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

The Wako-Saitô-Muñoz-Eaton Model for Predicting Protein Folding and Dynamics

Despite the recent advances in the prediction of protein structures by deep neutral networks, the elucidation of protein-folding mechanisms remains challenging. A promising theory for describing protein folding is a coarse-grained statistical mechanical model called the Wako-Saitô-Muñoz-Eaton (WSME) model. The model can calculate the free-energy landscapes of proteins based on a three-dimensional structure with low computational complexity, thereby providing a comprehensive understanding of the folding pathways and the structure and stability of the intermediates and transition states involved in the folding reaction. In this review, we summarize previous and recent studies on protein folding and dynamics performed using the WSME model and discuss future challenges and prospects. The WSME model successfully predicted the folding mechanisms of small single-domain proteins and the effects of amino-acid substitutions on protein stability and folding in a manner that was consistent with experimental results. Furthermore, extended versions of the WSME model were applied to predict the folding mechanisms of multi-domain proteins and the conformational changes associated with protein function. Thus, the WSME model may contribute significantly to solving the protein-folding problem and is expected to be useful for predicting protein folding, stability, and dynamics in basic research and in industrial and medical applications.

Network measures for protein folding state discrimination

Proteins fold using a two-state or multi-state kinetic mechanisms, but up to now there is not a first-principle model to explain this different behavior. We exploit the network properties of protein structures by introducing novel observables to address the problem of classifying the different types of folding kinetics. These observables display a plain physical meaning, in terms of vibrational modes, possible configurations compatible with the native protein structure, and folding cooperativity. The relevance of these observables is supported by a classification performance up to 90%, even with simple classifiers such as discriminant analysis.

Energy landscape and dynamics of proteins: an exact analysis of a simplified lattice model

We present the results of exact numerical studies of the energy landscape and the dynamics of a 12-monomer chain with contact interactions encoding the ground state on a square lattice. In spite of its simplicity, the model is shown to exhibit behavior at odds with the standard picture of proteins.

Nearly symmetrical proteins: folding pathways and transition states

The folding pathways of the B domain of protein A have been the subject of many experimental and computational studies. Based on a statistical mechanical model, it has been suggested that the native state symmetry leads to multiple pathways, highly dependent on temperature and denaturant concentration. Experiments, however, have not confirmed this scenario. By considering four nearly symmetrical proteins, one of them being the above molecule, here we show that, if contact energies are properly taken into account, a different picture emerges from kinetic simulations of the above-mentioned model. This is characterized by a dominant folding pathway, which is consistent with the most recent experimental results. Given the simplicity of the model, we also report on a direct sampling of the transition state.

Analysis of oligomeric proteins during unfolding by pH and temperature

During thermal transition and variation of pH, structural properties of 35 proteins and their complexes (bound with substrate and co-factor) were analyzed in detail. During pH alteration, these proteins were shown to have substantial differences in conformations. pH conformers were analyzed in detail. Free energy and other energy parameters were also estimated for these proteins at various pH and temperatures. Detailed structural analysis and binding interfaces of various substrates, inhibitors and cofactor of these proteins were also investigated using docking and molecular dynamic simulation.

Minimal folding pathways for coarse-grained biopolymer fragments

The minimal folding pathway or trajectory for a biopolymer can be defined as the transformation that minimizes the total distance traveled between a folded and an unfolded structure. This involves generalizing the usual Euclidean distance from points to one-dimensional objects such as a polymer. We apply this distance here to find minimal folding pathways for several candidate protein fragments, including the helix, the beta-hairpin, and a nonplanar structure where chain noncrossing is important. Comparing the distances traveled with root mean-squared distance and mean root-squared distance, we show that chain noncrossing can have large effects on the kinetic proximity of apparently similar conformations. Structures that are aligned to the beta-hairpin by minimizing mean root-squared distance, a quantity that closely approximates the true distance for long chains, show globally different orientation than structures aligned by minimizing root mean-squared distance.

