A ternary nucleation model for the nucleation pathway of protein folding

A novel approach to the theory of homogeneous and heterogeneous nucleation

A new approach to the theory of nucleation, formulated relatively recently by Ruckenstein, Narsimhan, and Nowakowski (see Refs. [7-16]) and developed further by Ruckenstein and other colleagues, is presented. In contrast to the classical nucleation theory, which is based on calculating the free energy of formation of a cluster of the new phase as a function of its size on the basis of macroscopic thermodynamics, the proposed theory uses the kinetic theory of fluids to calculate the condensation (W(+)) and dissociation (W(-)) rates on and from the surface of the cluster, respectively. The dissociation rate of a monomer from a cluster is evaluated from the average time spent by a surface monomer in the potential well as obtained from the solution of the Fokker-Planck equation in the phase space of position and momentum for liquid-to-solid transition and the phase space of energy for vapor-to-liquid transition. The condensation rates are calculated using traditional expressions. The knowledge of those two rates allows one to calculate the size of the critical cluster from the equality W(+)=W(-) as well as the rate of nucleation. The developed microscopic approach allows one to avoid the controversial application of classical thermodynamics to the description of nuclei which contain a few molecules. The new theory was applied to a number of cases, such as the liquid-to-solid and vapor-to-liquid phase transitions, binary nucleation, heterogeneous nucleation, nucleation on soluble particles and protein folding. The theory predicts higher nucleation rates at high saturation ratios (small critical clusters) than the classical nucleation theory for both solid-to-liquid as well as vapor-to-liquid transitions. As expected, at low saturation ratios for which the size of the critical cluster is large, the results of the new theory are consistent with those of the classical one. The present approach was combined with the density functional theory to account for the density profile in the cluster. This approach was also applied to protein folding, viewed as the evolution of a cluster of native residues of spherical shape within a protein molecule, which could explain protein folding/unfolding and their dependence on temperature.

A probabilistic approach to the effect of water hydrogen bonds on the kinetics of protein folding and protein denaturation

Previously, we presented a review of our kinetic models for the nucleation mechanism of protein folding and for the protein thermal denaturation in a barrierless way. A protein was treated as a random heteropolymer consisting of hydrophobic, hydrophilic, and neutral beads. As a crucial idea of the model, an overall potential around the cluster of native residues wherein a residue performs a chaotic motion was considered as the combination of the average dihedral, effective pairwise, and confining potentials. The overall potential as a function of the distance from the cluster has a double well shape. This allowed one to develop kinetic models for the nucleation mechanism of protein folding (NMPF) and barrierless protein denaturation (BPD) by using the mean first passage time analysis. In the original models, however, hydrogen bonding effects were taken into account only indirectly which affected the accuracy of the models because hydrogen bonding does play a crucial role in the folding, stability, and denaturation of proteins. To improve the NMPF and BPD models and explicitly take into account the hydrogen bonding "water-water" and "water-protein residue", we have developed a probabilistic hydrogen bond (PHB) model for the effect of hydrogen bond networks of water molecules around two solute particles (immersed in water) on their interaction, and have then combined the PHB model with the NMPF and BPD models. In this paper, that can be regarded as sequel of our previous review, we analyze the modified NMPF and BPD models that explicitly take into account the effect of water-water hydrogen bonding on these processes. As expected, the application of the modified models to the folding/unfolding of two model proteins (one short, consisting of 124 residues and the other large, consisting of 2500 residues) demonstrate that the hydrogen bond networks play a very important role in the protein folding/unfolding phenomena.

Effect of hydrogen bond networks on the nucleation mechanism of protein folding

We have recently developed a kinetic model for the nucleation mechanism of protein folding (NMPF) in terms of ternary nucleation by using the first passage time analysis. A protein was considered as a random heteropolymer consisting of hydrophobic, hydrophilic (some of which are negatively or positively ionizable), and neutral beads. The main idea of the NMPF model consisted of averaging the dihedral potential in which a selected residue is involved over all possible configurations of all neighboring residues along the protein chain. The combination of the average dihedral, effective pairwise (due to Lennard-Jones-type and electrostatic interactions), and confining (due to the polymer connectivity constraint) potentials gives rise to an overall potential around the cluster that, as a function of the distance from the cluster center, has a double-well shape. This allows one to evaluate the protein folding time. In the original NMPF model hydrogen bonding was not taken into account explicitly. To improve the NMPF model and make it more realistic, in this paper we modify our (previously developed) probabilistic hydrogen bond model and combine it with the former. Thus, a contribution due to the disruption of hydrogen bond networks around the interacting particles (cluster of native residues and residue in the protein unfolded part) appears in the overall potential field around a cluster. The modified model is applied to the folding of the same model proteins that were examined in the original model: a short protein consisting of 124 residues (roughly mimicking bovine pancreatic ribonuclease) and a long one consisting of 2500 residues (as a representative of large proteins with superlong polypeptide chains), at pH=8.3 , 7.3, and 6.3. The hydrogen bond contribution now plays a dominant role in the total potential field around the cluster (except for very short distances thereto where the repulsive energy tends to infinity). It is by an order of magnitude stronger for hydrophobic residues than for hydrophilic ones. The range of "residue-cluster" distances, at which the hydrogen bond effect exists, is twice as long for hydrophobic residues as for hydrophilic ones.

