Hue Sun Chan

From Levinthal to pathways to funnels

While the classical view of protein folding kinetics relies on phenomenological models, and regards folding intermediates in a structural way, the new view emphasizes the ensemble nature of protein conformations. Although folding has sometimes been regarded as a linear sequence of events, the new view sees folding as parallel microscopic multi-pathway diffusion-like processes. While the classical view invoked pathways to solve the problem of searching for the needle in the haystack, the pathway idea was then seen as conflicting with Anfinsens experiments showing that folding is pathway-independent (Levinthals paradox). In contrast, the new view sees no inherent paradox because it eliminates the pathway idea: folding can funnel to a single stable state by multiple routes in conformational space. The general energy landscape picture provides a conceptual framework for understanding both two-state and multi-state folding kinetics. Better tests of these ideas will come when new experiments become available for measuring not just averages of structural observables, but also correlations among their fluctuations. At that point we hope to learn much more about the real shapes of protein folding landscapes.

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‐exponental 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. Proteins 30:2–33, 1998.

The Protein Folding Problem

Thousands of different types of proteins occur in biological organisms. They are responsible for catalyzing and regulating biochemical reactions, transporting molecules, the chemistry of vision and of the photosynthetic conversion of light to growth, and they form the basis of structures such as skin, hair and tendon. Protein molecules have remarkable structures. A protein is a linear chain of a particular sequence of monomer units. A major class of proteins, globular proteins, ball up into compact configurations that can have much internal symmetry. (See figure 1.) Each globular protein has a unique folded state, determined by its sequence of monomers.

The effects of internal constraints on the configurations of chain molecules

We explore the three‐dimensional configurations of chain molecules containing one or more self‐contacts (constraints). We focus predominantly on the 1‐, 2‐, and 3‐constraint ensembles. We take into account excluded volume by exhaustive computer enumeration of the conformational spaces of short chains on three‐dimensional simple cubic lattices, and through use of the path integral approach of Edwards and Freed. We develop topological correlation functions to describe how the cyclization probability of one loop affects cyclization of another. There are two rather striking findings. (i) Considerable amounts of internal architecture (helices and antiparallel and parallel sheets) are predicted to arise in compact polymers due simply to steric restrictions. This appears to account for why there is so much internal organization in globular proteins. (ii) Several cyclization properties are remarkably ideal for chains which are relatively or highly compact in three dimensions. For example, in relatively compact molecules the correlation functions of loop pairs are well predicted by the random‐flight model of Jacobson and Stockmayer; the number of configurations of maximally compact chains is predicted relatively well by the Flory theory of excluded volume, which is found to be better than the Huggins theory in three dimensions; and the probability of cyclization within globular chains is well predicted by the Bragg–Williams approximation.We explore the three‐dimensional configurations of chain molecules containing one or more self‐contacts (constraints). We focus predominantly on the 1‐, 2‐, and 3‐constraint ensembles. We take into account excluded volume by exhaustive computer enumeration of the conformational spaces of short chains on three‐dimensional simple cubic lattices, and through use of the path integral approach of Edwards and Freed. We develop topological correlation functions to describe how the cyclization probability of one loop affects cyclization of another. There are two rather striking findings. (i) Considerable amounts of internal architecture (helices and antiparallel and parallel sheets) are predicted to arise in compact polymers due simply to steric restrictions. This appears to account for why there is so much internal organization in globular proteins. (ii) Several cyclization properties are remarkably ideal for chains which are relatively or highly compact in three dimensions. For example, in relatively compact mo...

Polyelectrostatic interactions of disordered ligands suggest a physical basis for ultrasensitivity

Regulation of biological processes often involves phosphorylation of intrinsically disordered protein regions, thereby modulating protein interactions. Initiation of DNA replication in yeast requires elimination of the cyclin-dependent kinase inhibitor Sic1 via the SCFCdc4 ubiquitin ligase. Intriguingly, the substrate adapter subunit Cdc4 binds to Sic1 only after phosphorylation of a minimum of any six of the nine cyclin-dependent kinase sites on Sic1. To investigate the physical basis of this ultrasensitive interaction, we consider a mean-field statistical mechanical model for the electrostatic interactions between a single receptor site and a conformationally disordered polyvalent ligand. The formulation treats phosphorylation sites as negative contributions to the total charge of the ligand and addresses its interplay with the strength of the favorable ligand–receptor contact. Our model predicts a threshold number of phosphorylation sites for receptor–ligand binding, suggesting that ultrasensitivity in the Sic1–Cdc4 system may be driven at least in part by cumulative electrostatic interactions. This hypothesis is supported by experimental affinities of Cdc4 for Sic1 fragments with different total charges. Thus, polyelectrostatic interactions may provide a simple yet powerful framework for understanding the modulation of protein interactions by multiple phosphorylation sites in disordered protein regions.

Intrachain loops in polymers: Effects of excluded volume

We develop a theory for the formation of loops and intrachain contacts in polymer molecules which are subject to excluded volume. We use two methods: (i) exhaustive simulations of chain conformations on two‐dimensional square lattices, and (ii) the Edwards path integral approach. The predictions are compared to those of the Jacobson–Stockmayer theory, which neglects excluded volume. Our results show that the cyclization probability in two dimensions depends on loop length to a power between −1.6 to −2.4, in contrast to the prediction of Jacobson–Stockmayer of a power of −1. In addition, the cyclization probability depends on the position in the chain, and end effects are significant. A principal result of the present work is the development of ‘‘topological’’ correlation functions among multiple loops in a chain. If two loops are far apart along the chain, they act independently, but as they approach each other, or if they are interlinked, then one can strongly hinder or enhance the likelihood of another....

