Peter G. Wolynes

Protein folding funnels: The nature of the transition state ensemble

BACKGROUND Energy landscape theory predicts that the folding funnel for a small fast-folding alpha-helical protein will have a transition state half-way to the native state. Estimates of the position of the transition state along an appropriate reaction coordinate can be obtained from linear free energy relationships observed for folding and unfolding rate constants as a function of denaturant concentration. The experimental results of Huang and Oas for lambda repressor, Fersht and collaborators for C12, and Gray and collaborators for cytochrome c indicate a free energy barrier midway between the folded and unfolded regions. This barrier arises from an entropic bottleneck for the folding process. RESULTS In keeping with the experimental results, lattice simulations based on the folding funnel description show that the transition state is not just a single conformation, but rather an ensemble of a relatively large number of configurations that can be described by specific values of one or a few order parameters (e.g. the fraction of native contacts). Analysis of this transition state or bottleneck region from our lattice simulations and from atomistic models for small alpha-helical proteins by Boczko and Brooks indicates a broad distribution for native contact participation in the transition state ensemble centered around 50%. Importantly, however, the lattice-simulated transition state ensemble does include some particularly hot contacts, as seen in the experiments, which have been termed by others a folding nucleus. CONCLUSIONS Linear free energy relations provide a crude spectroscopy of the transition state, allowing us to infer the values of a reaction coordinate based on the fraction of native contacts. This bottleneck may be thought of as a collection of delocalized nuclei where different native contacts will have different degrees of participation. The agreement between the experimental results and the theoretical predictions provides strong support for the landscape analysis.

The experimental survey of protein-folding energy landscapes.

We review what has been learned about the protein-folding problem from experimental kinetic studies. These studies reveal patterns of both great richness and surprising simplicity. The patterns can be interpreted in terms of proteins possessing an energy landscape which is largely, but not completely, funnel-like. Issues such as speed limitations of folding, the robustness of folding, the origin of barriers and cooperativity and the ensemble nature of transition states, intermediate and traps are assessed using the results from several experimental groups highlighting energy-landscape ideas as an interpretive framework.

Stochastic gene expression as a many-body problem

Gene expression has a stochastic component because of the single-molecule nature of the gene and the small number of copies of individual DNA-binding proteins in the cell. We show how the statistics of such systems can be mapped onto quantum many-body problems. The dynamics of a single gene switch resembles the spin-boson model of a two-site polaron or an electron transfer reaction. Networks of switches can be approximately described as quantum spin systems by using an appropriate variational principle. In this way, the concept of frustration for magnetic systems can be taken over into gene networks. The landscape of stable attractors depends on the degree and style of frustration, much as for neural networks. We show the number of attractors, which may represent cell types, is much smaller for appropriately designed weakly frustrated stochastic networks than for randomly connected networks.

Localizing frustration in native proteins and protein assemblies

We propose a method of quantifying the degree of frustration manifested by spatially local interactions in protein biomolecules. This method of localization smoothly generalizes the global criterion for an energy landscape to be funneled to the native state, which is in keeping with the principle of minimal frustration. A survey of the structural database shows that natural proteins are multiply connected by a web of local interactions that are individually minimally frustrated. In contrast, highly frustrated interactions are found clustered on the surface, often near binding sites. These binding sites become less frustrated upon complex formation.

The shapes of cooperatively rearranging regions in glass-forming liquids

The cooperative rearrangement of groups of many molecules has long been thought to underlie the dramatic slowing of liquid dynamics on cooling towards the glassy state. For instance, there exists experimental evidence for cooperatively rearranging regions (CRRs) on the nanometre length scale near the glass transition. The random first-order transition (RFOT) theory of glasses predicts that, near the glass-transition temperature, these regions are compact, but computer simulations and experiments on colloids suggest CRRs are string-like. Here, we present a microscopic theory within the framework of RFOT, which unites the two situations. We show that the shapes of CRRs in glassy liquids should change from being compact at low temperatures to fractal or ‘stringy’ as the dynamical crossover temperature from activated to collisional transport is approached from below. This theory predicts a correlation of the ratio of the dynamical crossover temperature to the laboratory glass-transition temperature, and the heat-capacity discontinuity at the glass transition. The predicted correlation quantitatively agrees with experimental results for 21 materials.

