non‐native interactions in the folding process), reveals significant differences in the corresponding locking scenarios—the collection of native contacts and their average locking times, which are largely ascribable to the dynamics of non‐native contacts. Furthermore, strong evidence for a positive role played by non‐native contacts at an early folding stage was also found. Interestingly, for topologically simple target structures, a positive interplay between native and non‐native contacts is observed also toward the end of the folding process, suggesting that non‐native contacts may indeed affect the overall folding process. For target models exhibiting clear two‐state kinetics, the relation between the nucleation mechanism of folding and the locking scenario is investigated. Our results suggest that the stabilization of the folding transition state can be achieved through the establishment of a very small network of native contacts that are the first to lock during the folding process.     

A model study of protein nascent chain and cotranslational folding using hydrophobic-polar residues

To study protein nascent chain folding during biosynthesis, we investigate the folding behavior of models of hydrophobic and polar (HP) chains at growing length using both two-dimensional square lattice model and an optimized three-dimensional 4-state discrete off-lattice model. After enumerating all possible sequences and conformations of HP heteropolymers up to length N = 18 and N = 15 in two and three-dimensional space, respectively, we examine changes in adopted structure, stability, and tolerance to single point mutation as the nascent chain grows. In both models, we find that stable model proteins have fewer folded nascent chains during growth, and often will only fold after reaching full length. For the few occasions where partial chains of stable proteins fold, these partial conformations on average are very similar to the corresponding parts of the final conformations at full length. Conversely, we find that sequences with fewer stable nascent chains and sequences with native-like folded nascent chains are more stable. In addition, these stable sequences in general can have many more point mutations and still fold into the same conformation as the wild type sequence. Our results suggest that stable proteins are less likely to be trapped in metastable conformations during biosynthesis, and are more resistant to point-mutations. Our results also imply that less stable proteins will require the assistance of chaperone and other factors during nascent chain folding. Taken together with other reported studies, it seems that cotranslational folding may not be a general mechanism of in vivo protein folding for small proteins, and in vitro folding studies are still relevant for understanding how proteins fold biologically.     

Population analyses of kinetic partitioning in protein folding

A simple lattice model of protein folding is studied in order to analyze the kinetic partitioning phenomena in the energy landscape perspective. By restricting the area of conformational space, it becomes possible to follow many Monte Carlo trajectories until they reach equilibrium. Alteration of population of trajectories is monitored and the relations between the energy landscape and kinetics are examined. Kinetic partitioning phenomena are categorized into different types in terms of characteristic time constants and partitioning ratio. In a specific partitioning process, refolding proceeds along the parallel pathways; the time constants have a temperature dependence similar to that observed in hen lysozyme. High-energy conformations are classified into groups according to the probability that the trajectories starting from those conformations will reach each energy valley. The partitioning ratio is determined by the way in which the conformational space is organized into these groups.     

Use of quantitative structure-property relationships to predict the folding ability of model proteins

We investigate the folding of a 125-bead heteropolymer model for proteins subject to Monte Carlo dynamics on a simple cubic lattice. Detailed study of a few sequences revealed a folding mechanism consisting of a rapid collapse followed by a slow search for a stable core that served as the transition state for folding to a near-native intermediate. Rearrangement from the intermediate to the native state slowed folding further because it required breaking native-like local structure between surface monomers so that those residues could condense onto the core. We demonstrate here the generality of this mechanism by a statistical analysis of a 200 sequence database using a method that employs a genetic algorithm to pick the sequence attributes that are most important for folding and an artificial neural network to derive the corresponding functional dependence of folding ability on the chosen sequence attributes [quantitative structure-property relationships (QSPRs)]. QSPRs that use three sequence attributes yielded substantially more accurate predictions than those that use only one. The results suggest that efficient search for the core is dependent on both the native state's overall stability and its amount of kinetically accessible, cooperative structure, whereas rearrangement from the intermediate is facilitated by destabilization of contacts between surface monomers. Implications for folding and design are discussed. Proteins 33:177–203, 1998. © 1998 Wiley-Liss, Inc.     

Analysis of fragments induced by simulated lattice protein folding

The folding process of a set of 42 proteins, representative of the various folds, has been simulated by means of a Monte Carlo method on a discrete lattice, using two different potentials of mean force. Multiple compact fragments of contiguous residues are formed in the simulation, stable in composition, but not in geometry. During time, the number of fragments decreases until one final compact globular state is reached. We focused on the early steps of the folding in order to evidence the maximum number of fragments, provided they are sufficiently stable in sequence. A correlation has been established between these proto fragments and regular secondary-structure elements, whatever their nature, alpha helices or beta strands. Quantitatively, this is revealed by an overall mean one-residue quality factor of nearly 60%, which is better for proteins mainly composed of alpha helices. The correspondence between the number of fragments and the number of secondary-structure elements is of 77% and the regions separating successive fragments are mainly located in loops. Besides, hydrophobic clusters deduced from HCA correspond to fragments with an equivalent accuracy. These results suggest that folding pathways do not contain structurally static intermediate. However, since the beginning of folding, most residues that will later form one given secondary structure are kept close in space by being involved in the same fragment. This aggregation may be a way to accelerate the formation of the native state and enforces the key role played by hydrophobic residues in the formation of the fragments, thus in the folding process itself.     

