Folding pathway of a lattice model for proteins

High Energy Channeling and Malleable Transition States: Molecular Dynamics Simulations and Free Energy Landscapes for the Thermal Unfolding of Protein U1A and 13 Mutants

The spliceosome protein U1A is a prototype case of the RNA recognition motif (RRM) ubiquitous in biological systems. The in vitro kinetics of the chemical denaturation of U1A indicate that the unfolding of U1A is a two-state process but takes place via high energy channeling and a malleable transition state, an interesting variation of typical two-state behavior. Molecular dynamics (MD) simulations have been applied extensively to the study of two-state unfolding and folding of proteins and provide an opportunity to obtain a theoretical account of the experimental results and a molecular model for the transition state ensemble. We describe herein all-atom MD studies including explicit solvent of up to 100 ns on the thermal unfolding (UF) of U1A and 13 mutants. Multiple MD UF trajectories are carried out to ensure accuracy and reproducibility. A vector representation of the MD unfolding process in RMSD space is obtained and used to calculate a free energy landscape for U1A unfolding. A corresponding MD simulation and free energy landscape for the protein CI2, well known to follow a simple two state folding/unfolding model, is provided as a control. The results indicate that the unfolding pathway on the MD calculated free energy landscape of U1A shows a markedly extended transition state compared with that of CI2. The MD results support the interpretation of the observed chevron plots for U1A in terms of a high energy, channel-like transition state. Analysis of the MDUF structures shows that the transition state ensemble involves microstates with most of the RRM secondary structure intact but expanded by ~14% with respect to the radius of gyration. Comparison with results on a prototype system indicates that the transition state involves an ensemble of molten globule structures and extends over the region of ~1-35 ns in the trajectories. Additional MDUF simulations were carried out for 13 U1A mutants, and the calculated φ-values show close accord with observed results and serve to validate our methodology.

The Early Phase of β2-Microglobulin Aggregation: Perspectives From Molecular Simulations

Protein β2-microglobulin is the causing agent of two amyloidosis, dialysis related amyloidosis (DRA), affecting the bones and cartilages of individuals with chronic renal failure undergoing long-term hemodialysis, and a systemic amyloidosis, found in one French family, which impairs visceral organs. The protein’s small size and its biomedical significance attracted the attention of theoretical scientists, and there are now several studies addressing its aggregation mechanism in the context of molecular simulations. Here, we review the early phase of β2-microglobulin aggregation, by focusing on the identification and structural characterization of monomers with the ability to trigger aggregation, and initial small oligomers (dimers, tetramers, hexamers etc.) formed in the so-called nucleation phase. We focus our analysis on results from molecular simulations and integrate our views with those coming from in vitro experiments to provide a broader perspective of this interesting field of research. We also outline directions for future computer simulation studies.

Comment on "Osmotic pressure of compressed lattice knots"

In a recent paper, E. J. Janse van Rensburg has presented computational data enumerating the conformations of closed circular self-avoiding lattice polymers with knots confined in a cubic box, and claimed to have observed a negative osmotic pressure in the system. The purpose of this comment is to state that osmotic pressure by a self-avoiding polymer, knotted or otherwise, is positive, which means a polymer pushes confining walls outwards, and the statement of the opposite is a mistake.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

Conformational Intermediate That Controls KPC-2 Catalysis and Beta-Lactam Drug Resistance

The KPC-2 carbapenemase enzyme is responsible for drug resistance in the majority of carbapenem-resistant gram-negative bacterial infections in the United States. A better understanding of what permits KPC-2 to hydrolyze carbapenem antibiotics and how this might be inhibited is thus of fundamental interest and great practical importance to development of better anti-infectives. By correlating molecular dynamics simulations with experimental enzyme kinetics, we have identified conformational changes that control KPC-2's ability to hydrolyze carbapenem antibiotics. Related beta-lactamase enzymes can interconvert between catalytically permissive and catalytically nonpermissive forms of an acylenzyme intermediate critical to drug hydrolysis. Using molecular dynamics simulations, we identify a similar equilibrium in KPC-2 and analyze the determinants of this conformational change. Because the conformational dynamics of KPC-2 are complex and sensitive to allosteric changes, we develop an information-theoretic approach to identify key determinants of this change. We measure unbiased estimators of the reaction coordinate between catalytically permissive and nonpermissive states, perform information-theoretic feature selection and, using restrained molecular dynamics simulations, validate the protein conformational changes predicted to control catalytically permissive geometry. We identify two binding-pocket residues that control the conformational transitions between catalytically active and inactive forms of KPC-2. Mutations to one of these residues, Trp105, lower the stability of the catalytically permissive state in simulations and have reduced experimental k _cat values that show a strong linear correlation with the simulated catalytically permissive state lifetimes. This understanding can be leveraged to predict the drug resistance of further KPC-2 mutants and help design inhibitors to combat extreme drug resistance.

