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.     

A quasi-physical algorithm for the structure optimization in an off-lattice protein model

Introduction Predicting the native structure of a protein from its amino acid sequence is one of the most challeng- ing problems in biophysics and bioinformatics. The difficulty of the problem comes from two aspects. One is the determination of the potential energy func- tion. The effective energy function can generally dis- tinguish the native states from non-native states of protein molecules. The other is that the potential en- ergy landscape of the system can be characterized by a multitu…     

Application of a chaperone-based refolding method to two- and three-dimensional off-lattice protein models

A model of protein-chaperone interaction as a two-phase (unfolding/refolding) iterative annealing mechanism able to promote structural segregation of hydrophobic and hydrophilic monomers and thereby facilitate access to nativelike states has recently been applied successfully to two 22-mers of the Honeycutt and Thirumalai BLN (hydrophobic, hydrophilic, neutral) heteropolymer model. This technique is here applied to a much wider data set: 94 8-mers of the off-lattice protein model originally presented in two dimensions by Stillinger and Head-Gordon, and later extended into three dimensions by Irb?ck and Potthast; the model chaperone is shown to be equally successful, and by progressive elaboration of the chaperone model as in the earlier BLN model work, to be utilizing very similar underlying mechanisms. It is demonstrated that on average, contacts with the model chaperone give rise to a consistent movement in structure space in the direction of more nativelike structures; this method of global minimization does not therefore rely fundamentally on random search. Insofar as the responses to the chaperone of the two- and three-dimensional forms of the substrate model do differ, this can be interpreted as reflecting the different handling of hydrophilic monomers in the models-in particular, whether there is active repulsion between these and monomers of hydrophobic character. The chaperone-induced refolding method is also tested on a set of 220 9-mer chains of each version of the substrate model, where it is seen that the two-dimensional model, with its more clearly distinguished roles for the hydrophobic and hydrophilic monomers, shows a more favorable scaling behavior.     

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.     

Molecular modeling of proteins: a strategy for energy minimization by molecular mechanics in the AMBER force field.

Energy minimization is an important step in molecular modeling of proteins. In this study, we sought to develop a minimization strategy which would give the best final structures with the shortest computer time in the AMBER force field. In the all-atom model, we performed energy minimization of the melittin (mostly alpha-helical) and cardiotoxin (mostly beta-sheet and beta-turns) crystal structures by both constrained and unconstrained pathways. In the constrained path, which has been recommended in the energy minimization of proteins, hydrogens were relaxed first, followed by the side chains of amino acid residues, and finally the whole molecule. Despite the logic of this approach, however, the structures minimized by the unconstrained path fit the experimental structures better than those minimized by constrained paths. Moreover, the unconstrained path saved considerable computer time. We also compared the effects of the steepest descents and conjugate gradients algorithms in energy minimization. Previously, steepest descents has been used in the initial stages of minimization and conjugate gradients in the final stages of minimization. We therefore studied the effect on the final structure of performing an initial minimization by steepest descents. The structures minimized by conjugate gradients alone resembled the structures minimized initially by the steepest descents and subsequently by the conjugate gradients algorithms. Thus an initial minimization using steepest descents is wasteful and unnecessary, especially when starting from the crystal structure. Based on these results, we propose the use of an unconstrained path and conjugate gradients for energy minimization of proteins. This procedure results in low energy structures closer to the experimental structures, and saves about 70-80% of computer time. This procedure was applied in building models of lysozyme mutants. The crystal structure of native T4 lysozyme was mutated to three different mutants and the structures were minimized. The minimized structures closely fit the crystal structures of the respective mutants (less than 0.3 A root-mean-square, RMS, deviation in the position of all heavy atoms). These results confirm the efficiency of the proposed minimization strategy in modeling closely related homologs. To determine the reliability of the united atom approximation, we also performed all of the above minimizations with united atom models. This approximation gave structures with similar but slightly higher RMS deviations than the all-atom model, but gave further savings of 60-70% in computer time. However, we feel further investigation is essential to determine the reliability of this approximation.(ABSTRACT TRUNCATED AT 400 WORDS)     

Molecular Modeling of Proteins: A Strategy for Energy Minimization by Molecular Mechanics in the AMBER Force Field