Sequence-based study of two related proteins with different folding behaviors

Z(SPA-1) is an engineered protein that binds to its parent, the three-helix-bundle Z domain of staphylococcal protein A. Uncomplexed Z(SPA-1) shows a reduced helix content and a melting behavior that is less cooperative, compared with the wild-type Z domain. Here we show that the difference in folding behavior between these two sequences can be partly understood in terms of an off-lattice model with 5-6 atoms per amino acid and a minimalistic potential, in which folding is driven by backbone hydrogen bonding and effective hydrophobic attraction.     

The role of atomic level steric effects and attractive forces in protein folding

Protein folding into tertiary structures is controlled by an interplay of attractive contact interactions and steric effects. We investigate the balance between these contributions using structure‐based models using an all‐atom representation of the structure combined with a coarse‐grained contact potential. Tertiary contact interactions between atoms are collected into a single broad attractive well between the C^β atoms between each residue pair in a native contact. Through the width of these contact potentials we control their tolerance for deviations from the ideal structure and the spatial range of attractive interactions. In the compact native state dominant packing constraints limit the effects of a coarse‐grained contact potential. During folding, however, the broad attractive potentials allow an early collapse that starts before the native local structure is completely adopted. As a consequence the folding transition is broadened and the free energy barrier is decreased. Eventually two‐state folding behavior is lost completely for systems with very broad attractive potentials. The stabilization of native‐like residue interactions in non‐perfect geometries early in the folding process frequently leads to structural traps. Global mirror images are a notable example. These traps are penalized by the details of the repulsive interactions only after further collapse. Successful folding to the native state requires simultaneous guidance from both attractive and repulsive interactions. Proteins 2012. © 2011 Wiley Periodicals, Inc.     

Large-scale prediction of protein geometry and stability changes for arbitrary single point mutations

We have developed a method to both predict the geometry and the relative stability of point mutants that may be used for arbitrary mutations. The geometry optimization procedure was first tested on a new benchmark of 2141 ordered pairs of X-ray crystal structures of proteins that differ by a single point mutation, the largest data set to date. An empirical energy function, which includes terms representing the energy contributions of the folded and denatured proteins and uses the predicted mutant side chain conformation, was fit to a training set consisting of half of a diverse set of 1816 experimental stability values for single point mutations in 81 different proteins. The data included a substantial number of small to large residue mutations not considered by previous prediction studies. After removing 22 (approximately 2%) outliers, the stability calculation gave a standard deviation of 1.08 kcal/mol with a correlation coefficient of 0.82. The prediction method was then tested on the remaining half of the experimental data, giving a standard deviation of 1.10 kcal/mol and covariance of 0.66 for 97% of the test set. A regression fit of the energy function to a subset of 137 mutants, for which both native and mutant structures were available, gave a prediction error comparable to that for the complete training set with predicted side chain conformations. We found that about half of the variation is due to conformation-independent residue contributions. Finally, a fit to the experimental stability data using these residue parameters exclusively suggests guidelines for improving protein stability in the absence of detailed structure information.     

Comparison of thermoluminescence characteristics in γ‐ray and C5+ ion beam‐irradiated LiCaAlF6:Ce phosphor

We compare the thermoluminescence (TL) behavior of Ce³⁺ ion‐activated LiCaAlF₆ exposed to γ‐rays and a carbon ion beam. The reported phosphor is synthesized using an in‐house precipitation method with varying concentrations of activator ion and is characterized by X‐ray diffraction (XRD) and TL. Rietveld refinement is performed to study the structural statistics. The TL glow curve consists of a prominent glow peak at 232°C with three shoulders at 115, 159 and 333°C when exposed to γ‐rays from a ⁶⁰Co source. When exposed to a C⁵⁺ ion beam, the TL glow curve consists of five peaks with peak temperatures near 156, 221, 250, 287 and 330°C, and is found to vary slightly with changing fluence. Glow curve convolution deconvolution (GCCD) functions are applied to the TL curves for complete analysis of the glow curve structure and TL traps. The order of kinetics (b), activation energy (E) and frequency factor are determined using Chen's peak shape method and theoretical curves are drawn using GCCD functions. A track interaction model (TIM) is used to explain the sublinearity/saturation at higher fluences. Ion beam parameters are analyzed using Monte‐Carlo simulation‐based SRIM‐2013 code. Copyright © 2016 John Wiley & Sons, Ltd.