Multi-scale simulations of biological systems using the OPEP coarse-grained model

Biomolecules are complex machines that are optimized by evolution to properly fulfill or contribute to a variety of biochemical tasks in the cellular environment. Computer simulations based on quantum mechanics and atomistic force fields have been proven to be a powerful microscope for obtaining valuable insights into many biological, physical, and chemical processes. Many interesting phenomena involve, however, a time scale and a number of degrees of freedom, notably if crowding is considered, that cannot be explored at an atomistic resolution. To bridge the gap between reality and simulation, many different advanced computational techniques and coarse-grained (CG) models have been developed. Here, we report some applications of the CG OPEP protein model to amyloid fibril formation, the response of catch-bond proteins to two types of fluid flow, and interactive simulations to fold peptides with well-defined 3D structures or with intrinsic disorder.

On the characterization and software implementation of general protein lattice models

models of proteins have been widely used as a practical means to computationally investigate general properties of the system. In lattice models any sterically feasible conformation is represented as a self-avoiding walk on a lattice, and residue types are limited in number. So far, only two- or three-dimensional lattices have been used. The inspection of the neighborhood of alpha carbons in the core of real proteins reveals that also lattices with higher coordination numbers, possibly in higher dimensional spaces, can be adopted. In this paper, a new general parametric lattice model for simplified protein conformations is proposed and investigated. It is shown how the supporting software can be consistently designed to let algorithms that operate on protein structures be implemented in a lattice-agnostic way. The necessary theoretical foundations are developed and organically presented, pinpointing the role of the concept of main directions in lattice-agnostic model handling. Subsequently, the model features across dimensions and lattice types are explored in tests performed on benchmark protein sequences, using a Python implementation. Simulations give insights on the use of square and triangular lattices in a range of dimensions. The trend of potential minimum for sequences of different lengths, varying the lattice dimension, is uncovered. Moreover, an extensive quantitative characterization of the usage of the so-called "move types" is reported for the first time. The proposed general framework for the development of lattice models is simple yet complete, and an object-oriented architecture can be proficiently employed for the supporting software, by designing ad-hoc classes. The proposed framework represents a new general viewpoint that potentially subsumes a number of solutions previously studied. The adoption of the described model pushes to look at protein structure issues from a more general and essential perspective, making computational investigations over simplified models more straightforward as well.

Evolution of the potential energy landscape with static pulling force for two model proteins

The energy landscape is analyzed for off-lattice bead models of protein L and protein G as a function of a static pulling force. Two different pairs of attachment points (pulling directions) are compared in each case, namely, residues 1/56 and 10/32. For the terminal residue pulling direction 1/56, the distinct global minimum structures are all extended, aside from the compact geometry that correlates with zero force. The helical turns finally disappear at the highest pulling forces considered. For the 10/32 pulling direction, the changes are more complicated, with a variety of competing arrangements for beads outside the region where the force is directly applied. These alternatives produce frustrated energy landscapes, with low-lying minima separated by high barriers. The calculated folding pathways in the absence of force are in good agreement with previous work. The N-terminal hairpin folds first for protein L and the C-terminal hairpin for protein G, which exhibits an intermediate. However, for a relatively low static force, where the global minimum retains its structure, the folding mechanisms change, sometimes dramatically, depending on the protein and the attachment points. The scaling relations predicted by catastrophe theory are found to hold in the limit of short path lengths.

Comparative modeling and protein-like features of hydrophobic-polar models on a two-dimensional lattice

Lattice models of proteins have been extensively used to study protein thermodynamics, folding dynamics, and evolution. Our study considers two different hydrophobic-polar (HP) models on the 2D square lattice: the purely HP model and a model where a compactness-favoring term is added. We exhaustively enumerate all the possible structures in our models and perform the study of their corresponding folds, HP arrangements in space and shapes. The two models considered differ greatly in their numbers of structures, folds, arrangements, and shapes. Despite their differences, both lattice models have distinctive protein-like features: (1) Shapes are compact in both models, especially when a compactness-favoring energy term is added. (2) The residue composition is independent of the chain length and is very close to 50% hydrophobic in both models, as we observe in real proteins. (3) Comparative modeling works well in both models, particularly in the more compact one. The fact that our models show protein-like features suggests that lattice models incorporate the fundamental physical principles of proteins. Our study supports the use of lattice models to study questions about proteins that require exactness and extensive calculations, such as protein design and evolution, which are often too complex and computationally demanding to be addressed with more detailed models.

