Structure optimization in a three-dimensional off-lattice protein model

We studied a three-dimensional off-lattice AB model with two species of monomers, hydrophobic (A) and hydrophilic (B), and present two optimization algorithms: face-centered-cubic (FCC)-lattice pruned-enriched-Rosenbluth method (PERM) and subsequent conjugate gradient (PERM++) minimization and heuristic conjugate gradient (HCG) simulation based on "off-trap" strategy. In PERM++, we apply the PERM to the FCC-lattice to produce the initial conformation, and conjugate gradient minimization is then used to reach the minimum energy state. Both algorithms have been tested in the three-dimensional AB model for all sequences with lengths 13 < or = n < or = 55. The numerical results show that the proposed methods are very promising for finding the ground states of proteins. In several cases, we renew the putative ground states energy values.     

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.     

Emergence of highly designable protein-backbone conformations in an off-lattice model

Despite the variety of protein sizes, shapes, and backbone configurations found in nature, the design of novel protein folds remains an open problem. Within simple lattice models it has been shown that all structures are not equally suitable for design. Rather, certain structures are distinguished by unusually high designability: the number of amino acid sequences for which they represent the unique lowest energy state; sequences associated with such structures possess both robustness to mutation and thermodynamic stability. Here we report that highly designable backbone conformations also emerge in a realistic off-lattice model. The highly designable conformations of a chain of 23 amino acids are identified and found to be remarkably insensitive to model parameters. Although some of these conformations correspond closely to known natural protein folds, such as the zinc finger and the helix-turn-helix motifs, others do not resemble known folds and may be candidates for novel fold design.     

Toward minimalist models of larger proteins: a ubiquitin-like protein

Our recently developed off-lattice bead model capable of simulating protein structures with mixed alpha/beta content has been extended to model the folding of a ubiquitin-like protein and provides a means for examining the more complex kinetics involved in the folding of larger proteins. Using trajectories generated from constant-temperature Langevin dynamics simulations and sampling with the multiple multi-histogram method over five-order parameters, we are able to characterize the free energy landscape for folding and find evidence for folding through compact intermediates. Our model reproduces the observation that the C-terminus loop structure in ubiquitin is the last to fold in the folding process and most likely plays a spectator role in the folding kinetics. The possibility of a productive metastable intermediate along the folding pathway consisting of collapsed states with no secondary structure, and of intermediates or transition structures involving secondary structural elements occurring early in the sequence, is also supported by our model. The kinetics of folding remain multi-exponential below the folding temperature, with glass-like kinetics appearing at T/T(f) approximately 0.86. This new physicochemical model, designed to be predictive, helps validate the value of modeling protein folding at this level of detail for genomic-scale studies, and motivates further studies of other protein topologies and the impact of more complex energy functions, such as the addition of solvation forces.     

Principles of protein folding--a perspective from simple exact models.

General principles of protein structure, stability, and folding kinetics have recently been explored in computer simulations of simple exact lattice models. These models represent protein chains at a rudimentary level, but they involve few parameters, approximations, or implicit biases, and they allow complete explorations of conformational and sequence spaces. Such simulations have resulted in testable predictions that are sometimes unanticipated: The folding code is mainly binary and delocalized throughout the amino acid sequence. The secondary and tertiary structures of a protein are specified mainly by the sequence of polar and nonpolar monomers. More specific interactions may refine the structure, rather than dominate the folding code. Simple exact models can account for the properties that characterize protein folding: two-state cooperativity, secondary and tertiary structures, and multistage folding kinetics--fast hydrophobic collapse followed by slower annealing. These studies suggest the possibility of creating "foldable" chain molecules other than proteins. The encoding of a unique compact chain conformation may not require amino acids; it may require only the ability to synthesize specific monomer sequences in which at least one monomer type is solvent-averse.     

Effect of denaturant and protein concentrations upon protein refolding and aggregation: a simple lattice model.