The effect of hydrogen bonding on the solvent-mediated interaction of composite plates

When two solute particles in water sufficiently approach each other, the disruption of water-water hydrogen bonds in their first hydration layers gives rise to an additional contribution to their overall interaction. Here we present a probabilistic approach to examining interactions between two identical parallel plates whereof the surfaces are covered with uniformly distributed hydrophobic and hydrophilic sites. The proposed approach allows one to determine the average number of hydrogen bonds per water molecule in the first hydration shell of a plate. Because of the constraint imposed by the proximity to the plate, a water molecule forms less hydrogen bond in this shell than in the bulk medium. As a result, the water molecules prefer the latter to the former, even though a bond is stronger in the former than in the latter. The interplay of these factors results in an additional contribution to the overall plate interaction which is attractive and naturally short-range, appearing only when the distance between the plates is smaller than five lengths of a hydrogen bond. At a given distance, it monotonically increases from 0 to its maximum value as the fraction of hydrophobic surface area on a plate increases from 0 to 1. When this fraction is 0.5, this contribution can be up to two orders of magnitude larger than the van der Waals interaction (depending on the water density in the vicinity of a plate).

First passage time analysis of protein folding via nucleation and of barrierless protein denaturation

A review of the kinetic models, recently developed by the authors for the nucleation mechanism of protein folding and for the barrierless thermal denaturation, is presented. Both models are based on the mean first passage time analysis. A protein is treated as a random heteropolymer consisting of hydrophobic, hydrophilic, or neutral beads. As a crucial idea of the model, an overall potential around the cluster of native residues wherein a residue performs a chaotic motion is considered as the combination of the average dihedral, effective pairwise, and confining potentials. The overall potential as a function of the distance from the cluster center has a double well shape which allows one to determine its emission and absorption rates by the first passage time analysis. One can thus develop a theory for the nucleation mechanism of protein folding and calculate the temperature dependence of the folding time. A kinetic model for protein denaturation occurring in a barrierless way has been also developed by using the same approach. The numerical calculations for two model proteins (one consisting of 124 amino acids and the other of 2500 amino acids) demonstrate that the models can predict folding and unfolding times consistent with experimental data.

Thermal denaturation of a native protein via spinodal decomposition in the framework of first-passage-time analysis

In this paper we present a kinetic model for the thermal unfolding of a native protein. Due to a sufficiently large temperature increase or decrease, the rate with which a cluster of native residues within a protein emits residues becomes larger than the rate of absorption of residues from the unfolded part of the protein in the whole range of cluster sizes up to the size of the whole protein. This leads to the unfolding of the protein in a barrierless way, i.e., as spinodal decomposition. Using the formalism of the first passage time analysis [previously applied also to the problem of protein folding via nucleation by the authors, J. Phys. Chem. B 111, 886 (2007); J. Chem. Phys. 126, 175103 (2007)], one can determine the temperature dependence of the rates of emission and absorption of residues by the cluster. Knowing these rates as functions of temperature and cluster size, one can find the threshold temperatures of cold and hot barrierless denaturation as well as the unfolding times at temperatures lower and higher, respectively, than those threshold values. For a numerical illustration, the method is applied to the thermal unfolding of a model protein consisting of 2500 residues.

Intermediates in the refolding of urea-denatured dimeric arginine kinase from Stichopus japonicus

The refolding of urea-denatured dimeric AK was investigated by both equilibrium and kinetic measurements. Both studies indicated that the refolding of dimeric AK is a multiphasic process. The equilibrium studies, monitored by enzyme activity, intrinsic protein fluorescence, circular dichroism (CD), 1-anilinonaphtalene-8-sulfonate (ANS) binding, size-exclusion chromatography and glutaraldehyde cross-linking showed that there were at least two intermediates involved in this process: I(1) (existing in 1.8-1.4M urea) and I(2) (existing in 0.8-0.4M urea). I(1) was a monomeric intermediate and possessed characteristic similar to the globular folding intermediates described in the literature. I(2) was an active native-like intermediate. The kinetic studies suggested that the refolding of AK possessed a burst phase, fast phase and slow phase, which involved at least the burst phase intermediates (I(B)). Comparison of the properties of these intermediates suggested that I(B) in the kinetic process corresponded to I(1) in the equilibrium process. Based on these results, a scheme for refolding of urea-denatured AK was proposed.