Abstract

Energy minimization is an important step in molecular modeling of proteins. In this study, we sought to develop a minimization strategy which would give the best final structures with the shortest computer time in the AMBER force field. In the all-atom model, we performed energy minimization of the melittin (mostly α-helical) and cardiotoxin (mostly β-sheet and β-turns) crystal structures by both constrained and unconstrained pathways. In the constrained path, which has been recommended in the energy minimization of proteins, hydrogens were relaxed first, followed by the side chains of amino acid residues, and finally the whole molecule. Despite the logic of this approach, however, the structures minimized by the unconstrained path fit the experimental structures better than those minimized by constrained paths. Moreover, the unconstrained path saved considerable computer time. We also compared the effects of the steepest descents and conjugate gradients algorithms in energy minimization. Previously, steepest descents has been used in the initial stages of minimization and conjugate gradients in the final stages of minimization. We therefore studied the effect on the final structure of performing an initial minimization by steepest descents. The structures minimized by conjugate gradients alone resembled the structures minimized initially by the steepest descents and subsequently by the conjugate gradients algorithms. Thus an initial minimization using steepest descents is wasteful and unnecessary, especially when starting from the crystal structure. Based on these results, we propose the use of an unconstrained path and conjugate gradients for energy minimization of proteins. This procedure results in low energy structures closer to the experimental structures, and saves about 70–80% of computer time. This procedure was applied in building models of lysozyme mutants. The crystal structure of native T₄ lysozyme was mutated to three different mutants and the structures were minimized. The minimized structures closely fit the crystal structures of the respective mutants (< 0.3 Å root-mean-square, RMS, deviation in the position of all heavy atoms). These results confirm the efficiency of the proposed minimization strategy in modeling closely related homologs. To determine the reliability of the united atom approximation, we also performed all of the above minimizations with united atom models. This approximation gave structures with similar but slightly higher RMS deviations than the all-atom model, but gave further savings of60-70% in computer time. However, we feel further investigation is essential to determine the reliability of this approximation. Finally, to determine the limitation of the procedure, we built the melittin molecule interactively in an α-helical conformation and this model showed an RMS deviation greater than 2.8 Å when compared to the melittin crystal structure. This model was minimized by various strategies. None of the minimized structures converged towards the crystal structure. Thus, although the proposed method seems to give valid structures starting from closely related crystal structures, it cannot predict the native structure when the starting structure is far from the native structure. From these results, we recommend the use of the proposed strategy of minimizing by an unconstrained path using the conjugate gradients algorithm, but only for modeling of closely related structural homologs of proteins.     

Ligand binding to proteins: the binding landscape model.

Models of ligand binding are often based on four assumptions: (1) steric fit: that binding is determined mainly by shape complementarity; (2) native binding: that ligands mainly bind to native states; (3) locality: that ligands perturb protein structures mainly at the binding site; and (4) continuity: that small changes in ligand or protein structure lead to small changes in binding affinity. Using a generalization of the 2D HP lattice model, we study ligand binding and explore these assumptions. We first validate the model by showing that it reproduces typical binding behaviors. We observe ligand-induced denaturation, ANS and heme-like binding, and "lock-and-key" and "induced-fit" specific binding behaviors characterized by Michaelis-Menten or more cooperative types of binding isotherms. We then explore cases where the model predicts violations of the standard assumptions. For example, very different binding modes can result from two ligands of identical shape. Ligands can sometimes bind highly denatured states more tightly than native states and yet have Michaelis-Menten isotherms. Even low-population binding to denatured states can cause changes in global stability, hydrogen-exchange rates, and thermal B-factors, contrary to expectations, but in agreement with experiments. We conclude that ligand binding, similar to protein folding, may be better described in terms of energy landscapes than in terms of simpler mass-action models.     

Identifying proteins of high designability via surface-exposure patterns

Using an off-lattice model, we fully enumerate folded conformations of polypeptide chains of up to N = 19 monomers. Structures are found to differ markedly in designability, defined as the number of sequences with that structure as a unique lowest-energy conformation. We find that designability is closely correlated with the pattern of surface exposure of the folded structure. For longer chains, complete enumeration of structures is impractical. Instead, structures can be randomly sampled, and relative designability estimated either from designability within the random sample, or directly from surface-exposure pattern. We compare the surface-exposure patterns of those structures identified as highly designable to the patterns of naturally occurring proteins.     