We present a study of the competition between protein refolding and aggregation for simple lattice model proteins. The effect of solvent conditions (i.e., the denaturant concentration and the protein concentration) on the folding and aggregation behavior of a system of simple, two-dimensional lattice protein molecules has been investigated via (dynamic Monte Carlo simulations. The population profiles and aggregation propensities of the nine most populated intermediate configurations exhibit a complex dependence on the solution conditions that can be understood by considering the competition between intra- and interchain interactions. Some of these configurations are not even seen in isolated chain simulations; they are observed to be highly aggregation prone and are stabilized primarily by the aggregation reaction in multiple-chain systems. Aggregation arises from the association of partially folded intermediates rather than from the association of denatured random-coil states. The aggregation reaction dominates over the folding reaction at high protein concentration and low denaturant concentration, resulting in low refolding yields at those conditions. However, optimum folding conditions exist at which the refolding yield is a maximum, in agreement with some experimental observations.     

Rubredoxin refolding on nanostructured hydrophobic surfaces: Evidence for a new type of biomimetic chaperones

Rubredoxins (Rds) are small proteins containing a tetrahedral Fe(SCys)₄ site. Folded forms of metal free Rds (apoRds) show greatly impaired ability to incorporate iron compared with chaotropically unfolded apoRds. In this study, formation of the Rd holoprotein (holoRd) on addition of iron to a structured, but iron‐uptake incompetent apoRd was investigated in the presence of polystyrene nanoparticles (NP). In our rationale, hydrophobic contacts between apoRd and the NP surface would expose protein regions (including ligand cysteines) buried in the structured apoRd, allowing iron incorporation and folding to the native holoRd. Burial of the hydrophobic regions in the folded holoRd would allow its detachment from the NP surface. We found that both rate and yield of holoRd formation increased significantly in the presence of NP and were influenced by the NP concentration and size. Rates and yields had an optimum at “catalytic” NP concentrations (0.2 g/L NP) when using relatively small NP (46 nm diameter). At these optimal conditions, only a fraction of the apoRd was bound to the NP, consistent with the occurrence of turnover events on the NP surface. Lower rates and yields at higher NP concentrations or when using larger NP (200 nm) suggest that steric effects and molecular crowding on the NP surface favor specific “iron‐uptake‐competent” conformations of apoRd on the NP surface. This bio‐mimetic chaperone system may be applicable to other proteins requiring an unfolding step before cofactor‐triggered refolding, particularly when over‐expressed under limited cofactor accessibility. Proteins 2014; 82:3154–3162. © 2014 Wiley Periodicals, Inc.     

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.     

Effect of solvent conditions upon refolding pathways and intermediates for a simple lattice protein

Folding pathways and intermediates for a two-dimensional lattice protein have been investigated via computer simulation at various denaturant concentrations. The protein is represented as a chain of 8 hydrophobic (H) and 12 polar (P) beads on a square lattice sequenced in such a way that the native state is a compact hydrophobic core surrounded by a shell of polar beads. Two nonbonded H beads are said to attract each other with a potential of mean force of strength ϵ. Increasing |ϵ/kT| mimics decreasing the denaturant concentration in the solution. Dynamic Monte Carlo simulations have been performed in order to investigate the folding transition and the folding pathways. Sharp folding—unfolding transitions are observed and the folding process proceeds along well-defined pathways that are populated by partially folded intermediates. The folding pathways as well as the populations of the intermediates are strongly dependent upon the denaturant concentration. Generally, intermediates containing long open stretches of H beads are more populated at high denaturant concentration, whereas compact intermediates containing a substantial number of hydrophobic contacts are more populated at low denaturant concentrations. The folding process is also observed to be cooperative in nature in that the chain does not start folding until a key fold in the middle section of the chain is formed correctly. © 1997 John Wiley & Sons, Inc. Biopoly 42: 399–409, 1997     

Transient interactions of a slow-folding protein with the Hsp70 chaperone machinery