Structured pathway across the transition state for peptide folding revealed by molecular dynamics simulations

Small globular proteins and peptides commonly exhibit two-state folding kinetics in which the rate limiting step of folding is the surmounting of a single free energy barrier at the transition state (TS) separating the folded and the unfolded states. An intriguing question is whether the polypeptide chain reaches, and leaves, the TS by completely random fluctuations, or whether there is a directed, stepwise process. Here, the folding TS of a 15-residue β-hairpin peptide, Peptide 1, is characterized using independent 2.5 μs-long unbiased atomistic molecular dynamics (MD) simulations (a total of 15 μs). The trajectories were started from fully unfolded structures. Multiple (spontaneous) folding events to the NMR-derived conformation are observed, allowing both structural and dynamical characterization of the folding TS. A common loop-like topology is observed in all the TS structures with native end-to-end and turn contacts, while the central segments of the strands are not in contact. Non-native sidechain contacts are present in the TS between the only tryptophan (W11) and the turn region (P7-G9). Prior to the TS the turn is found to be already locked by the W11 sidechain, while the ends are apart. Once the ends have also come into contact, the TS is reached. Finally, along the reactive folding paths the cooperative loss of the W11 non-native contacts and the formation of the central inter-strand native contacts lead to the peptide rapidly proceeding from the TS to the native state. The present results indicate a directed stepwise process to folding the peptide.

The folding dynamics of many small protein/peptides investigated recently are in terms of simple two-state model in which only two populations exist (folded and unfolded), separated by a single free energy barrier with only one kinetically important transition state (TS). However, dynamical characterization of the folding TS is challenging. We have used independent unbiased atomistic molecular dynamics simulations with clear folding-unfolding transitions to characterize structural and dynamical features of transition state ensemble of Peptide 1. A common loop-like topology is observed in all TS structures extracted from multiple simulations. The trajectories were used to examine the mechanism by which the TS is reached and subsequent events in folding pathways. The folding TS is reached and crossed in a directed stagewise process rather than through random fluctuations. Specific structures are formed before, during, and after the transition state, indicating a clear structured folding pathway.

Effect of solvation-related interaction on the low-temperature dynamics of proteins

The effect of solvation-related interaction on the low-temperature dynamics of proteins is studied by taking into account the desolvation barriers in the interactions of native contacts. It is found out that about the folding transition temperature, the protein folds in a cooperative manner, and the water molecules are expelled from the hydrophobic core at the final stage in the folding process. At low temperature, however, the protein would generally be trapped in many metastable conformations with some water molecules frozen inside the protein. The desolvation takes an important role in these processes. The number of frozen water molecules and that of frozen states of proteins are further analyzed with the methods based on principal component analysis (PCA) and the clustering of conformations. It is found out that both the numbers of frozen water molecules and the frozen states of the protein increase quickly below a certain temperature. Especially, the number of frozen states of the protein increases exponentially following the decrease in the temperature, which resembles the basic features of glassy dynamics. Interestingly, it is observed that the freezing of water molecules and that of protein conformations happen at almost the same temperature. This suggests that the solvation-related interaction performs an important role for the low-temperature dynamics of the model protein.

Nucleation process in the folding of a domain-swapped dimer

Nucleation processes are important for the understanding in protein dynamics. To evaluate the effect of nucleation mechanism in dimerization process, a domain-swapped dimer (Esp8) is simulated with the symmetrized Gō model and the classical Gō model. The pathways of the dimerization are analyzed with computational phi -analysis method. It is found out that some nuclei are observed in the kinetic steps of the dimeric association though the whole pathway is a process with multiple intermediate states. The key residues in the nuclei are rather similar to those observed in the monomeric folding. The differences with the monomeric cases are also discussed. These differences illustrate the effects of dimeric feature on the nucleation process. Besides, manual mutations are carried out to illustrate the importance of the interactions related to the nuclei. It is observed that the mutations in the nuclei-related interactions apparently change the dynamics while other mutations have little effect on the kinetics. All of these results outline a picture that the nucleation processes act as the fundamental steps of high-order organization of protein systems.

