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

Rare-event trajectory ensemble analysis reveals metastable dynamical phases in lattice proteins

We explore the dynamical large deviations of a lattice heteropolymer model of a protein by means of path sampling of trajectories. We uncover the existence of nonequilibrium dynamical phase transitions in ensembles of trajectories between active and inactive dynamical phases, whose nature depends on the properties of the interaction potential. We consider three potentials: two heterogeneous interaction potentials and a homogeneous Gō potential. When preserving the full heterogeneity of interactions due to a given amino acid sequence, either in a fully interacting model or in a native contacts interacting model (heterogeneous Gō model), the observed dynamic transitions occur between equilibrium highly native states and highly native but kinetically trapped states. A native activity is defined that allows us to distinguish these dynamic phases. In contrast, for the homogeneous Gō model, where all native interaction energies are uniform and the amino acid sequence plays no role, the dynamical transition is a direct consequence of the static bistability between the unfolded and the native state. In the two heterogeneous interaction models the native-active and native-inactive states, despite their thermodynamic similarity, have widely varying dynamical properties, and the transition between them occurs even in lattice proteins whose sequences are designed to make them optimal folders.

Balancing bond, nonbond, and gō-like terms in coarse grain simulations of conformational dynamics

Characterization of the protein conformational landscape remains a challenging problem, whether it concerns elucidating folding mechanisms, predicting native structures or modeling functional transitions. Coarse-grained molecular dynamics simulation methods enable exhaustive sampling of the energetic landscape at resolutions of biological interest. The general utility of structure-based models is reviewed along with their differing levels of approximation. Simple Gō models incorporate attractive native interactions and repulsive nonnative contacts, resulting in an ideal smooth landscape. Non-Gō coarse-grained models reduce the parameter set as needed but do not include bias to any desired native structure. While non-Gō models have achieved limited success in protein coarse-graining, they can be combined with native structured-based potentials to create a balanced and powerful force field. Recent applications of such Gō-like models have yielded insight into complex folding mechanisms and conformational transitions in large macromolecules. The accuracy and usefulness of reduced representations are also revealed to be a function of the mathematical treatment of the intrinsic bonded topology.

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

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

Roles of non-native hydrogen-bonding interaction in helix-coil transition of a single polypeptide as revealed by comparison between Gō-like and non-Gō models

To explore the role of non-native interactions in the helix-coil transition, a detailed comparison between a Gō-like model and a non-Gō model has been performed via lattice Monte Carlo simulations. Only native hydrogen bonding interactions occur in the Gō-like model, and the non-native ones with sequence interval more than 4 is also included into the non-Gō model. Some significant differences between the results from those two models have been found. The non-native hydrogen bonds were found most populated at temperature around the helix-coil transition. The rearrangement of non-native hydrogen bonds into native ones in the formation of alpha-helix leads to the increase of susceptibility of chain conformation, and even two peaks of susceptibility of radius of gyration versus temperature exist in the case of non-Gō model for a non-short peptide, while just a single peak exists in the case of Gō model for a single polypeptide chain with various chain lengths. The non-native hydrogen bonds have complicated the temperature-dependence of Zimm-Bragg nucleation constant. The increase of relative probability of non-native hydrogen bonding for long polypeptide chains leads to non-monotonous chain length effect on the transition temperature.

The origin of nonmonotonic complex behavior and the effects of nonnative interactions on the diffusive properties of protein folding

We present a method for calculating the configurational-dependent diffusion coefficient of a globular protein as a function of the global folding process. Using a coarse-grained structure-based model, we determined the diffusion coefficient, in reaction coordinate space, as a function of the fraction of native contacts formed Q for the cold shock protein (TmCSP). We find nonmonotonic behavior for the diffusion coefficient, with high values for the folded and unfolded ensembles and a lower range of values in the transition state ensemble. We also characterized the folding landscape associated with an energetically frustrated variant of the model. We find that a low-level of frustration can actually stabilize the native ensemble and increase the associated diffusion coefficient. These findings can be understood from a mechanistic standpoint, in that the transition state ensemble has a more homogeneous structural content when frustration is present. Additionally, these findings are consistent with earlier calculations based on lattice models of protein folding and more recent single-molecule fluorescence measurements.

Native atomic burials, supplemented by physically motivated hydrogen bond constraints, contain sufficient information to determine the tertiary structure of small globular proteins