Most known proteins have at least one local Hsp70 chaperone binding site. Does this mean that all proteins interact with Hsp70 as they fold? This study makes an initial step to address the above question by examining the interaction of the E.coli Hsp70 chaperone (known as DnaK) and its co-chaperones DnaJ and GrpE with a slow-folding E.coli substrate, RNase H^D. Importantly, this protein is a nonobligatory client, and it is able to fold in vitro even in the absence of chaperones. We employ stopped-flow mixing, chromatography, and activity assays to analyze the kinetic perturbations induced by DnaK/DnaJ/GrpE (K/J/E) on the folding of RNase H^D. We find that K/J/E slows down RNase H^D''s apparent folding, consistent with the presence of transient chaperone-substrate interactions. However, kinetic retardation is moderate for this slow-folding client and it is expected to be even smaller for faster-folding substrates. Given that the interaction of folding-competent substrates such as RNase H^D with the K/J/E chaperones is relatively short-lived, it does not significantly interfere with the timely production of folded biologically active substrate. The above mode of action is important because it preserves K/J/E bioavailability, enabling this chaperone system to act primarily by assisting the folding of other misfolded and (or) aggregation-prone cellular proteins that are unable to fold independently. When refolding is carried out in the presence of K/J and absence of the nucleotide exchange factor GrpE, some of the substrate population becomes trapped as a chaperone-bound partially unfolded state.     

Accelerated simulation of unfolding and refolding of a large single chain globular protein

We have developed novel strategies for contracting simulation times in protein dynamics that enable us to study a complex protein with molecular weight in excess of 34 kDa. Starting from a crystal structure, we produce unfolded and then refolded states for the protein. We then compare these quantitatively using both established and new metrics for protein structure and quality checking. These include use of the programs Concoord and Darvols. Simulation of protein-folded structure well beyond the molten globule state and then recovery back to the folded state is itself new, and our results throw new light on the protein-folding process. We accomplish this using a novel cooling protocol developed for this work.     

Folding of a small helical protein using hydrogen bonds and hydrophobicity forces

A reduced protein model with five to six atoms per amino acid and five amino acid types is developed and tested on a three-helix-bundle protein, a 46-amino acid fragment from staphylococcal protein A. The model does not rely on the widely used Go approximation, which ignores non-native interactions. We find that the collapse transition is considerably more abrupt for the protein A sequence than for random sequences with the same composition. The chain collapse is found to be at least as fast as helix formation. Energy minimization restricted to the thermodynamically favored topology gives a structure that has a root-mean-square deviation of 1.8 A from the native structure. The sequence-dependent part of our potential is pairwise additive. Our calculations suggest that fine-tuning this potential by parameter optimization is of limited use.     

基于新杂合进化算法的蛋白质折叠计算

蛋白质能量最小化是蛋白质折叠的重要内容。用于蛋白质折叠的新的杂合进化算法结合了交叉和柯西变异。基于toy模型的蛋白质能量最小化算例表明,这个新的杂合进化算法是有效的。     

Folding pathway dependence on energetic frustration and interaction heterogeneity for a three-dimensional hydrophobic protein model

Monte Carlo simulations of a hydrophobic protein model of 40 monomers in the cubic lattice are used to explore the effect of energetic frustration and interaction heterogeneity on its folding pathway. The folding pathway is described by the dependence of relevant conformational averages on an appropriate reaction coordinate, pfold, defined as the probability for a given conformation to reach the native structure before unfolding. We compare the energetically frustrated and heterogeneous hydrophobic potential, according to which individual monomers have a higher or lower tendency to form contacts unspecifically depending on their hydrophobicities, to an unfrustrated homogeneous Go-type potential with uniformly attractive native interactions and neutral non-native interactions (called Go1 in this study), and to an unfrustrated heterogeneous potential with neutral non-native interactions and native interactions having the same energy as the hydrophobic potential (called Go2 in this study). Folding kinetics are slowed down dramatically when energetic frustration increases, as expected and previously observed in a two-dimensional model. Contrary to our previous results in two dimensions, however, it appears that the folding pathway and transition state ensemble can be significantly dependent on the energy function used to stabilize the native structure. The sequence of events along the reaction coordinate, or the order along this coordinate in which different regions of the native conformation become structured, turns out to be similar for the hydrophobic and Go2 potentials, but with analogous events tending to occur at lower pfold values in the first case. In particular, the transition state obtained from the ensemble around pfold = 0.5 is more structured for the hydrophobic potential. For Go1, not only the transition state ensemble but the order of events itself is modified, suggesting that interaction heterogeneity, in addition to energetic frustration, can have significant effects on the folding mechanism, most likely by modifying the probability of different contacts in the unfolded state, the starting point for the folding reaction. Although based on a simple model, these results provide interesting insight into how sequence-dependent switching between folding pathways might occur in real proteins.     