Cooperativity principles in protein folding.

Knowledge of the physical driving forces in proteins is essential for understanding their structures and functions. As polymers, proteins have remarkable thermodynamic and kinetic properties. A well-known observation is that the folding and unfolding of many small single-domain proteins, of which chymotrypsin inhibitor 2 is a prime example, appear to involve only two main states—N (native) and D (denatured). These proteins’ folding/unfolding transitions are often referred to as ‘‘cooperative’’ because of their phenomenological similarity to ‘‘all-or-none’’ processes. Traditionally, only N, D, and a small number of postulated intermediate states were invoked to account for experimental protein folding data. Under such an interpretative framework, two-state folding is described by the reaction N Ð D, and different properties are ascribed to N and D to account for different proteins. Although useful, this approach does not address the microscopic origins of experimentally observed two-state–like behavior. Traditional analyses simply assume that there are a small number of conformational states. But proteins are chain molecules. Physically, it is obvious that a polymer chain can adopt many conformations, ranging from the most open to maximally compact, and all intermediate compactness in between. Thus, whether and how the multitude of conformations available to a protein may be grouped into two or more ‘‘states’’—as traditionally assumed— should be ascertained through a fundamental understanding of the effective intrachain interactions involved. In the protein literature, however, folding energetics are often discussed in terms of the sum of contactlike energies of a fully folded native structure versus that of a random-coil–like state or a certain other prespecified unfolded conformational ensemble. Such analyses have yielded important insight. But they obscure the remarkable nature of protein cooperativities. This is because cooperativity has already been presumed in these discourses by their preclusion of many a priori possible conformations—notably compact nonnative conformations—from the energetic equation. To gain a consistent understanding

Sequence space soup of proteins and copolymers

To study the protein folding problem, we use exhaustive computer enumeration to explore ‘‘sequence space soup,’’ an imaginary solution containing the ‘‘native’’ conformations (i.e., of lowest free energy) under folding conditions, of every possible copolymer sequence. The model is of short self‐avoiding chains of hydrophobic (H) and polar (P) monomers configured on the two‐dimensional square lattice. By exhaustive enumeration, we identify all native structures for every possible sequence. We find that random sequences of H/P copolymers will bear striking resemblance to known proteins: Most sequences under folding conditions will be approximately as compact as known proteins, will have considerable amounts of secondary structure, and it is most probable that an arbitrary sequence will fold to a number of lowest free energy conformations that is of order one. In these respects, this simple model shows that proteinlike behavior should arise simply in copolymers in which one monomer type is highly solvent ave...

Solvation Effects and Driving Forces for Protein Thermodynamic and Kinetic Cooperativity: How Adequate is Native-centric Topological Modeling?

What energetic and solvation effects underlie the remarkable two-state thermodynamics and folding/unfolding kinetics of small single-domain proteins? To address this question, we investigate the folding and unfolding of a hierarchy of continuum Langevin dynamics models of chymotrypsin inhibitor 2. We find that residue-based additive Gō-like contact energies, although native-centric, are by themselves insufficient for protein-like calorimetric two-state cooperativity. Further native biases by local conformational preferences are necessary for protein-like thermodynamics. Kinetically, however, even models with both contact and local native-centric energies do not produce simple two-state chevron plots. Thus a model proteins thermodynamic cooperativity is not sufficient for simple two-state kinetics. The models tested appear to have increasing internal friction with increasing native stability, leading to chevron rollovers that typify kinetics that are commonly referred to as non-two-state. The free energy profiles of these models are found to be sensitive to the choice of native contacts and the presumed spatial ranges of the contact interactions. Motivated by explicit-water considerations, we explore recent treatments of solvent granularity that incorporate desolvation free energy barriers into effective implicit-solvent intraprotein interactions. This additional feature reduces both folding and unfolding rates vis-à-vis that of the corresponding models without desolvation barriers, but the kinetics remain non-two-state. Taken together, our observations suggest that interaction mechanisms more intricate than simple Gō-like constructs and pairwise additive solvation-like contributions are needed to rationalize some of the most basic generic protein properties. Therefore, as experimental constraints on protein chain models, requiring a consistent account of protein-like thermodynamic and kinetic cooperativity can be more stringent and productive for some applications than simply requiring a model heteropolymer to fold to a target structure.

Cooperativity, Local-Nonlocal Coupling, and Nonnative Interactions: Principles of Protein Folding from Coarse-Grained Models

Coarse-grained, self-contained polymer models are powerful tools in the study of protein folding. They are also essential to assess predictions from less rigorous theoretical approaches that lack an explicit-chain representation. Here we review advances in coarse-grained modeling of cooperative protein folding, noting in particular that the Levinthal paradox was raised in response to the experimental discovery of two-state-like folding in the late 1960s, rather than to the problem of conformational search per se. Comparisons between theory and experiment indicate a prominent role of desolvation barriers in cooperative folding, which likely emerges generally from a coupling between local conformational preferences and nonlocal packing interactions. Many of these principles have been elucidated by native-centric models, wherein nonnative interactions may be treated perturbatively. We discuss these developments as well as recent applications of coarse-grained chain modeling to knotted proteins and to intrinsically disordered proteins.