Mechanical unfolding and refolding pathways of ubiquitin

Mechanical unfolding and refolding of ubiquitin are studied by Monte Carlo simulations of a Gō model with binary variables. The exponential dependence of the time constants on the force is verified, and folding and unfolding lengths are computed, with good agreement with experimental results. Furthermore, the model exhibits intermediate kinetic states, as observed in experiments. Unfolding and refolding pathways and intermediate states, obtained by tracing single secondary structure elements, are consistent with simulations of previous all-atom models and with the experimentally observed step sizes.

A simple and exact Laplacian clustering of complex networking phenomena: application to gene expression profiles

Unraveling of the unified networking characteristics of complex networking phenomena is of great interest yet a formidable task. There is currently no simple strategy with a rigorous framework. Using an analogy to the exact algebraic property for a transition matrix of a master equation in statistical physics, we propose a method based on a Laplacian matrix for the discovery and prediction of new classes in the unsupervised complex networking phenomena where the class of each sample is completely unknown. Using this proposed Laplacian approach, we can simultaneously discover different classes and determine the identity of each class. Through an illustrative test of the Laplacian approach applied to real datasets of gene expression profiles, leukemia data [Golub TR, et al. (1999) Science 286:531-537], and lymphoma data [Alizadeh AA, et al. (2000) Nature 403:503-511], we demonstrate that this approach is accurate and robust with a mathematical and physical realization. It offers a general framework for characterizing any kind of complex networking phenomenon in broad areas irrespective of whether they are supervised or unsupervised.

Cooperative folding kinetics of BBL protein and peripheral subunit-binding domain homologues

Recent experiments claiming that Naf-BBL protein follows a global downhill folding raised an important controversy as to the folding mechanism of fast-folding proteins. Under the global downhill folding scenario, not only do proteins undergo a gradual folding, but folding events along the continuous folding pathway also could be mapped out from the equilibrium denaturation experiment. Based on the exact calculation using a free energy landscape, relaxation eigenmodes from a master equation, and Monte Carlo simulation of an extended Muñoz-Eaton model that incorporates multiscale-heterogeneous pairwise interactions between amino acids, here we show that the very nature of a two-state cooperative transition such as a bimodal distribution from an exact free energy landscape and biphasic relaxation kinetics manifest in the thermodynamics and folding-unfolding kinetics of BBL and peripheral subunit-binding domain homologues. Our results provide an unequivocal resolution to the fundamental controversy related to the global downhill folding scheme, whose applicability to other proteins should be critically reexamined.

Parallel folding pathways in the SH3 domain protein

The transition-state ensemble (TSE) is the set of protein conformations with an equal probability to fold or unfold. Its characterization is crucial for an understanding of the folding process. We determined the TSE of the src-SH3 domain protein by using extensive molecular dynamics simulations of the Go model and computing the folding probability of a generated set of TSE candidate conformations. We found that the TSE possesses a well-defined hydrophobic core with variable enveloping structures resulting from the superposition of three parallel folding pathways. The most preferred pathway agrees with the experimentally determined TSE, while the two least preferred pathways differ significantly. The knowledge of the different pathways allows us to design the interactions between amino acids that guide the protein to fold through the least preferred pathway. This particular design is akin to a circular permutation of the protein. The finding motivates the hypothesis that the different experimentally observed TSEs in homologous proteins and circular permutants may represent potentially available pathways to the wild-type protein.

Rate determining factors in protein model structures

Previous research has shown a strong correlation of protein folding rates to the native state geometry, yet a complete explanation for this dependence is still lacking. Here we study the rate-geometry relationship with a simple statistical physics model, and focus on two classes of model geometries, representing ideal parallel and antiparallel structures. We find that the logarithm of the rate shows an almost perfect linear correlation with the "absolute contact order", but the slope depends on the particular class considered. We discuss these findings in the light of experimental results.