Direction and contrast tuning of macaque MSTd neurons during saccades

Saccades are rapid eye movements that change the direction of gaze, although the full-field image motion associated with these movements is rarely perceived. The attenuation of visual perception during saccades is referred to as saccadic suppression. The mechanisms that produce saccadic suppression are not well understood. We recorded from neurons in the dorsal medial superior temporal area (MSTd) of alert macaque monkeys and compared the neural responses produced by the retinal slip associated with saccades (active motion) to responses evoked by identical motion presented during fixation (passive motion). We provide evidence for a neural correlate of saccadic suppression and expand on two contentious results from previous studies. First, we confirm the finding that some neurons in MSTd reverse their preferred direction during saccades. We quantify this effect by calculating changes in direction tuning index for a large cell population. Second, it has been noted that neural activity associated with saccades can arrive in the parietal cortex <or=30 ms earlier than activity produced by similar visual stimulation during fixation. This led to the question of whether the saccade-related responses were visual in origin or were motor signals arising from saccade-planning areas of the brain. By comparing the responses to saccades made over textured backgrounds of different contrasts, we provide strong evidence that saccade-related responses were visual in origin. Refinements of the possible models of saccadic suppression are discussed.

Effect of the ordered water on protein folding: an off-lattice Gō-like model study

Recent experiments and numerical simulations have shown that the water molecules confined on the surfaces of some substrates, including the surfaces of cellular components in tissues and cells, form icelike ordered structures. If a protein folds in an environment with those icelike ordered water molecules, its behavior may be different from that in bulk water. Here, the effect of this ordered water environment on protein folding is studied by using an off-lattice Gō-like model. It is found that the ordered water environment significantly improves the native state stability and greatly speeds up the folding rate of the proteins.

Mutational robustness and geometrical form in protein structures

Theoretical studies of RNA and lattice protein models suggest that mutationally robust or the so-called designable phenotypes tend to have special geometric features such as being more compact and more geometrically regular. Such geometrical forms have been also linked to speed of folding and stability properties that may also assist in promoting mutational robustness. Here we test these theoretical predictions on a non-redundant collection of 2,660 experimentally determined structures from the PDB (Protein Data Bank) and CATH (Class Architecture Topology Homologous superfamily) database. We first developed an index summarizing the geometrical regularity of the structures and then used this index to show that the statistical pattern of empirical data is consistent with the theoretical predictions relating geometry to mutational robustness. Mutationally robust proteins tend to be more symmetric and compact. But, the relationship between compactness and robustness cannot be explained simply by the geometrical packing of individual amino acids in proteins; rather, it is the property of the whole system that is related to the statistical characteristics of the folding landscape. Finally, we hypothesize that a triplet relationship between mutational robustness, stability and form is a general properties of objects that optimize real-valued relationships between sequences and discrete structures.

An improved replica-exchange sampling method: temperature intervals with global energy reassignment

In a molecular dynamics (MD) simulation, representative sampling over the entire phase space is desired to obtain an accurate canonical distribution at a given temperature. For large molecules, such as proteins, this is problematic because systems tend to become trapped in local energy minima. The extensively used replica-exchange molecular dynamics (REMD) simulation technique overcomes this kinetic-trapping problem by allowing Boltzmann-weighted configuration exchange processes to occur between numerous thermally adjacent and compositionally identical simulations that are thermostated at sequentially higher temperatures. While the REMD method provides much better sampling than conventional MD, there are two substantial difficulties that are inherent in its application: (1) the large number of replicas that must be used to span a designated temperature range and (2) the subsequent long time required for configurations sampled at high temperatures to exchange down for potential inclusion within the low-temperature ensemble of interest. In this work, a new method based on temperature intervals with global energy reassignment (TIGER) is presented that overcomes both of these problems. A TIGER simulation is conducted as a series of short heating-sampling-quenching cycles. At the end of each cycle, the potential energies of all replicas are simultaneously compared at the same temperature using a Metropolis sampling method and then globally reassigned to the designated temperature levels. TIGER is compared with regular MD and REMD methods for the alanine dipeptide in water. The results indicate that TIGER increases sampling efficiency while substantially reducing the number of central processing units required for a comparable conventional REMD simulation.

