Rotamer strain energy in protein helices - quantification of a major force opposing protein folding

It is widely believed that the dominant force opposing protein folding is the entropic cost of restricting internal rotations. The energetic changes from restricting side-chain torsional motion are more complex than simply a loss of conformational entropy, however. A second force opposing protein folding arises when a side-chain in the folded state is not in its lowest-energy rotamer, giving rotameric strain. chi strain energy results from a dihedral angle being shifted from the most stable conformation of a rotamer when a protein folds. We calculated the energy of a side-chain as a function of its dihedral angles in a poly(Ala) helix. Using these energy profiles, we quantify conformational entropy, rotameric strain energy and chi strain energy for all 17 amino acid residues with side-chains in alpha-helices. We can calculate these terms for any amino acid in a helix interior in a protein, as a function of its side-chain dihedral angles, and have implemented this algorithm on a web page. The mean change in rotameric strain energy on folding is 0.42 kcal mol-1 per residue and the mean chi strain energy is 0.64 kcal mol-1 per residue. Loss of conformational entropy opposes folding by a mean of 1.1 kcal mol-1 per residue, and the mean total force opposing restricting a side-chain into a helix is 2.2 kcal mol-1. Conformational entropy estimates alone therefore greatly underestimate the forces opposing protein folding. The introduction of strain when a protein folds should not be neglected when attempting to quantify the balance of forces affecting protein stability. Consideration of rotameric strain energy may help the use of rotamer libraries in protein design and rationalise the effects of mutations where side-chain conformations change.     

Structural bases of lectin-carbohydrate affinities: comparison with protein-folding energetics.

We have made a comparative structure based analysis of the thermodynamics of lectin-carbohydrate (L-C) binding and protein folding. Examination of the total change in accessible surface area in those processes revealed a much larger decrease in free energy per unit of area buried in the case of L-C associations. According to our analysis, this larger stabilization of L-C interactions arises from a more favorable enthalpy of burying a unit of polar surface area, and from higher proportions of polar areas. Hydrogen bonds present at 14 L-C interfaces were identified, and their overall characteristics were compared to those reported before for hydrogen bonds in protein structures. Three major factors might explain why polar-polar interactions are stronger in L-C binding than in protein folding: (1) higher surface density of hydrogen bonds; (2) better hydrogen-bonding geometry; (3) larger proportion of hydrogen bonds involving charged groups. Theoretically, the binding entropy can be partitioned into three main contributions: entropy changes due to surface desolvation, entropy losses arising from freezing rotatable bonds, and entropic effects that result from restricting translation and overall rotation motions. These contributions were estimated from structural information and added up to give calculated binding entropies. Good correlation between experimental and calculated values was observed when solvation effects were treated according to a parametrization developed by other authors from protein folding studies. Finally, our structural parametrization gave calculated free energies that deviate from experimental values by 1.1 kcal/mol on the average; this amounts to an uncertainty of one order of magnitude in the binding constant.     

15N backbone dynamics of the S-peptide from ribonuclease A in its free and S-protein bound forms: toward a site-specific analysis of entropy changes upon folding.

Backbone 15N relaxation parameters (R1, R2, 1H-15N NOE) have been measured for a 22-residue recombinant variant of the S-peptide in its free and S-protein bound forms. NMR relaxation data were analyzed using the "model-free" approach (Lipari & Szabo, 1982). Order parameters obtained from "model-free" simulations were used to calculate 1H-15N bond vector entropies using a recently described method (Yang & Kay, 1996), in which the form of the probability density function for bond vector fluctuations is derived from a diffusion-in-a-cone motional model. The average change in 1H-15N bond vector entropies for residues T3-S15, which become ordered upon binding of the S-peptide to the S-protein, is -12.6+/-1.4 J/mol.residue.K. 15N relaxation data suggest a gradient of decreasing entropy values moving from the termini toward the center of the free peptide. The difference between the entropies of the terminal and central residues is about -12 J/mol residue K, a value comparable to that of the average entropy change per residue upon complex formation. Similar entropy gradients are evident in NMR relaxation studies of other denatured proteins. Taken together, these observations suggest denatured proteins may contain entropic contributions from non-local interactions. Consequently, calculations that model the entropy of a residue in a denatured protein as that of a residue in a di- or tri-peptide, might over-estimate the magnitude of entropy changes upon folding.     

