Ken A. Dill

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.

The Protein-Folding Problem, 50 Years On

Protein Folding: Past and Future Fifty years ago the Nobel Prize in chemistry was awarded to Max Perutz and John Kendrew for determining the structure of globular proteins. Since first viewing their structure of myoglobin, scientists have sought to understand protein folding. Dill and MacCallum (p. 1042) review the progress that has been made on three central questions: What is the code that relates sequence to structure? How do proteins fold so fast? Can protein structure be computationally predicted? While we have come some way toward answering these questions, new questions have been gene rated. It is no longer useful to talk about “solving the protein-folding problem”—protein folding has grown into a field of research where the next 50 years promise to be as exciting as the last. The protein-folding problem was first posed about one half-century ago. The term refers to three broad questions: (i) What is the physical code by which an amino acid sequence dictates a protein’s native structure? (ii) How can proteins fold so fast? (iii) Can we devise a computer algorithm to predict protein structures from their sequences? We review progress on these problems. In a few cases, computer simulations of the physical forces in chemically detailed models have now achieved the accurate folding of small proteins. We have learned that proteins fold rapidly because random thermal motions cause conformational changes leading energetically downhill toward the native structure, a principle that is captured in funnel-shaped energy landscapes. And thanks in part to the large Protein Data Bank of known structures, predicting protein structures is now far more successful than was thought possible in the early days. What began as three questions of basic science one half-century ago has now grown into the full-fledged research field of protein physical science.

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.

Hydrogen bonding in globular proteins

A global census of the hydrogen bonds in 42 X-ray-elucidated proteins was taken and the following demographic trends identified: (1) Most hydrogen bonds are local, i.e. between partners that are close in sequence, the primary exception being hydrogen-bonded ion pairs. (2) Most hydrogen bonds are between backbone atoms in the protein, an average of 68%. (3) All proteins studied have extensive hydrogen-bonded secondary structure, an average of 82%. (4) Almost all backbone hydrogen bonds are within single elements of secondary structure. An approximate rule of thirds applies: slightly more than one-third (37%) form i----i--3 hydrogen bonds, almost one-third (32%) form i----i--4 hydrogen bonds, and slightly less than one-third (26%) reside in paired strands of beta-sheet. The remaining 5% are not wholly within an individual helix, turn or sheet. (5) Side-chain to backbone hydrogen bonds are clustered at helix-capping positions. (6) An extensive network of hydrogen bonds is present in helices. (7) To a close approximation, the total number of hydrogen bonds is a simple function of a proteins helix and sheet content. (8) A unique quantity, termed the reduced number of hydrogen bonds, is defined as the maximum number of hydrogen bonds possible when every donor:acceptor pair is constrained to be 1:1. This quantity scales linearly with chain length, with 0.71 reduced hydrogen bond per residue. Implications of these results for pathways of protein folding are discussed.

Use of the Weighted Histogram Analysis Method for the Analysis of Simulated and Parallel Tempering Simulations

The growing adoption of generalized-ensemble algorithms for biomolecular simulation has resulted in a resurgence in the use of the weighted histogram analysis method (WHAM) to make use of all data generated by these simulations. Unfortunately, the original presentation of WHAM by Kumar et al. is not directly applicable to data generated by these methods. WHAM was originally formulated to combine data from independent samplings of the canonical ensemble, whereas many generalized-ensemble algorithms sample from mixtures of canonical ensembles at different temperatures. Sorting configurations generated from a parallel tempering simulation by temperature obscures the temporal correlation in the data and results in an improper treatment of the statistical uncertainties used in constructing the estimate of the density of states. Here we present variants of WHAM, STWHAM and PTWHAM, derived with the same set of assumptions, that can be directly applied to several generalized ensemble algorithms, including simulated tempering, parallel tempering (better known as replica-exchange among temperatures), and replica-exchange simulated tempering. We present methods that explicitly capture the considerable temporal correlation in sequentially generated configurations using autocorrelation analysis. This allows estimation of the statistical uncertainty in WHAM estimates of expectations for the canonical ensemble. We test the method with a one-dimensional model system and then apply it to the estimation of potentials of mean force from parallel tempering simulations of the alanine dipeptide in both implicit and explicit solvent.

Additivity Principles in Biochemistry

We cannot yet reliably fold proteins or RNA molecules by computer, predict ligand binding affinities, compute conformational transitions, or use the sequence information in the Human Genome very effectively to understand biomolecular function and disease. Why not? Perhaps some of our models in computational biology are based on flawed assumptions. Thermodynamic additivity principles are the foundations of chemistry, but few additivity principles have yet been found successful in biochemistry.

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.

Using quaternions to calculate RMSD

A widely used way to compare the structures of biomolecules or solid bodies is to translate and rotate one structure with respect to the other to minimize the root‐mean‐square deviation (RMSD). We present a simple derivation, based on quaternions, for the optimal solid body transformation (rotation‐translation) that minimizes the RMSD between two sets of vectors. We prove that the quaternion method is equivalent to the well‐known formula due to Kabsch. We analyze the various cases that may arise, and give a complete enumeration of the special cases in terms of the arrangement of the eigenvalues of a traceless, 4 × 4 symmetric matrix. A key result here is an expression for the gradient of the RMSD as a function of model parameters. This can be useful, for example, in finding the minimum energy path of a reaction using the elastic band methods or in optimizing model parameters to best fit a target structure.

Are Proteins Well-Packed?

The average packing density inside proteins is as high as in crystalline solids. Does this mean proteins are well-packed? We go beyond average densities, and look at the full distribution functions of free volumes inside proteins. Using a new and rigorous Delaunay triangulation method for parsing space into empty and filled regions, we introduce formal definitions of interior and surface packing densities. Although proteins look like organic crystals by the criterion of average density, they look more like liquids and glasses by the criterion of their free volume distributions. The distributions are broad, and the scalings of volume-to-surface, volume-to-cluster-radius, and numbers of void versus volume show that the interiors of proteins are more like randomly packed spheres near their percolation threshold than like jigsaw puzzles. We find that larger proteins are packed more loosely than smaller proteins. And we find that the enthalpies of folding (per amino acid) are independent of the packing density of a protein, indicating that van der Waals interactions are not a dominant component of the folding forces.

A kinematic view of loop closure

We consider the problem of loop closure, i.e., of finding the ensemble of possible backbone structures of a chain segment of a protein molecule that is geometrically consistent with preceding and following parts of the chain whose structures are given. We reduce this problem of determining the loop conformations of six torsions to finding the real roots of a 16th degree polynomial in one variable, based on the robotics literature on the kinematics of the equivalent rotator linkage in the most general case of oblique rotators. We provide a simple intuitive view and derivation of the polynomial for the case in which each of the three pair of torsional axes has a common point. Our method generalizes previous work on analytical loop closure in that the torsion angles need not be consecutive, and any rigid intervening segments are allowed between the free torsions. Our approach also allows for a small degree of flexibility in the bond angles and the peptide torsion angles; this substantially enlarges the space of solvable configurations as is demonstrated by an application of the method to the modeling of cyclic pentapeptides. We give further applications to two important problems. First, we show that this analytical loop closure algorithm can be efficiently combined with an existing loop‐construction algorithm to sample loops longer than three residues. Second, we show that Monte Carlo minimization is made severalfold more efficient by employing the local moves generated by the loop closure algorithm, when applied to the global minimization of an eight‐residue loop. Our loop closure algorithm is freely available at http://dillgroup. ucsf.edu/loop_closure/.