Folding kinetics of a lattice protein via a forward flux sampling approach

We implement a forward flux sampling approach [R. J. Allen et al., J. Chem. Phys. 124, 194111 (2006)] for calculating transition rate constants and for sampling paths of protein folding events. The algorithm generates trajectories for the transition between the unfolded and folded states as chains of partially connected paths, which can be used to obtain the transition-state ensemble and the properties that characterize these intermediates. We apply this approach to Monte Carlo simulations of a model lattice protein in open space and in confined spaces of varying dimensions. We study the effect of confinement on both protein thermodynamic stability and folding kinetics; the former by mapping free-energy landscapes and the latter by the determination of rate constants and mechanistic details of the folding pathway. Our results show that, for the range of temperatures where the native state is stable, confinement of a protein destabilizes the unfolded state by reducing its entropy, resulting in increased thermodynamic stability of the folded state. Relative to the folding in open space, we find that the kinetics can be accelerated at temperatures above the temperature at which the unconfined protein folds fastest and that the rate constant increases with the number of constrained dimensions. By examining the statistical properties of the transition-state ensemble, we detect signs of a classical nucleation folding mechanism for a core of native contacts formed at an early stage of the process. This nucleus acts as folding foci and is composed of those residues that have higher probability to form native contacts in the transition-state intermediates, which can vary depending on the confinement conditions of the system.

One-dimensional reaction coordinate and the corresponding potential of mean force from commitment probability distribution

In general, finding a one-dimensional representation of the kinetics of a high-dimensional system is a great simplification for the study of complex systems. Here, we propose a method to obtain a reaction coordinate whose potential of the mean force can reproduce the commitment probability distribution from the multidimensional surface. We prove that such a relevant one-dimensional representation can be readily calculated from the equilibrium distribution of commitment probabilities, which can be obtained with simulations. Also, it is shown that this representation is complementary to a previously proposed one-dimensional representation based on a quadratic approximation of the potential energy surface. The usefulness of the method is examined with dynamics in a two-dimensional system, showing that the one-dimensional surface thus obtained can predict the existence of an intermediate and the occurrence of path switching without a priori knowledge of the morphology of the original surface. The applicability of the method to more complex and realistic reactions such as protein folding is also discussed.

Identification of characteristic protein folding channels in a coarse-grained hydrophobic-polar peptide model

Folding channels and free-energy landscapes of hydrophobic-polar heteropolymers are discussed on the basis of a minimalistic off-lattice coarse-grained model. We investigate how rearrangements of hydrophobic and polar monomers in a heteropolymer sequence lead to completely different folding behaviors. Studying three exemplified sequences with the same content of hydrophobic and polar residues, we can reproduce within this simple model two-state folding, folding through intermediates, as well as metastability.

Two-state folding, folding through intermediates, and metastability in a minimalistic hydrophobic-polar model for proteins

Within the frame of an effective, coarse-grained hydrophobic-polar protein model, we employ multicanonical Monte Carlo simulations to investigate free-energy landscapes and folding channels of exemplified heteropolymer sequences, which are permutations of each other. Despite the simplicity of the model, the knowledge of the free-energy landscape in dependence of a suitable system order parameter enables us to reveal complex folding characteristics known from real bioproteins and synthetic peptides, such as two-state folding, folding through weakly stable intermediates, and glassy metastability.

Denaturation of jack-bean urease by sodium n-dodecyl sulphate: A kinetic study below the critical micelle concentration

Kinetics of urease denaturation by anionic surfactant (sodium n-dodecyl sulphate, SDS) at concentrations below the critical micelle concentration (CMC) is investigated spectrophotometrically at neutral pH and the corresponding two-phase kinetic parameters of the process are estimated from a three-state reversible process using a binomial exponential relation based on the relaxation time method as: Using a prepared computer program, the experimental data are properly fitted into a binomial exponential relation, considering a two-phase denaturation pathway including a kinetically stable folded intermediate formed at SDS concentration of 1.1 mM. Forward and backward rate constants are estimated as: k(1)=0.2141+/-4.5 x 10(-3), k(2)=5.173 x 10(-3)+/-8.3 x 10(-5), k(-1)=0.09432+/-3.6 x 10(-4) and k(-2)=2.079 x 10(-3)+/-5.6 x 10(-5)s(-1) for the proposed mechanism. The rate-limiting step as well as the reaction coordinates in the denaturation mechanism are established. The mechanism involves formation of a kinetically stable folded native like intermediate through the electrostatic interactions. The intermediate was found to be more stable even than the native form (by about 9 kJmol(-1)) and still hexamer, because no loss of amplitude was observed. Electrophoresis experiments on the native and surfactant/urease complexes indicated a higher mobility for the kinetically folded native like intermediate.