Energy landscapes and solved protein-folding problems

Energy–landscape theory has led to much progress in protein folding kinetics, protein structure prediction and protein design. Funnel landscapes describe protein folding and binding and explain how protein topology determines kinetics. Landscape–optimized energy functions based on bioinformatic input have been used to correctly predict low–resolution protein structures and also to design novel proteins automatically.

Microscopic theory of protein folding rates. II. Local reaction coordinates and chain dynamics

The motions involved in barrier crossing for protein folding are investigated in terms of the chain dynamics of the polymer backbone, completing the microscopic description of protein folding presented in the preceding paper. Local reaction coordinates are identified as collective growth modes of the unstable fluctuations about the saddle points in the free energy surface. The description of the chain dynamics incorporates internal friction (independent of the solvent viscosity) arising from the elementary isomerization of the backbone dihedral angles. We find that the folding rate depends linearly on the solvent friction for high viscosity, but saturates at low viscosity because of internal friction. For λ-repressor, the calculated folding rate prefactor, along with the free energy barrier from the variational theory, gives a folding rate that agrees well with the experimentally determined rate under highly stabilizing conditions, but the theory predicts too large a folding rate at the transition midpoin...

Chemical physics of protein folding

In 2008, to the consternation of some, one of the editors of this special issue on the “Chemical Physics of Protein Folding,” was quoted as saying, “What was called the protein folding problem 20 years ago is solved” (1). One purpose of this special issue is to drive home this point. The other, more important purpose is to illustrate how workers on the protein folding problem, by moving beyond their early obsession with seeming paradoxes (2), are developing a quantitative understanding of how the simpler biological structures assemble both in vitro and in vivo. The emerging quantitative understanding reveals simultaneously the richness of folding phenomena and the elegant simplicity of the underlying principles of spontaneous biomolecular assembly. The appreciation of these contrasting aspects of the folding problem has come about through the cooperation of theorists and experimentalists, a theme common to all the contributions to this special issue. Although the basic ideas about the folding energy landscape have turned out to be quite simple, entering even into some undergraduate textbooks (3), exploring their consequences in real systems has required painstaking intellectual analysis, as well as detailed computer simulations and experiments that still stretch the bounds of what is feasible. The backgrounds of the contributors to this issue reflect the breadth of the folding field and range from computer science and theoretical physics to molecular biology and organic chemistry. A great deal of the progress in the field can thus be traced to a fairly successful effort to develop a common language and conceptual framework for describing folding.

Recent successes of the energy landscape theory of protein folding and function.

Protein folding and binding can be understood using energy landscape theory. When seeming deviations from the predictions of the funnel hypothesis are found, landscape theory helps us locate the cause. Sometimes the deviation reflects symmetry effects, allowing extra degeneracies to occur. Such effects seem to explain some kinetic anomalies in helical bundles. When binding processes were found to use apparently non-funneled landscapes this was traced to an inadequate understanding of biomolecular forces. The discrepancy allowed the discovery of new water-mediated forces - some of which act between hydrophilic residues. Introducing such forces into the algorithms greatly improves the quality of structure predictions.

The physics and bioinformatics of binding and folding: An energy landscape perspective

It has been recognized in the last few years that unstructured proteins play an important role in biological organisms, often participating in signal transduction, transcriptional regulation, and a variety of other regulatory activities. Various hypotheses have been put forward for the ubiquity of the unfolded state; rapid turnover, faster or more specific binding kinetics, multifunctionality may all possibly explain apparent ubiquitousness of unfolded proteins in eukaryotic cells. In this paper we extend the energy landscape theory of protein folding to construct an analytical model of how binding and folding are coupled thermodynamically when the energy landscape is partially rugged. To deduce the parameters that enter the theory, which is based on Generalized Random Energy Model, we have analyzed in a bioinformatic sense a large structural database of more than 500 protein complexes. We find that Miyazawa–Jernigan contact potential shows similar energy gaps for folding for both hydrophobic and hydrophilic proteins, but that for binding contacts hydrophobic interfaces turn out to be funneled while hydrophilic ones are antifunneled. This suggests evolution has found a mechanism for avoiding frustration between folding and binding by making use of indirect water‐mediated interactions. By juxtaposing the monomeric protein folding free energy profile in the protein complex database with another database consisting of only well‐folded monomers, we estimate that at least 15% of monomers in the former database are unfolded in the absence of partner protein interface interactions. When employing the parameters characteristic of these unfolded monomers to construct binding/folding phase diagrams, we find that these monomers would indeed fold if sufficiently stabilizing binding contacts, consistent with that fold, are formed.