Optimal relationship between average conformational entropy and average energy of interactions between residues for fast protein folding

Based on the known experimental data and using the theoretical modeling of protein folding, we demonstrate that there exists an optimal relationship between the average conformational entropy and the average energy of contacts per residue, that is an entropy capacity, for fast protein folding. Statistical analysis of conformational entropy and the number of contacts per residue for 5829 protein structures from four general structural classes (all-alpha, all-beta, +/-/beta, alpha+beta) demonstrates that each class of proteins has its own class-specific average number of contacts and average conformational entropy per residue. These class-specific features determine the folding rates: a proteins are the fastest folding proteins, then follow beta and alpha+beta proteins, and finally alpha/beta proteins are the slowest ones.     

An empirical energy function for threading protein sequence through the folding motif

In this paper we present a new residue contact potantial derived by statistical analysis of protein crystal structures. This gives mean hydrophobic and pairwise contact energies as a function of residue type and distance interval. To test the accuracy of this potential we generate model structures by “threading” different sequences through backbone folding motifs found in the structural data base. We find that conformational energies calculated by summing contact potentials show perfect specificity in matching the correct sequences with each globular folding motif in a 161-protcin data set. They also identify correct models with the core folding motifs of heme-rythrin and immunoglobulin McPC603 V₁-do- main, among millions of alternatives possible when we align subsequences with α-helices and β-strands, and allow for variation in the lengths of intervening loops. We suggest that contact potentials reflect important constraints on nonbonded interaction in native proteins, and that “threading” may be useful for structure prediction by recognition of folding motif. © 1993 Wiley-Liss, Inc.     

Fractal structure and conformational entropy of protein chain

The structures of 25 proteins arbitrarily chosen are investigated by fractal geometry, and their fractal dimensions (D_f) and conformational entropies S(N₀) are calculated by Havlin—Ben Avraham and Monte Carlo method, respectively. Comparison of the D_f and S(N₀) gives the relation: D_f = 1.532 - 3.00 × 10⁻⁴ S(N₀). The entropy data obtained by Monte Carlo method for the chain of random self-avoiding walks confirm the prediction of renormalization group: S(N₀) = 1.544N₀ + 0.1667 In N₀ + 0.1570 where N₀ is the number of residues in a protein chain. Both the D_f and S(N₀) reflect the conformational properties of a protein molecular chain. The idea resulting from the present communication suggests that the thermodynamic behaviours of proteins may be related to multifractals.     

Conformational stability of biological polyelectrolytes: Evaluation of enthalpy and entropy changes of conformational transitions

A new method is proposed for the determination of the enthalpy and entropy changes of nonionic origin upon conformational transition of linear biopolyelectrolytes in solution. For all transition midpoints, defined by given temperature and ionic strength, the total free energy change of the system is zero, which means that the nonionic contribution to the free energy change is equal in value and opposite in sign to the polyelectrolytic one. The counterion condensation theory of linear polyelectrolytes provides for the appropriate analytical expression to be used in such calculations. Linear plots of the proper functions of the calculated free energy changes vs the proper functions of temperature allows for the determination of the enthalpic and entropic terms of the nonionic free energy change of transition. The method has been applied to the extensive available data of the ion-induced conformational change of κ-carrageenan, a linear sulfated galactan extracted from seaweeds. The method has proved very successful, with the results showing a remarkable convergency of the enthalpy values for different monovalent counterions. On the other hand, the above approach has made it possible to explain the known effect of counterion specificity on the transition by a small difference in the nonionic entropic contributions. © 1998 John Wiley & Sons, Inc. Biopoly 45: 203–216, 1998     

The backbone conformational entropy of protein folding: experimental measures from atomic force microscopy