Hydrophobic collapse in (in silico) protein folding

A model of hydrophobic collapse, which is treated as the driving force for protein folding, is presented. This model is the superposition of three models commonly used in protein structure prediction: (1) 'oil-drop' model introduced by Kauzmann, (2) a lattice model introduced to decrease the number of degrees of freedom for structural changes and (3) a model of the formation of hydrophobic core as a key feature in driving the folding of proteins. These three models together helped to develop the idea of a fuzzy-oil-drop as a model for an external force field of hydrophobic character mimicking the hydrophobicity-differentiated environment for hydrophobic collapse. All amino acids in the polypeptide interact pair-wise during the folding process (energy minimization procedure) and interact with the external hydrophobic force field defined by a three-dimensional Gaussian function. The value of the Gaussian function usually interpreted as a probability distribution is treated as a normalized hydrophobicity distribution, with its maximum in the center of the ellipsoid and decreasing proportionally with the distance versus the center. The fuzzy-oil-drop is elastic and changes its shape and size during the simulated folding procedure.

Fuzzy-oil-drop hydrophobic force field--a model to represent late-stage folding (in silico) of lysozyme

A model of hydrophobic collapse (in silico), which is generally considered to be the driving force for protein folding, is presented in this work. The model introduces the external field in the form of a fuzzy-oil-drop assumed to represent the environment. The drop is expressed in the form of a three-dimensional Gauss function. The usual probability value is assumed to represent the hydrophobicity distribution in the three-dimensional space of the virtual environment. The differences between this idealized hydrophobicity distribution and the one represented by the folded polypeptide chain is the parameter to be minimized in the structure optimization procedure. The size of fuzzy-oil-drop is critical for the folding process. A strong correlation between protein length and the dimension of the native and early-stage molecular form was found on the basis of single-domain proteins analysis. A previously presented early-stage folding (in silico) model was used to create the starting structure for the procedure of late-stage folding of lysozyme. The results of simulation were found to be promising, although additional improvements for the formation of beta-structure and disulfide bonds as well as the participation of natural ligand in folding process seem to be necessary.

Hydrophobic collapse in late-stage folding (in silico) of bovine pancreatic trypsin inhibitor

Hydrophobic collapse is commonly considered as a process of significance for protein folding. Many models have been applied for description of this event. A model introducing an external force field mimicking the hydrophobic environment and simultaneously the driving force for the folding process is presented in this paper. Bovine pancreatic trypsin inhibitor (BPTI) was taken as a test protein. An early-stage folding (in silico) model presented elsewhere was used to create the starting structure for hydrophobic collapse. The resulting structure was energy-refined using molecular dynamics simulation in an explicit solvent. The similarity versus the crystal structure of BPTI is estimated using visual analysis, residue-residue contacts, phi, psi angle distributions, RMSD, accessible solvent area, radii of gyration and hydrodynamic radii. A program allowing creation of early-stage folding structural forms to be created for any protein is available from http://bioinformatics.cm-uj.krakow.pl/earlystage. The program for late-stage folding simulation is available on request.

Folding with downhill behavior and low cooperativity of proteins

The downhill folding observed experimentally for a small protein BBL is studied using off-lattice Gō-like model. Our simulations show that the downhill folding has low cooperativity and is barrierless, which is consistent with the experimental findings. As an example of comparison in detail, the two-state folding behavior of proteins, for example, protein CI2, is also simulated. By observing the formation of contacts between the residues for these two proteins, it is found that the physical origin of the downhill folding is due to the deficiency of nonlocal contacts which determine the folding cooperatively. From a statistics on contacts of the native structures of 17 well-studied proteins and the calculation of their cooperativity factors kappa2 based on folding simulations, a strong correlation between the number of nonlocal contacts per residue NN and the factors kappa2 is obtained. Protein BBL with a value of NN = 0.73 has the lowest cooperativity factor kappa2 = 0.34 among all 17 proteins. A crossover around NNc approximately 0.9 could be defined to separate the two-state folders and the downhill folder roughly. A protein would behave downhill folding when its NN = NNc. For proteins with their NN values are about (or slightly larger than) NNc, the folding behaves with low cooperativity and the barriers are small, showing a weak two-state behavior or a downhill-like behavior. Furthermore, simulations on mutants of a two-state folder show that a mutant becomes a downhill folder when its NN is reduced to a value smaller than NNc. These could enable us to identify the downhill folding or the cooperative two-state folding behavior solely from the native structures of proteins.

Multicanonical study of coarse-grained off-lattice models for folding heteropolymers

We have performed multicanonical simulations of hydrophobic-hydrophilic heteropolymers with two simple effective, coarse-grained off-lattice models to study the influence of specific interactions in the models on conformational transitions of selected sequences with 20 monomers. Another aspect of the investigation was the comparison with the purely hydrophobic homopolymer and the study of general conformational properties induced by the "disorder" in the sequence of a heteropolymer. Furthermore, we applied an optimization algorithm to sequences with up to 55 monomers and compared the global-energy minimum found with lowest-energy states identified within the multicanonical simulation. This was used to find out how reliable the multicanonical method samples the free-energy landscape, in particular for low temperatures.