IRECS: a new algorithm for the selection of most probable ensembles of side-chain conformations in protein models

We introduce a new algorithm, IRECS (Iterative REduction of Conformational Space), for identifying ensembles of most probable side-chain conformations for homology modeling. On the basis of a given rotamer library, IRECS ranks all side-chain rotamers of a protein according to the probability with which each side chain adopts the respective rotamer conformation. This ranking enables the user to select small rotamer sets that are most likely to contain a near-native rotamer for each side chain. IRECS can therefore act as a fast heuristic alternative to the Dead-End-Elimination algorithm (DEE). In contrast to DEE, IRECS allows for the selection of rotamer subsets of arbitrary size, thus being able to define structure ensembles for a protein. We show that the selection of more than one rotamer per side chain is generally meaningful, since the selected rotamers represent the conformational space of flexible side chains. A knowledge-based statistical potential ROTA was constructed for the IRECS algorithm. The potential was optimized to discriminate between side-chain conformations of native and rotameric decoys of protein structures. By restricting the number of rotamers per side chain to one, IRECS can optimize side chains for a single conformation model. The average accuracy of IRECS for the chi1 and chi1+2 dihedral angles amounts to 84.7% and 71.6%, respectively, using a 40 degrees cutoff. When we compared IRECS with SCWRL and SCAP, the performance of IRECS was comparable to that of both methods. IRECS and the ROTA potential are available for download from the URL http://irecs.bioinf.mpi-inf.mpg.de.     

Comparison of straight chain and cyclic unnatural amino acids embedded in the core of staphylococcal nuclease.

We have determined by X-ray crystallography the structures of several variants of staphylococcal nuclease with long flexible straight chain and equivalent length cyclic unnatural amino acid side chains embedded in the protein core. The terminal atoms in the straight side chains are not well defined by the observed electron density even though they remain buried within the protein interior. We have previously observed this behavior and have suggested that it may arise from the addition of side-chain vibrational and oscillational motions with each bond as a side chain grows away from the relatively rigid protein main chain and/or the population of multiple rotamers (Wynn R, Harkins P, Richards FM. Fox RO. 1996. Mobile unnatural amino acid side chains in the core of staphylococcal nuclease. Protein Sci 5:1026-1031). Reduction of the number of degrees of freedom by cyclization of a side chain would be expected to constrain these motions. These side chains are in fact well defined in the structures described here. Over-packing of the protein core results in a 1.0 A shift of helix 1 away from the site of mutation. Additionally, we have determined the structure of a side chain containing a single hydrogen to fluorine atom replacement on a methyl group. A fluorine atom is intermediate in size between methyl group and a hydrogen atom. The fluorine atom is observed in a single position indicating it does not rotate like methyl hydrogen atoms. This change also causes subtle differences in the packing interactions.     

Thermodynamics of amino acid side-chain internal rotations

The absolute Gibbs energy, enthalpy and entropy of each of the internal rotations found in protein side chains has been calculated. The calculation requires the moments of inertia of the side chains about each bond, the potential energy barrier and the symmetry number and gives the maximum possible thermodynamic consequences of restricting side chain motion when a protein folds. Hindering side chain internal rotations is unfavourable in terms of Gibbs energy and entropy; it is enthalpically favourable at 0 K. At room temperature, it is estimated that the adverse entropy of hindering buried side chain internal rotation is only 25% of the absolute entropy. The difference between absolute entropies in the folded and unfolded states gives the entropy change for folding. The estimated Gibbs energy change for restricting each residue correlates moderately well with the probability of that residue being found on the folded protein surface, rather than in the protein interior (where motion is restricted).     

Stability of yeast iso-1-ferricytochrome c as a function of pH and temperature.

Absorbance-detected thermal denaturation studies of the C102T variant of Saccharomyces cerevisiae iso-1-ferricytochrome c were performed between pH 3 and 5. Thermal denaturation in this pH range is reversible, shows no concentration dependence, and is consistent with a 2-state model. Values for free energy (delta GD), enthalpy (delta HD), and entropy (delta SD) of denaturation were determined as functions of pH and temperature. The value of delta GD at 300 K, pH 4.6, is 5.1 +/- 0.3 kcal mol-1. The change in molar heat capacity upon denaturation (delta Cp), determined by the temperature dependence of delta HD as a function of pH (1.37 +/- 0.06 kcal mol-1 K-1), agrees with the value determined by differential scanning calorimetry. pH-dependent changes in the Soret region indicate that a group or groups in the heme environment of the denatured protein, probably 1 or both heme propionates, ionize with a pK near 4. The C102T variant exhibits both enthalpy and entropy convergence with a delta HD of 1.30 kcal mol-1 residue-1 at 373.6 K and a delta SD of 4.24 cal mol-1 K-1 residue-1 at 385.2 K. These values agree with those for other single-domain, globular proteins.     