The energy dissipated during the atomic force microscopy-based mechanical unfolding and extension of proteins is typically an order of magnitude greater than their folding free energy. The vast majority of the "excess" energy dissipated is thought to arise due to backbone conformational entropy losses as the solvated, random-coil unfolded state is stretched into an extended, low-entropy conformation. We have investigated this hypothesis in light of recent measurements of the energy dissipated during the mechanical unfolding of "polyproteins" comprised of multiple, homogeneous domains. Given the assumption that backbone conformational entropy losses account for the vast majority of the energy dissipated (an assumption supported by numerous lines of experimental evidence), we estimate that approximately 19(+/-2)J/(mol K residue) of entropy is lost during the extension of three mechanically stable beta-sheet polyproteins. If, as suggested by measured peak-to-peak extension distances, pulling proceeds to near completion, this estimate corresponds to the absolute backbone conformational entropy of the unfolded state. As such, it is exceedingly close to previous theoretical and semi-empirical estimates that place this value at approximately 20J/(mol K residue). The estimated backbone conformational entropy lost during the extension of two helical polyproteins, which, in contrast to the mechanically stable beta-sheet polyproteins, rupture at very low applied forces, is three- to sixfold less. Either previous estimates of the backbone conformational entropy are significantly in error, or the reduced mechanical strength of the helical proteins leads to the rupture of a subsequent domain before full extension (and thus complete entropy loss) is achieved.     

Investigations into sequence and conformational dependence of backbone entropy,inter-basin dynamics and the Flory isolated-pair hypothesis for peptides

The populations and transitions between Ramachandran basins are studied for combinations of the standard 20 amino acids in monomers, dimers and trimers using an implicit solvent Langevin dynamics algorithm and employing seven commonly used force-fields. Both the basin populations and inter-conversion rates are influenced by the nearest neighbor's conformation and identity, contrary to the Flory isolated-pair hypothesis. This conclusion is robust to the choice of force-field, even though the use of different force-fields produces large variations in the populations and inter-conversion rates between the dominant helical, extended beta, and polyproline II basins. The computed variation of conformational and dynamical properties with different force-fields exceeds the difference between explicit and implicit solvent calculations using the same force-field. For all force-fields, the inter-basin transitions exhibit a directional dependence, with most transitions going through extended beta conformation, even when it is the least populated basin. The implications of these results are discussed in the context of estimates for the backbone entropy of single residues, and for the ability of all-atom simulations to reproduce experimental protein folding data.     

Calculation of the total electrostatic energy of a macromolecular system: solvation energies, binding energies, and conformational analysis

In this report we describe an accurate numerical method for calculating the total electrostatic energy of molecules of arbitrary shape and charge distribution, accounting for both Coulombic and solvent polarization terms. In addition to the solvation energies of individual molecules, the method can be used to calculate the electrostatic energy associated with conformational changes in proteins as well as changes in solvation energy that accompany the binding of charged substrates. The validity of the method is examined by calculating the hydration energies of acetate, methyl ammonium, ammonium, and methanol. The method is then used to study the relationship between the depth of a charge within a protein and its interaction with the solvent. Calculations of the relative electrostatic energies of crystal and misfolded conformations of Themiste dyscritum hemerythrin and the VL domain of an antibody are also presented. The results indicate that electrostatic charge-solvent interactions strongly favor the crystal structures. More generally, it is found that charge-solvent interactions, which are frequently neglected in protein structure analysis, can make large contributions to the total energy of a macromolecular system.     

Probing the conformational changes and peroxidase-like activity of cytochrome c upon interaction with iron nanoparticles

Herein, the interaction of iron nanoparticle (Fe-NP) with cytochrome c (Cyt c) was investigated, and a range of techniques such as dynamic light scattering (DLS), zeta potential measurements, static and synchronous fluorescence spectroscopy, near and far circular dichroism (CD) spectroscopy, and ultraviolet–visible (UV–vis) spectroscopy were used to analyze the interaction between Cyt c and Fe-NP. DLS and zeta potential measurements showed that the values of hydrodynamic radius and charge distribution of Fe-NP are 83.95 ± 3.7 nm and 4.5 ± .8 mV, respectively. The fluorescence spectroscopy results demonstrated that the binding of Fe-NP with Cyt c is mediated by hydrogen bonds and van der Waals interactions. Also Fe-NP induced conformational changes in Cyt c and reduced the melting temperature value of Cyt c from 79.18 to 71.33°C. CD experiments of interaction between Fe-NP and Cyt c revealed that the secondary structure of Cyt c with the dominant α-helix structures remained unchanged whereas the tertiary structure and heme position of Cyt c are subjected to remarkable changes. Absorption spectroscopy at 695 nm revealed that Fe-NP considerably disrupt the Fe…S(Met80) bond. In addition, the UV–vis experiment showed the peroxidase-like activity of Cyt c upon interaction with Fe-NP. Hence, the data indicate the Fe-NP results in unfolding of Cyt c and subsequent peroxidase-like activity of denatured species. It was concluded that a comprehensive study of the interaction of Fe-NP with biological system is a crucial step for their potential application as intracellular delivery carriers and medicinal agents.     