Computational protein design with side-chain conformational entropy

Recent advances in modeling protein structures at the atomic level have made it possible to tackle "de novo" computational protein design. Most procedures are based on combinatorial optimization using a scoring function that estimates the folding free energy of a protein sequence on a given main-chain structure. However, the computation of the conformational entropy in the folded state is generally an intractable problem, and its contribution to the free energy is not properly evaluated. In this article, we propose a new automated protein design methodology that incorporates such conformational entropy based on statistical mechanics principles. We define the free energy of a protein sequence by the corresponding partition function over rotamer states. The free energy is written in variational form in a pairwise approximation and minimized using the Belief Propagation algorithm. In this way, a free energy is associated to each amino acid sequence: we use this insight to rescore the results obtained with a standard minimization method, with the energy as the cost function. Then, we set up a design method that directly uses the free energy as a cost function in combination with a stochastic search in the sequence space. We validate the methods on the design of three superficial sites of a small SH3 domain, and then apply them to the complete redesign of 27 proteins. Our results indicate that accounting for entropic contribution in the score function affects the outcome in a highly nontrivial way, and might improve current computational design techniques based on protein stability.     

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.     

Structure optimization of the two-dimensional off-lattice hydrophobic–hydrophilic model

A two-dimensional off-lattice protein model with two species of monomers, hydrophobic and hydrophilic, was studied. Low-energy configurations in the model were optimized using the improved energy landscape paving (ELP+) method. In ELP+, the energy landscape paving (ELP) was first applied to search for the low-energy states. After the ELP led to the basins of the local energy minima, the additional degree-of-freedom of bond length was introduced, and the gradient descent method was then used to search for lower energy states near the local minima. Numerical results show that the proposed methods are quite effective for finding the ground states of proteins. A comparison between ELP+ and other methods is made.     

Relationship between Personality Disorder Functioning Styles and the Emotional States in Bipolar I and II Disorders

Background

Bipolar disorder types I (BD I) and II (BD II) behave differently in clinical manifestations, normal personality traits, responses to pharmacotherapies, biochemical backgrounds and neuroimaging activations. How the varied emotional states of BD I and II are related to the comorbid personality disorders remains to be settled.

Methods

We therefore administered the Plutchick – van Praag Depression Inventory (PVP), the Mood Disorder Questionnaire (MDQ), the Hypomanic Checklist-32 (HCL-32), and the Parker Personality Measure (PERM) in 37 patients with BD I, 34 BD II, and in 76 healthy volunteers.

Results

Compared to the healthy volunteers, patients with BD I and II scored higher on some PERM styles, PVP, MDQ and HCL-32 scales. In BD I, the PERM Borderline style predicted the PVP scale; and Antisocial predicted HCL-32. In BD II, Borderline, Dependant, Paranoid (-) and Schizoid (-) predicted PVP; Borderline predicted MDQ; Passive-Aggressive and Schizoid (-) predicted HCL-32. In controls, Borderline and Narcissistic (-) predicted PVP; Borderline and Dependant (-) predicted MDQ.

Conclusion

Besides confirming the different predictability of the 11 functioning styles of personality disorder to BD I and II, we found that the prediction was more common in BD II, which might underlie its higher risk of suicide and poorer treatment outcome.     

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.     

Enumerating Designing Sequences in the HP Model

The hydrophobic/polar HP model on the square lattice has been widely used toinvestigate basics of protein folding. In the cases where all designing sequences (sequences with unique ground states) were enumerated without restrictions on the number of contacts, the upper limit on the chain length N has been 18–20 because of the rapid exponential growth of thenumbers of conformations and sequences. We show how a few optimizations push this limit by about 5 units. Based on these calculations, we study the statistical distribution of hydrophobicity along designing sequences. We find that the average number of hydrophobic and polar clumps along the chains is larger for designing sequences than for random ones, which is in agreement with earlier findings for N 18 and with results for real enzymes. We also show that this deviation from randomness disappears if the calculations are restricted to maximally compact structures.     