The nature of the free energy barriers to two-state folding

We introduce a simple procedure to analyze the temperature dependence of the folding and unfolding rates of two-state proteins. We start from the simple transition-state-like rate expression: k = D(eff)exp(-DeltaG(TS)/RT), in which upper and lower bounds for the intra-chain effective diffusion coefficient (D(eff)) are obtained empirically using the timescales of elementary processes in protein folding. From the changes in DeltaG(TS) as a function of temperature, we calculate enthalpies and heat capacities of activation, together with the more elusive entropies of activation. We then estimate the conformational entropy of the transition state by extrapolation to the temperature at which the solvation entropy vanishes by cancellation between polar and apolar terms. This approach is based on the convergence temperatures for the entropy of solvating apolar (approximately 385 K) and polar groups (approximately 335 K), the assumption that the structural properties of the transition state are somewhere in between the unfolded and folded states, and the established relationship between observed heat capacity and solvent accessibility.1 To circumvent the lack of structural information about transition states, we use the empirically determined heat capacities of activation as constraints to identify the extreme values of the transition state conformational entropy that are consistent with experiment. The application of this simple approach to six two-state folding proteins for which there is temperature-dependent data available in the literature provides important clues about protein folding. For these six proteins, we obtain an average equilibrium cost in conformational entropy of -4.3 cal x mol(-1)K(-1)per residue, which is in close agreement to previous empirical and computational estimates of the same quantity. Furthermore, we find that all these proteins have a conformationally diverse transition state, with more than half of the conformational entropy of the unfolded state. In agreement with predictions from theory and computer simulations, the transition state signals the change from a regime dominated by loss in conformational entropy to one driven by the gain in stabilization free energy (i.e., including protein interactions and solvation effects). Moreover, the height of the barrier is determined by how much stabilization free energy is realized at that point, which is related to the relative contribution of local versus non-local interactions. A remarkable observation is that the fraction of conformational entropy per residue that is present in the transition state is very similar for the six proteins in this study. Based on this commonality, we propose that the observed change in thermodynamic regime is connected to a change in the pattern of structure formation: from one driven by formation of pairwise interactions to one dominated by coupling of the networks of interactions involved in forming the protein core. In this framework, the barrier to two-state folding is crossed when the folding protein reaches a "critical native density" that allows expulsion of remaining interstitial water and consolidation of the core. The principle of critical native density should be general for all two-state proteins, but can accommodate different folding mechanisms depending on the particularities of the structure and sequence.     

Statistical and conformational analysis of the electron density of protein side chains

Protein side chains make most of the specific contacts between proteins and other molecules, and their conformational properties have been studied for many years. These properties have been analyzed primarily in the form of rotamer libraries, which cluster the observed conformations into groups and provide frequencies and average dihedral angles for these groups. In recent years, these libraries have improved with higher resolution structures and using various criteria such as high thermal factors to eliminate side chains that may be misplaced within the crystallographic model coordinates. Many of these side chains have highly non-rotameric dihedral angles. The origin of side chains with high B-factors and/or with non-rotameric dihedral angles is of interest in the determination of protein structures and in assessing the prediction of side chain conformations. In this paper, using a statistical analysis of the electron density of a large set of proteins, it is shown that: (1) most non-rotameric side chains have low electron density compared to rotameric side chains; (2) up to 15% of chi1 non-rotameric side chains in PDB models can clearly be fit to density at a single rotameric conformation and in some cases multiple rotameric conformations; (3) a further 47% of non-rotameric side chains have highly dispersed electron density, indicating potentially interconverting rotameric conformations; (4) the entropy of these side chains is close to that of side chains annotated as having more than one chi(1) rotamer in the crystallographic model; (5) many rotameric side chains with high entropy clearly show multiple conformations that are not annotated in the crystallographic model. These results indicate that modeling of side chains alternating between rotamers in the electron density is important and needs further improvement, both in structure determination and in structure prediction.     

Backbone Flexibility of CDR3 and Immune Recognition of Antigens

Conformational entropy is an important component of protein–protein interactions; however, there is no reliable method for computing this parameter. We have developed a statistical measure of residual backbone entropy in folded proteins by using the ?–ψ distributions of the 20 amino acids in common secondary structures. The backbone entropy patterns of amino acids within helix, sheet or coil form clusters that recapitulate the branching and hydrogen bonding properties of the side chains in the secondary structure type. The same types of residues in coil and sheet have identical backbone entropies, while helix residues have much smaller conformational entropies. We estimated the backbone entropy change for immunoglobulin complementarity-determining regions (CDRs) from the crystal structures of 34 low-affinity T-cell receptors and 40 high-affinity Fabs as a result of the formation of protein complexes. Surprisingly, we discovered that the computed backbone entropy loss of only the CDR3, but not all CDRs, correlated significantly with the kinetic and affinity constants of the 74 selected complexes. Consequently, we propose a simple algorithm to introduce proline mutations that restrict the conformational flexibility of CDRs and enhance the kinetics and affinity of immunoglobulin interactions. Combining the proline mutations with rationally designed mutants from a previous study led to 2400-fold increase in the affinity of the A6 T-cell receptor for Tax-HLAA2. However, this mutational scheme failed to induce significant binding changes in the already-high-affinity C225–Fab/huEGFR interface. Our results will serve as a roadmap to formulate more effective target functions to design immune complexes with improved biological functions.