Evidence for an internal entropy contribution to phosphoryl transfer: a study of domain closure, backbone flexibility, and the catalytic cycle of cAMP-dependent protein kinase.

While there is no question that ligands can induce large-scale domain movements that narrow (close) the active-site cleft of the catalytic (C) subunit of cAMP-dependent protein kinase (cAPK), the results from small-angle X-ray scattering, protein footprinting, and thermostability studies are inconsistent with regard to which ligands induce these movements. This inconsistency suggests a greater complexity of cAPK conformational dynamics than is generally recognized. As an initial step to study this issue in relation to the catalysis, a new method to measure cAPK domain closure was developed, and the state of domain closure and the local segmental flexibility at major steps of the cAPK catalytic cycle were examined with site-directed labeling and fluorescence spectroscopy. To achieve this, a C subunit mutant (F239C/C199A) was engineered that allowed for fluorescein 5-maleimide (donor) conjugation of F239C in the large lobe and tetramethylrhodamine (acceptor) conjugation of C343 in the small lobe. Domain closure was assessed as an increase in the efficiency of energy transfer between donor and acceptor. The anisotropy decay of fluoroscein 5-maleimide, conjugated to a site of cysteine substitution (K81C) in the small lobe of the C subunit was used to assess the local backbone flexibility around the B helix. The effects of substrate/pseudosubstrate (ATP and PKI(5-24)), a fragment of protein kinase inhibitor) and products (ADP and phosphorylated PKS) on domain closure and B helix flexibility were measured. The results show that domain closure is not tightly coupled to the flexibility around K81C. Moreover, although substrates/pseudosubstrate and products independently close the active-site cleft, only the substrates substantially decreased the backbone flexibility around the B helix. Because this order-to-disorder transition coincides with the phosphoryl transfer transition, the results suggest the existence of an internal entropy contribution to catalysis.     

Perfect temperature for protein structure prediction and folding

We have investigated the influence of the “noise” of inevitable errors in energetic parameters on-protein structure prediction. Because of this noise, only a part of all the interactions operating in a protein chain can be taken into account, and therefore a search for the energy minimum becomes inadequate for protein structure prediction. One can rather rely on statistical mechanics: a calculation carried out at a temperature T_* somewhat below that of protein melting gives the best possible, though always approximate prediction. The early stages of protein folding also “take into account” only a part of all the interactions; consequently, the same temperature T_* is favorable for the self-organization of native-like intermediates in protein folding. © 1995 Wiley-Liss, Inc.     

Conformational equilibria and free energy profiles for the allosteric transition of the ribose-binding protein

The ribose-binding protein (RBP) is a sugar-binding bacterial periplasmic protein whose function is associated with a large allosteric conformational change from an open to a closed conformation upon binding to ribose. The crystal structures of RBP in open and closed conformations have been solved. It has been hypothesized that the open and closed conformations exist in a dynamic equilibrium in solution, and that sugar binding shifts the population from open conformations to closed conformations. Here, we study by computer simulations the thermodynamic changes that accompany this conformational change, and model the structural changes that accompany the allosteric transition, using umbrella sampling molecular dynamics and the weighted histogram analysis method. The open state is comprised of a diverse ensemble of conformations; the open ribose-free X-ray crystal conformations being representative of this ensemble. The unligated open form of RBP is stabilized by conformational entropy. The simulations predict detectable populations of closed ribose-free conformations in solution. Additional interdomain hydrogen bonds stabilize this state. The predicted shift in equilibrium from the open to the closed state on binding to ribose is in agreement with experiments. This is driven by the energetic stabilization of the closed conformation due to ribose-protein interactions. We also observe a significant population of a hitherto unobserved ribose-bound partially open state. We believe that this state is the one that has been suggested to play a role in the transfer of ribose to the membrane-bound permease complex.     