All-atom Monte Carlo simulation of GCAA RNA folding

We report a detailed all-atom simulation of the folding of the GCAA RNA tetraloop. The GCAA tetraloop motif is a very common and thermodynamically stable secondary structure in natural RNAs. We use our simulation methods to study the folding behavior of a 12-base GCAA tetraloop structure with a four-base helix adjacent to the tetraloop proper. We implement an all-atom Monte Carlo (MC) simulation of RNA structural dynamics using a Go potential. Molecular dynamics (MD) simulation of RNA and protein has realistic energetics and sterics, but is extremely expensive in terms of computational time. By coarsely treating non-covalent energetics, but retaining all-atom sterics and entropic effects, all-atom MC techniques are a useful method for the study of protein and now RNA. We observe a sharp folding transition for this structure, and in simulations at room temperature the state histogram shows three distinct minima: an unfolded state (U), a more narrow intermediated state (I), and a narrow folded state (F). The intermediate consists primarily of structures with the GCAA loop and some helix hydrogen bonds formed. Repeated kinetic folding simulations reveal that the number of helix base-pairs forms a simple 1D reaction coordinate for the I-->N transition.

Parallel replica dynamics with a heterogeneous distribution of barriers: Application ton-hexadecane pyrolysis

Parallel replica dynamics simulation methods appropriate for the simulation of chemical reactions in molecular systems with many conformational degrees of freedom have been developed and applied to study the microsecond-scale pyrolysis of n-hexadecane in the temperature range of 2100-2500 K. The algorithm uses a transition detection scheme that is based on molecular topology, rather than energetic basins. This algorithm allows efficient parallelization of small systems even when using more processors than particles (in contrast to more traditional parallelization algorithms), and even when there are frequent conformational transitions (in contrast to previous implementations of the parallel replica algorithm). The parallel efficiency for pyrolysis initiation reactions was over 90% on 61 processors for this 50-atom system. The parallel replica dynamics technique results in reaction probabilities that are statistically indistinguishable from those obtained from direct molecular dynamics, under conditions where both are feasible, but allows simulations at temperatures as much as 1000 K lower than direct molecular dynamics simulations. The rate of initiation displayed Arrhenius behavior over the entire temperature range, with an activation energy and frequency factor of E(a) = 79.7 kcal/mol and log A/s(-1) = 14.8, respectively, in reasonable agreement with experiment and empirical kinetic models. Several interesting unimolecular reaction mechanisms were observed in simulations of the chain propagation reactions above 2000 K, which are not included in most coarse-grained kinetic models. More studies are needed in order to determine whether these mechanisms are experimentally relevant, or specific to the potential energy surface used.

Explicit-chain model of native-state hydrogen exchange: Implications for event ordering and cooperativity in protein folding

Native-state hydrogen exchange experiments on several proteins have revealed partially unfolded conformations with diverse stabilities. These equilibrium observations have been used to support kinetic arguments that folding proceeds via a sequential "pathway." This interpretative logic is evaluated here by analyzing the relationship between thermodynamic behavior and folding kinetics in a class of simplified lattice protein models. The chain models studied have varying degrees of cooperative interplay (coupling) between local helical conformational preference and favorable nonlocal interactions. When model cooperativity is high, as native conditions are weakened, "isotherms" of free energy of exchange for residues belonging to the same helix merge together before global unfolding. The point of merger depends on the model energetic favorability of the helix. This trend is similar to the corresponding experimental observations. Kinetically, we find that the ordering of helix formation in the very last stage of native core assembly tends to follow the stabilities of their converged isotherms. In a majority (but not all) of folding trajectories, the final assembly of helices that are thermodynamically more stable against exchange precedes that of helices that are less stable against exchange. These model features are in partial agreement with common experimental interpretations. However, the model results also underscore the ensemble nature of the folding process: the kinetics of helix formation is not a discrete, strictly "all-or-none" process as that envisioned by certain non-explicit-chain models. Helices generally undergo many cycles of partial formation and dissolution before their conformations are fixed in the final assembly stage of folding, a kinetic stage that takes up only approximately 2% of the average folding time in the present model; and the ordering of the helices' final assembly in some trajectories can be different from the dominant ordering stipulated by the exchange isotherms.