The importance of explicit chain representation in protein folding models: an examination of Ising-like models

A class of models that represents a protein chain as a sequence of "folded" and "unfolded" residues has recently been used to correlate rates and mechanisms of protein folding with the protein native structure. In order to better understand the conditions under which these "Ising-like" models apply, we compare results from this model to those obtained from an off-lattice model which uses the same potential function. We find that Ising-like models by construction impose folding via a highly sequential nucleation-condensation mechanism, which in turn leads to more rugged energy landscapes, fewer "pathways" to the native state, and in the specific case examined here, the cold shock protein A from Escherichia coli, a qualitative difference in the most likely order of events in folding.     

Computing the transition state populations in simple protein models

We describe the master equation method for computing the kinetics of protein folding. We illustrate the method using a simple Go model. Presently most models of two-state fast-folding protein folding kinetics invoke the classical idea of a transition state to explain why there is a single exponential decay in time. However, if proteins fold via funnel-shaped energy landscapes, as predicted by many theoretical studies, then it raises the question of what is the transition state. Is it a specific structure, or a small ensemble of structures, as is expected from classical transition state theory? Or is it more like the denatured states of proteins, a very broad ensemble? The answer that is usually obtained depends on the assumptions made about the transition state. The present method is a rigorous way to find transition states, without assumptions or approximations, even for very nonclassical shapes of energy landscapes. We illustrate the method here, showing how the transition states in two-state protein folding can be very broad ensembles. © 2002 Wiley Periodicals, Inc. Biopolymers 68: 35–46, 2003     

Exhaustive enumeration of protein conformations using experimental restraints.

We present an efficient new algorithm that enumerates all possible conformations of a protein that satisfy a given set of distance restraints. Rapid growth of all possible self-avoiding conformations on the diamond lattice provides construction of alpha-carbon representations of a protein fold. We investigated the dependence of the number of conformations on pairwise distance restraints for the proteins crambin, pancreatic trypsin inhibitor, and ubiquitin. Knowledge of between one and two contacts per monomer is shown to be sufficient to restrict the number of candidate structures to approximately 1,000 conformations. Pairwise RMS deviations of atomic position comparisons between pairs of these 1,000 structures revealed that these conformations can be grouped into about 25 families of structures. These results suggest a new approach to assessing alternative protein folds given a very limited number of distance restraints. Such restraints are available from several experimental techniques such as NMR, NOESY, energy transfer fluorescence spectroscopy, and crosslinking experiments. This work focuses on exhaustive enumeration of protein structures with emphasis on the possible use of NOESY-determined distance restraints.     

Fast tree search for a triangular lattice model of protein folding

Using a triangular lattice model to study the designability of protein folding, we overcame the parity problem of previous cubic lattice model and enumerated all the sequences and compact structures on a simple two-dimensional triangular lattice model of size 4 5 6 5 4. We used two types of amino acids, hydrophobic and polar, to make up the sequences, and achieved 223W212 different sequences excluding the reverse symmetry sequences. The total string number of distinct compact structures was 219,093, excluding reflection symmetry in the self-avoiding path of length 24 triangular lattice model. Based on this model, we applied a fast search algorithm by constructing a cluster tree. The algorithm decreased the computation by computing the objective energy of non-leaf nodes. The parallel experiments proved that the fast tree search algorithm yielded an exponential speed-up in the model of size 4 5 6 5 4. Designability analysis was performed to understand the search result.