Rapid collapse precedes the fast two-state folding of the cold shock protein

The cold shock protein Bc-Csp folds very rapidly in a reaction that is well described by a kinetic two-state mechanism without intermediates. We measured the shortening of six intra-protein distances during folding by F?rster resonance energy transfer (FRET) in combination with stopped-flow experiments. Single tryptophan residues were engineered into the protein as the donors, and single 5-(((acetylamino)ethyl)amino)naphthalene-1-sulfonate (AEDANS) residues were placed as the acceptors at solvent-exposed sites of Bc-Csp. Their R0 value of about 22 A was well suited for following distance changes during the folding of this protein with a high sensitivity. The mutagenesis and the labeling did not alter the refolding kinetics. The changes in energy transfer during folding were monitored by both donor and acceptor emission and reciprocal effects were found. In two cases the donor-acceptor distances were similar in the unfolded and the folded state and, as a consequence, the kinetic changes in energy transfer upon folding were very small. For four donor/acceptor pairs we found that > or =50% of the increase in energy transfer upon folding occurred prior to the rate-limiting step of folding. This reveals that about half of the shortening of the intra-molecular distances upon folding has occurred already before the rate-limiting step and suggests that the fast two-state folding reaction of Bc-Csp is preceded by a very rapid collapse.     

Effect of polyols on the conformational stability and biological activity of a model protein lysozyme

The purpose of this study was to investigate the stabilizing action of polyols against various protein degradation mechanisms (eg, aggregation, deamidation, oxidation), using a model protein lysozyme. Differential scanning calorimeter (DSC) was used to measure the thermodynamic parameters, mid point transition temperature and calorimetric enthalpy, in order to evaluate conformational stability. Enzyme activity assay was used to corroborate the DSC results. Mannitol, sucrose, lactose, glycerol, and propylene glycol were used as polyols to stabilize lysozyme against aggregation, deamidation, and oxidation. Mannitol was found to stabilize lysozyme against aggregation, sucrose against deamidation both at neutral pH and at acidic pH, and lactose against oxidation. Stabilizers that provided greater conformational stability of lysozyme against various degradation mechanisms also protected specific enzyme activity to a greater extent. It was concluded that DSC and bioassay could be valuable tools for screening stabilizers in protein formulations.     

Exploring the conformational space of protein side chains using dead-end elimination and the A* algorithm

We describe an algorithm which enables us to search the conformational space of the side chains of a protein to identify the global minimum energy combination of side chain conformations as well as all other conformations within a specified energy cutoff of the global energy minimum. The program is used to explore the side chain conformational energy surface of a number of proteins, to investigate how this surface varies with the energy model used to describe the interactions within the system and the rotamer library. Enumeration of the rotamer combinations enables us to directly evaluate the partition function, and thus calculate the side chain contribution to the conformational entropy of the folded protein. An investigation of these conformations and the relationships between them shows that most of the conformations near to the global energy minimum arise from changes in side chain conformations that are essentially independent; very few result from a concerted change in conformation of two or more residues. Some of the limitations of the approach are discussed. Proteins 33:227–239, 1998. © 1998 Wiley-Liss, Inc.     

Structural basis of hierarchical multiple substates of a protein. II: Monte Carlo simulation of native thermal fluctuations and energy minimization

Conformational fluctuations in a globular protein, bovine pancreatic trypsin inhibitor, in the time range between picoseconds and nanoseconds are studied by a Monte Carlo simulation method. Multiple energy minima are derived from sampled conformations by minimizing their energy. They are distributed in clusters in the conformational space. A hierarchical structure is observed in the simulated dynamics. In the time range between 10(-14) and 10(-10) seconds dynamics is well represented by a superposition of vibrational motions within an energy well with transitions among minima within each cluster. Transitions among clusters take place in the time range of nanoseconds or longer.     