Solvent-amino acid interaction energies in three-dimensional-lattice Monte Carlo simulations of a model 27-mer protein: Folding thermodynamics and kinetics

Amino acid residue-solvent interactions are required for lattice Monte Carlo simulations of model proteins in water. In this study, we propose an interaction-energy scale that is based on the interaction scale by Miyazawa and Jernigan. It permits systematic variation of the amino acid-solvent interactions by introducing a contrast parameter for the hydrophobicity, C(s), and a mean attraction parameter for the amino acids, omega. Changes in the interaction energies strongly affect many protein properties. We present an optimized energy parameter set for best representing realistic behavior typical for many proteins (fast folding and high cooperativity for single chains). Our optimal parameters feature a much weaker hydrophobicity contrast and mean attraction than does the original interaction scale. The proposed interaction scale is designed for calculating the behavior of proteins in bulk and at interfaces as a function of solvent characteristics, as well as protein size and sequence.

3D-Lattice Monte Carlo simulations of model proteins. Size effects on folding thermodynamics and kinetics

Recently, we devised an energy scale to vary systematically amino-acid residue-solvent interactions for Monte Carlo simulations of lattice-model proteins in water. For 27-mer proteins, the folding behavior varies appreciably with the choice of interaction parameters. We now perform similar simulations with 64-mers to study the size dependence of the optimal energy parameter set for representing realistic behavior typical of many real proteins (i.e. fast folding and high cooperativity for single chains). We find that 64-mers are considerably more stable and more cooperative compared to 27-mers. The optimal interfacial-interaction-energy parameter set, however, is relatively size independent.

What can one learn from experiments about the elusive transition state?

We present the results of an exact analysis of a model energy landscape of a protein to clarify the idea of the transition state and the physical meaning of the phi values determined in protein engineering experiments. We benchmark our findings to various theoretical approaches proposed in the literature for the identification and characterization of the transition state.

Towards realistic description of collective motions in the lattice protein folding models

Collective motions and the formation of clusters of residues play an important role in the folding of real proteins. However, existing Monte Carlo (MC) techniques of the protein folding simulations based on highly popular lattice models provide only a schematic representation of collective motions, which is rather far from physical reality. The Clustering Monte Carlo (CMC) algorithm was developed with particular aim to provide a realistic description of collective motions on the lattice. CMC allows modeling the cluster dynamics and the effects of the solvent viscosity, which is impossible in conventional algorithms. In this study two 2D lattice peptides, with the ground states of hierarchical and non-hierarchical design, were investigated comparatively using three methods: Metropolis MC with the local move set, Metropolis MC with unspecific rigid rotations and the CMC algorithm. We present evidence that the folding pathways and kinetics of hierarchically folding clustered sequence are not adequately described in conventional MC simulations, and the account for cluster dynamics provided by CMC allows to capture essential features of the folding process. Our data suggest that the methods, which enable specific cluster motions, such as CMC, should be used for a more realistic description of protein folding.

Improved Gō-like Models Demonstrate the Robustness of Protein Folding Mechanisms Towards Non-native Interactions

The use of simple theoretical models has provided a considerable contribution to our present understanding of the means by which proteins adopt their native fold from the plethora of available unfolded states. A common assumption in building computationally tractable models has been the neglect of stabilizing non-native interactions in the class of models described as "Gō-like." The focus of this study is the characterization of the folding of a number of proteins via a Gō-like model, which aims to map a maximal amount of information reflecting the protein sequence onto a "minimalist" skeleton. This model is shown to contain sufficient information to reproduce the folding transition states of a number of proteins, including topologically analogous proteins that fold via different transition states. Remarkably, these models also demonstrate consistency with the general features of folding transition states thought to be stabilized by non-native interactions. This suggests that native interactions are the primary determinant of most protein folding transition states, and that non-native interactions lead only to local structural perturbations. A prediction is also included for an asymmetrical folding transition state of bacteriophage lambda protein W, which has yet to be subjected to experimental characterization.

Origins of chevron rollovers in non-two-state protein folding kinetics

Chevron rollovers of some proteins imply that their logarithmic folding rates are nonlinear in native stability. This is predicted by lattice and continuum Gō models to arise from diminished accessibilities of the ground state from transiently populated compact conformations under strongly native conditions. Despite these models' native-centric interactions, the slowdown is due partly to kinetic trapping caused by some of the folding intermediates' non-native topologies. Notably, simple two-state folding kinetics of small single-domain proteins are not reproduced by common Gō-like schemes.