We investigate the possibility that atomic burials, as measured by their distances from the structural geometrical center, contain sufficient information to determine the tertiary structure of globular proteins. We report Monte Carlo simulated annealing results of all-atom hard-sphere models in continuous space for four small proteins: the all-beta WW-domain 1E0L, the alpha/beta protein-G 1IGD, the all-alpha engrailed homeo-domain 1ENH, and the alpha + beta engineered monomeric form of the Cro protein 1ORC. We used as energy function the sum over all atoms, labeled by i, of |R(i) - R(i) (*)|, where R(i) is the atomic distance from the center of coordinates, or central distance, and R(i) (*) is the "ideal" central distance obtained from the native structure. Hydrogen bonds were taken into consideration by the assignment of two ideal distances for backbone atoms forming hydrogen bonds in the native structure depending on the formation of a geometrically defined bond, independently of bond partner. Lowest energy final conformations turned out to be very similar to the native structure for the four proteins under investigation and a strong correlation was observed between energy and distance root mean square deviation (DRMS) from the native in the case of all-beta 1E0L and alpha/beta 1IGD. For all alpha 1ENH and alpha + beta 1ORC the overall correlation between energy and DRMS among final conformations was not as high because some trajectories resulted in high DRMS but low energy final conformations in which alpha-helices adopted a non-native mutual orientation. Comparison between central distances and actual accessible surface areas corroborated the implicit assumption of correlation between these two quantities. The Z-score obtained with this native-centric potential in the discrimination of native 1ORC from a set of random compact structures confirmed that it contains a much smaller amount of native information when compared to a traditional contact Go potential but indicated that simple sequence-dependent burial potentials still need some improvement in order to attain a similar discriminability. Taken together, our results suggest that central distances, in conjunction to physically motivated hydrogen bond constraints, contain sufficient information to determine the native conformation of these small proteins and that a solution to the folding problem for globular proteins could arise from sufficiently accurate burial predictions from sequence followed by minimization of a burial-dependent energy function.

Alternative hydrogen bond implementations produce opposite effects on collapse cooperativity of lattice homopolypeptide models

We use complete enumeration of self-avoiding chains of up to N=26 monomers in two-dimensional lattices to investigate the effect of alternative implementations of backbone hydrogen bonds on the cooperativity of homopolypeptide collapse. Following a recent study on protein folding models, we use the square lattice with z=3 local conformations per monomer and lattice extensions containing diagonal steps which result in z=5 or z=7 and assume that only a subset of zh<z local conformations is compatible with hydrogen bond formation. As previously observed in heteropolymeric folding, a significant increase in cooperativity, as measured by kappa2 values, results from the coupling between hydrogen bonds and hydrophobic interactions, in such a way that hydrophobic contacts are favorable only when contacting monomers are involved in hydrogen bond formation. For some z/zh combinations the energy distribution is bimodal at the collapse transition temperature. The situation can be regarded as if all hydrophobic contacts actually decrease the energy by the same amount, 2h , with the addition of an energetic increase, epsilon2=h, as a penalty for each contacting monomer not satisfying the hydrogen bond condition. Cooperativity is little affected and might even decrease, however, when hydrogen bonds produce a decrease in energy by the same amount, epsilon1=h, for each bonding monomer. For the more general situation when the hydrogen bond effect is not equal, in modulus, to the hydrophobic interaction, i.e., epsilon2 not equalh or epsilon1 not equal h, we observe a pronounced increase in kappa2 for small epsilon2, with a maximum around epsilon2/h approximately 1.5, followed by a gradual decrease to a limiting value at large epsilon2. The opposite behavior is observed when epsilon1 is varied. The observed qualitative difference is shown to arise from opposite effects on the convexity of the total density of states of the system when subdensities corresponding to different numbers of hydrogen bonds are differently favored as opposed to the case when subdensities corresponding to different numbers of contacting monomers not forming hydrogen bonds are differently unfavored. Potential implications for the cooperativity of protein folding and protein unspecific collapse are discussed.

Description of atomic burials in compact globular proteins by Fermi-Dirac probability distributions

We perform a statistical analysis of atomic distributions as a function of the distance R from the molecular geometrical center in a nonredundant set of compact globular proteins. The number of atoms increases quadratically for small R, indicating a constant average density inside the core, reaches a maximum at a size-dependent distance R(max), and falls rapidly for larger R. The empirical curves turn out to be consistent with the volume increase of spherical concentric solid shells and a Fermi-Dirac distribution in which the distance R plays the role of an effective atomic energy epsilon(R) = R. The effective chemical potential mu governing the distribution increases with the number of residues, reflecting the size of the protein globule, while the temperature parameter beta decreases. Interestingly, betamu is not as strongly dependent on protein size and appears to be tuned to maintain approximately half of the atoms in the high density interior and the other half in the exterior region of rapidly decreasing density. A normalized size-independent distribution was obtained for the atomic probability as a function of the reduced distance, r = R/R(g), where R(g) is the radius of gyration. The global normalized Fermi distribution, F(r), can be reasonably decomposed in Fermi-like subdistributions for different atomic types tau, F(tau)(r), with Sigma(tau)F(tau)(r) = F(r), which depend on two additional parameters mu(tau) and h(tau). The chemical potential mu(tau) affects a scaling prefactor and depends on the overall frequency of the corresponding atomic type, while the maximum position of the subdistribution is determined by h(tau), which appears in a type-dependent atomic effective energy, epsilon(tau)(r) = h(tau)r, and is strongly correlated to available hydrophobicity scales. Better adjustments are obtained when the effective energy is not assumed to be necessarily linear, or epsilon(tau)*(r) = h(tau)*r(alpha,), in which case a correlation with hydrophobicity scales is found for the product alpha(tau)h(tau)*. These results indicate that compact globular proteins are consistent with a thermodynamic system governed by hydrophobic-like energy functions, with reduced distances from the geometrical center, reflecting atomic burials, and provide a conceptual framework for the eventual prediction from sequence of a few parameters from which whole atomic probability distributions and potentials of mean force can be reconstructed.