Model of a complex between the tetrahemic cytochrome c3 and the ferredoxin I from Desulfovibrio desulfuricans (Norway strain)

A three-dimensional model of an electron-transfer complex between the tetrahemic cytochrome c3 and the ferredoxin I from the sulfate-reducing bacterium Desulfovibrio desulfuricans (Norway strain) has been generated through computer graphics methods. The model is based on the known X-ray structure of the cytochrome and on a model of the ferredoxin that has been derived through computer graphics modeling and energy minimization methods, from the X-ray structure of the homologous ferredoxin from Peptococcus aerogenes. Four possible models of interaction between the two molecules were examined by bringing in close proximity each of the four hemes and the redox center (4Fe-4S) of the ferredoxin and by optimizing the ion pairs interactions. One of these models shows by far the "best" structure in terms of charges, interactions, and complementarity of the topology of the contact surfaces. In this complex, the distance between the iron atoms of the ferredoxin redox center and the hemic iron atom is 11.8 A, which compares well with those found between redox centers in other complexes. The contact surface area between the two molecules is 170 A2.     

Bayesian Hierarchical Clustering for Studying Cancer Gene Expression Data with Unknown Statistics

Clustering analysis is an important tool in studying gene expression data. The Bayesian hierarchical clustering (BHC) algorithm can automatically infer the number of clusters and uses Bayesian model selection to improve clustering quality. In this paper, we present an extension of the BHC algorithm. Our Gaussian BHC (GBHC) algorithm represents data as a mixture of Gaussian distributions. It uses normal-gamma distribution as a conjugate prior on the mean and precision of each of the Gaussian components. We tested GBHC over 11 cancer and 3 synthetic datasets. The results on cancer datasets show that in sample clustering, GBHC on average produces a clustering partition that is more concordant with the ground truth than those obtained from other commonly used algorithms. Furthermore, GBHC frequently infers the number of clusters that is often close to the ground truth. In gene clustering, GBHC also produces a clustering partition that is more biologically plausible than several other state-of-the-art methods. This suggests GBHC as an alternative tool for studying gene expression data.The implementation of GBHC is available at https://sites.google.com/site/gaussianbhc/     

First-Principles Protein Folding Simulations

Abstract

It is widely believed that the prediction of the three-dimensional structures of proteins from the first principles is impossible. This view is based on the fact that the number of possible structures for each protein is astronomically large. The question is then why a protein folds into its native structure with the proper biological functions in the time scale of milliseconds to minutes, and this is called Levinthal's paradox. In this article I will discuss our strategy for attacking the protein folding problem. Our approach consists of two elements: the inclusion of accurate solvent effects and the development of powerful simulation algorithms that can avoid getting trapped in states of energy local minima. For the former, we discuss several models varying in nature from crude (distance-dependent dielectric function) to rigorous (reference interaction site model). For the latter, we show the effectiveness of Monte Carlo simulated annealing and generalized-ensemble algorithms.     

Selected-fit versus induced-fit protein binding: kinetic differences and mutational analysis

The binding of a ligand molecule to a protein is often accompanied by conformational changes of the protein. A central question is whether the ligand induces the conformational change (induced-fit), or rather selects and stabilizes a complementary conformation from a pre-existing equilibrium of ground and excited states of the protein (selected-fit). We consider here the binding kinetics in a simple four-state model of ligand-protein binding. In this model, the protein has two conformations, which can both bind the ligand. The first conformation is the ground state of the protein when the ligand is off, and the second conformation is the ground state when the ligand is bound. The induced-fit mechanism corresponds to ligand binding in the unbound ground state, and the selected-fit mechanism to ligand binding in the excited state. We find a simple, characteristic difference between the on- and off-rates in the two mechanisms if the conformational relaxation into the ground states is fast. In the case of selected-fit binding, the on-rate depends on the conformational equilibrium constant, whereas the off-rate is independent. In the case of induced-fit binding, in contrast, the off-rate depends on the conformational equilibrium, while the on-rate is independent. Whether a protein binds a ligand via selected-fit or induced-fit thus may be revealed by mutations far from the protein's binding pocket, or other "perturbations" that only affect the conformational equilibrium. In the case of selected-fit, such mutations will only change the on-rate, and in the case of induced-fit, only the off-rate.