Downhill versus two-state protein folding in a statistical mechanical model

The authors address the problem of downhill protein folding in the framework of a simple statistical mechanical model, which allows an exact solution for the equilibrium and a semianalytical treatment of the kinetics. Focusing on protein 1BBL, a candidate for downhill folding behavior, and comparing it to the WW domain of protein PIN1, a two-state folder of comparable size, the authors show that there are qualitative differences in both the equilibrium and kinetic properties of the two molecules. However, the barrierless scenario which would be expected if 1BBL were a true downhill folder is observed only at low enough temperature.

Ising-like model for protein mechanical unfolding

The mechanical unfolding of proteins is studied by extending the Wako-Saitô-Muñoz-Eaton model. This model is generalized by including an external force, and its thermodynamics turns out to be exactly solvable. We consider two molecules, the 27th immunoglobulin domain of titin and protein PIN1. We determine equilibrium force-extension curves for the titin and study the mechanical unfolding of this molecule, finding good agreement with experiments. By using an extended form of the Jarzynski equality, we compute the free energy landscape of the PIN1 as a function of the molecule length.

Protein folding dynamics in lattice model with physical movement

We study folding dynamics of protein-like sequences on square lattice by constructing a physically realizable move set that exhausts all possible conformational changes for a structure. By solving the master equation of 16-mer characterized by a 802,075times802,075 transition matrix, we monitor the time-dependent probabilities of occupancy of all conformations over 9-orders of time scale from the first kinetic move until reaching Boltzmann equilibrium. We find that folding rates of protein-like sequences adopting the same ground state conformation differ as much as 200 times, and parameters of the native structures, designability, and thermodynamic properties are weak predictors of the folding rates in our model systems. Instead, we show that properties of the kinetic energy landscape defined by the connection graph of physical moves can provide excellent account for observed folding rates. Without the approximation of macrostates, we show how transiently accumulating intermediate states can be identified by basin analysis of the kinetic energy landscape.

Transition-states in protein folding kinetics: the structural interpretation of Phi values

Phi values are experimental measures of the effects of mutations on the folding kinetics of a protein. A central question is what structural information Phi values give about the transition-state of folding. Traditionally, a Phi value is interpreted as representing the "nativeness" of a mutated residue in the transition-state. However, this interpretation is often problematic. We present here a better structural interpretation of Phi values for mutations within a given helix. Our interpretation is based on a simple physical model that distinguishes between secondary and tertiary free energy contributions of helical residues. From a linear fit of the model to experimental data, we obtain two structural parameters: the extent of helix formation in the transition-state, and the nativeness of tertiary interactions in the transition-state. We apply the model to all proteins with well-characterized helices for which more than 10 Phi values are available: protein A, CI2, and protein L. The model is simple to apply to experimental data, captures nonclassical Phi values <0 or >1 in these helices, and explains how different mutations at a given site can lead to different Phi values.

Protein folding dynamics via quantification of kinematic energy landscape

We study folding dynamics of proteinlike sequences on a square lattice using a physical move set that exhausts all possible conformational changes. By analytically solving the master equation, we follow the time-dependent probabilities of occupancy of all 802 075 conformations of 16 mers over 7 orders of time span. We find that (i) folding rates of these proteinlike sequences of the same length can differ by 4 orders of magnitude, (ii) folding rates of sequences of the same conformation can differ by a factor of 190, and (iii) parameters of the native structures, designability, and thermodynamic properties are weak predictors of the folding rates; rather, a basin analysis of the kinematic energy landscape defined by the moves can provide an excellent account of the observed folding rates.

Thermal unfolding of proteins

Thermal unfolding of proteins is compared to folding and mechanical stretching in a simple topology-based dynamical model. We define the unfolding time and demonstrate its low-temperature divergence. Below a characteristic temperature, contacts break at separate time scales and unfolding proceeds approximately in a way reverse to folding. Features in these scenarios agree with experiments and atomic simulations on titin.