Kinematics and thermodynamics of a folding heteropolymer

Thermal stability of hydrophobic helical oligomers: a lattice simulation study in explicit water

We investigate the thermal stability of helical hydrophobic oligomers using a three-dimensional, water-explicit lattice model and the Wang-Landau Monte Carlo method. The degree of oligomer helicity is controlled by the parameter ε(mm) < 0, which mimics monomer-monomer hydrogen bond interactions leading to the formation of helical turns in atomistic proteins. We vary |ε(mm)| between 0 and 4.5 kcal/mol and therefore investigate systems ranging from flexible homopolymers (i.e., those with no secondary structure) to helical oligomers that are stable over a broad range of temperatures. We find that systems with |ε(mm)| ≤ 2.0 kcal/mol exhibit a broad thermal unfolding transition at high temperature, leading to an ensemble of random coils. In contrast, the structure of conformations involved in a second, low-temperature, transition is strongly dependent on |ε(mm)|. Weakly helical oligomers are observed when |ε(mm)| ≤ 1.0 kcal/mol and exhibit a low-temperature, cold-unfolding-like transition to an ensemble of strongly water-penetrated globular conformations. For higher |ε(mm)| (1.7 kcal/mol ≤ |ε(mm)| ≤ 2.0 kcal/mol), cold unfolding is suppressed, and the low-temperature conformational transition becomes a "crystallization", in which a "molten" helix is transformed into a defect-free helix. The molten helix preserves ≥50% of the helical contacts observed in the "crystal" at a lower temperature. When |ε(mm)| = 4.5 kcal/mol, we find that conformational transitions are largely suppressed within the range of temperatures investigated.

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.

Finite size effects on locating conformational transitions for macromolecules

It has been shown from simulation and experiment that locations of peaks in structural and thermodynamic quantities accompanying "phase" transitions of a single macromolecule (collapse or crystallization/melting) do not coincide. Thus, for chains with finite lengths these different measures yield apparently different results for transition temperatures. To resolve this issue we use scaling, verified by computer simulations, to conclusively show that these different locations for peak positions are simply a consequence of the finite chain length, as has been conjectured previously.

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

In this paper, we study an off-lattice protein AB model with two species of monomers, hydrophobic and hydrophilic, and present a heuristic quasi-physical algorithm. First, by elaborately simulating the movement of the smooth solids in the physical world, we find low-energy conformations for a given monomer chain. A subsequent off-trap strategy is then proposed to trigger a jump for a stuck situation in order to get out of the local minima. The algorithm has been tested in the three-dimensional AB model for all sequences with lengths of 13-55 monomers. In several cases, we renew the putative ground state energy values. The numerical results show that the proposed methods are very promising for finding the ground states of proteins.

Optimizing energy potentials for success in protein tertiary structure prediction

BACKGROUND

Success in solving the protein structure prediction problem relies on the choice of an accurate potential energy function. for a single protein sequence, it has been shown that the potential energy function can be optimized for predictive success by maximizing the energy gap between the correct structure and the ensemble of random structures relative to the distribution of the energies of these random structures (the Z-score). Different methods have been described for implementing this procedure for an ensemble of database proteins. Here, we demonstrate a new approach.

RESULTS

For a single protein sequence, the probability of success (i.e the probability that the folded state is the lowest energy state) is derived. We then maximize the average probability of success for a set of proteins to obtain the optimal potential energy function. This results in maximum attention being focused on the proteins whose structures are difficult but not impossible to predict.

CONCLUSIONS

Using a lattice model of proteins, we show that the optimal interaction potentials obtained by our method are both more accurate and more likely to produce successful predictions than those obtained by other averaging procedures.

Cooperativity in protein folding: from lattice models with sidechains to real proteins

BACKGROUND

Over the past few years novel folding mechanisms of globular proteins have been proposed using minimal lattice and off-lattice models. The factors determining the cooperativity of folding in these models and especially their explicit relation to experiments have not been fully established, however.

RESULTS

We consider equilibrium folding transitions in lattice models with and without sidechains. A dimensionless measure, omega c, is introduced to quantitatively assess the degree of cooperativity in lattice models and in real proteins. We show that larger values of omega c resembling the values seen in proteins are obtained in lattice models with sidechains. The enhanced cooperativity of such models results from possible denser packing of sidechains in the interior of the model polypeptide chain. We also establish that omega c correlates extremely well with sigma T = (T o - T f) /T o, where T o and T f are collapse and folding transition temperatures, respectively. These theoretical ideas are used to analyze folding transitions in two-state folders (RNase A, chymotrypsin inhibitor 2, fibronectin type III modules and tendamistat) and three-state folders (apomyoglobin and lysozyme). The values of omega c extracted from experiments show a correlation with sigma T (suitably generalized when folding is induced by denaturants or acid).

CONCLUSIONS

A quantitative description of the cooperative transition of real proteins can be made by lattice models with sidechains. The degree of cooperativity in minimal models and real proteins can be expressed in terms of the single parameter sigma, which can be estimated from experimental data.

The nucleation-collapse mechanism in protein folding: evidence for the non-uniqueness of the folding nucleus

BACKGROUND

Recent experimental and theoretical studies have shown that several small proteins reach the native state by a nucleation-collapse mechanism. Studies based on lattice models have been used to suggest that the critical nucleus is specific, leading to the notion that the transition state may be unique. On the other hand, results of studies using off-lattice models show that the critical nuclei should be viewed as fluctuating mobile structures, thus implying non-unique transition states.