Comparing the energy landscapes for native folding and aggregation of PrP

Protein sequences are evolved to encode generally one folded structure, out of a nearly infinite array of possible folds. Underlying this code is a funneled free energy landscape that guides folding to the native conformation. Protein misfolding and aggregation are also a manifestation of free-energy landscapes. The detailed mechanisms of these processes are poorly understood, but often involve rare, transient species and a variety of different pathways. The inherent complexity of misfolding has hampered efforts to measure aggregation pathways and the underlying energy landscape, especially using traditional methods where ensemble averaging obscures important rare and transient events. We recently studied the misfolding and aggregation of prion protein by examining 2 monomers tethered in close proximity as a dimer, showing how the steps leading to the formation of a stable aggregated state can be resolved in the single-molecule limit and the underlying energy landscape thereby reconstructed. This approach allows a more quantitative comparison of native folding versus misfolding, including fundamental differences in the dynamics for misfolding. By identifying key steps and interactions leading to misfolding, it should help to identify potential drug targets. Here we describe the importance of characterizing free-energy landscapes for aggregation and the challenges involved in doing so, and we discuss how single-molecule studies can help test proposed structural models for PrP aggregates.     

The contribution of cross-links to protein stability: A normal mode analysis of the configurational entropy of the native state

The vibrational entropy of native BPTI, with three disulfide bonds, was determined by use of normal mode calculations and compared with that of folded variants having either one less disulfide bond or lacking a peptide bond at the trypsin-reactive site. Favorable contributions to the free energy of 2.5–5.1 kcal/mol at 300 K were calculated for the reduction of disulfide bonds in the folded state, whereas no favorable contribution was found for the hydrolysis of the peptide bond cleaved by trypsin. This is on the order of the effect of disulfides in the unfolded state. The implications of these results for the stabilization of a folded protein by the introduction of crosslinks are discussed. © 1993 Wiley-Liss, Inc.     

Comparison of the predicted structure for the activated form of the P21 protein with the X-ray crystal structure

The predicted conformation and position of the central transforming region (residues 55–67) of the p21 protein are compared with the conformation and position of this segment in a recently determined X-ray crystal structure of residues 1–166 of this protein in the activated state bound to a nonhydrolyzable GTP derivative. We previously predicted that this segment of the protein would adopt a roughly extended conformation from Ile 55-Thr 58, a reverse turn at Ala 59-Gln 61, followed by an -helix from Glu 62-Met 67. We further predicted that this region of the activated protein occupies a position that is virtually identical to corresponding regions in the homologous purine nucleotide-binding proteins, bacterial elongation factor (EF-tu), and adenylate kinase (ADK). We find that there is a close correspondence between the conformation and position of our predicted structure and those found in the X-ray crystal structure. A mechanism for activation of the protein is proposed and is corroborated by X-ray crystallographic data.     

Shape of the free energy barriers for protein folding probed by multiple perturbation analysis

The characterization of the free energy barriers has been a major goal in studies on the mechanism of protein folding. Testing the effect of mutations or denaturants on protein folding reactions revealed that transition state movement is rare, suggesting that folding barriers are robust and narrow maxima on the free energy landscape. Here we demonstrate that the application of multiple perturbations allows the observation of small transition state movements that escape detection in single perturbation experiments. We used tendamistat as a model protein to test the broadness of the free energy barriers. Tendamistat folds over two consecutive transition states and through a high-energy intermediate. Measuring the combined effect of temperature and denaturant on the position of the transition state in the wild-type protein and in several mutants revealed that the early transition state shows significant transition state movement. Its accessible surface area state becomes more native-like with destabilization of the native state by temperature. To the same extent, the entropy of the early transition state becomes more native-like with increasing denaturant concentration, in accordance with Hammond behavior. The position of the late transition state, in contrast, is much less sensitive to the applied perturbations. These results suggest that the barriers in protein folding become increasingly narrow as the folding polypeptide chain approaches the native state.