RESULTS

The microscopic underpinnings of the nucleation-collapse mechanism in protein folding are probed using minimal off-lattice models and Langevin dynamics. We consider a 46-mer continuum model which has a native beta-barrel-like structure. The fast-folding trajectories reach the native state by a nucleation-collapse process. An algorithm based on the self-organized neural nets is used to identify the critical nuclei for a large number of rapidly folding trajectories. This method, which reduces the determination of the critical nucleus to one of 'pattern recognition', unambiguously shows that the folding nucleus is not unique. The only common characteristics of the mobile critical nuclei are that they are small (containing on average 15-22 residues) and are largely composed of residues near the loop regions of the molecule. The structures of the transition states, corresponding to the critical nuclei, show the existence of spatially localized ordered regions that are largely made up of residues that are close to each other. These structures are stabilized by a few long-range contacts. The structures in the ensemble of transition states exhibit a rather diverse degree of similarity to the native conformation.

CONCLUSIONS

The multiplicity of delocalized nucleation regions can explain the two-state folding by a nucleation-collapse mechanism for small single-domain proteins (such as chymotrypsin inhibitor 2) and their mutants. Because there are many distinct critical nuclei, we predict that the folding kinetics of fast-folding proteins will not be drastically changed even if some of the residues in a 'typical' nucleus are altered.

Protein folding in the landscape perspective: chevron plots and non-Arrhenius kinetics

We use two simple models and the energy landscape perspective to study protein folding kinetics. A major challenge has been to use the landscape perspective to interpret experimental data, which requires ensemble averaging over the microscopic trajectories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experiments show that two-state relaxation times have "chevron plot" dependences on denaturant and non-Arrhenius dependences on temperature. We find that HP and HP+ models can account for these behaviors. The HP model often gives bumpy landscapes with many kinetic traps and multiple-exponential behavior, whereas the HP+ model gives more smooth funnels and two-state behavior. Multiple-exponential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often involves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process.

Evolution of model proteins on a foldability landscape

We model the evolution of simple lattice proteins as a random walk in a fitness landscape, where the fitness represents the ability of the protein to fold. At higher selective pressure, the evolutionary trajectories are confined to neutral networks where the native structure is conserved and the dynamics are non self-averaging and nonexponential. The optimizability of the corresponding native structure has a strong effect on the size of these neutral networks and thus on the nature of the evolutionary process.

Thermodynamics of protein folding: a statistical mechanical study of a small all-beta protein

The thermodynamic properties of a 46-mer beta-barrel protein model are investigated using Langevin dynamics and the histogram analysis method. By obtaining the density of states distribution and using the methods of statistical mechanics, we are able to identify the thermodynamic transitions for this model protein and characterize the nature of these transitions. Consistent with an earlier study of this model, we find that the transition from a random coil state to a manifold of collapsed but nonnative states is a continuous transition, and the transition from the manifold of collapsed states to the native state is first order-like. However, our calculations indicate that the folding transition is only weakly first order. Most importantly, we are able to characterize the free energy surface of the protein model, as well as the processes of compaction and native structure formation, from a statistical point of view. We also examined the thermodynamic transition state. By combining the earlier kinetic analysis for the same protein model, we provide a more complete description of this model protein and propose possible further modifications of the model to improve its stability and foldability.

Unraveling principles of lead discovery: from unfrustrated energy landscapes to novel molecular anchors

The search for novel leads is a critical step in the drug discovery process. Computational approaches to identify new lead molecules have focused on discovering complete ligands by evaluating the binding affinity of a large number of candidates, a task of considerable complexity. A new computational method is introduced in this work based on the premise that the primary molecular recognition event in the protein binding site may be accomplished by small core fragments that serve as molecular anchors, providing a structurally stable platform that can be subsequently tailored into complete ligands. To fulfill its role, we show that an effective molecular anchor must meet both the thermodynamic requirement of relative energetic stability of a single binding mode and its consistent kinetic accessibility, which may be measured by the structural consensus of multiple docking simulations. From a large number of candidates, this technique is able to identify known core fragments responsible for primary recognition by the FK506 binding protein (FKBP-12), along with a diverse repertoire of novel molecular cores. By contrast, absolute energetic criteria for selecting molecular anchors are found to be promiscuous. A relationship between a minimum frustration principle of binding energy landscapes and receptor-specific molecular anchors in their role as "recognition nuclei" is established, thereby unraveling a mechanism of lead discovery and providing a practical route to receptor-biased computational combinatorial chemistry.

Why are some proteins structures so common?

Many biological proteins are observed to fold into one of a limited number of structural motifs. By considering the requirements imposed on proteins by their need to fold rapidly, and the ease with which such requirements can be fulfilled as a function of the native structure, we can explain why certain structures are repeatedly observed among proteins with negligible sequence similarity. This work has implications for the understanding of protein sequence structure relationships as well as protein evolution.

Elucidating the folding problem of helical peptides using empirical parameters

Using an empirical analysis of experimental data we have estimated a set of energy contributions which accounts for the stability of isolated alpha-helices. With this database and an algorithm based on statistical mechanics, we describe the average helical behaviour in solution of 323 peptides and the helicity per residue of those peptides analyzed by nuclear magnetic resonance. Moreover the algorithm successfully detects the alpha-helical tendency, in solution, of a peptide corresponding to a beta-strand of ubiquitin.