Population analyses of kinetic partitioning in protein folding

A Study of the Boltzmann Sequence-Structure Channel

We rigorously study a channel that maps sequences from a finite alphabet to self-avoiding walks in the two-dimensional grid, inspired by a model of protein folding from statistical physics and studied empirically by biophysicists. This channel, which we call the Boltzmann sequence-structure channel, is characterized by a Boltzmann/Gibbs distribution with a free parameter corresponding to temperature. In our previous work, we verified empirically that the channel capacity appears to have a phase transition for small temperature and decays to zero for high temperature. In this paper, we make some progress toward theoretically explaining these phenomena. We first estimate the conditional entropy between the input sequence and the output fold, giving an upper bound which exhibits a phase transition with respect to temperature. Next, we formulate a class of parameter settings under which the dependence between walk energies is governed by their number of shared contacts. In this setting, we derive a lower bound on the conditional entropy. This lower bound allows us to conclude that the mutual information tends to zero in a nontrivial regime of high temperature, giving some support to the empirical fact regarding capacity. Finally, we construct an example setting of the parameters of the model for which the conditional entropy is exactly calculable and which does not exhibit a phase transition.

On the origin of protein superfamilies and superfolds

Distributions of protein families and folds in genomes are highly skewed, having a small number of prevalent superfamiles/superfolds and a large number of families/folds of a small size. Why are the distributions of protein families and folds skewed? Why are there only a limited number of protein families? Here, we employ an information theoretic approach to investigate the protein sequence-structure relationship that leads to the skewed distributions. We consider that protein sequences and folds constitute an information theoretic channel and computed the most efficient distribution of sequences that code all protein folds. The identified distributions of sequences and folds are found to follow a power law, consistent with those observed for proteins in nature. Importantly, the skewed distributions of sequences and folds are suggested to have different origins: the skewed distribution of sequences is due to evolutionary pressure to achieve efficient coding of necessary folds, whereas that of folds is based on the thermodynamic stability of folds. The current study provides a new information theoretic framework for proteins that could be widely applied for understanding protein sequences, structures, functions, and interactions.

A lattice protein with an amyloidogenic latent state: stability and folding kinetics

We have designed a model lattice protein that has two stable folded states, the lower free energy native state and a latent state of somewhat higher energy. The two states have a sizable part of their structures in common (two "alpha-helices") and differ in the content of "alpha-helices" and "beta-strands" in the rest of their structures; i.e. for the native state, this part is alpha-helical, and for the latent state it is composed of beta-strands. Thus, the lattice protein free energy surface mimics that of amyloidogenic proteins that form well organized fibrils under appropriate conditions. A Go-like potential was used and the folding process was simulated with a Monte Carlo method. To gain insight into the equilibrium free energy surface and the folding kinetics, we have combined standard approaches (reduced free energy surfaces, contact maps, time-dependent populations of the characteristic states, and folding time distributions) with a new approach. The latter is based on a principal coordinate analysis of the entire set of contacts, which makes possible the introduction of unbiased reaction coordinates and the construction of a kinetic network for the folding process. The system is found to have four characteristic basins, namely a semicompact globule, an on-pathway intermediate (the bifurcation basin), and the native and latent states. The bifurcation basin is shallow and consists of the structure common to the native and latent states, with the rest disorganized. On the basis of the simulation results, a simple kinetic model describing the transitions between the characteristic states was developed, and the rate constants for the essential transitions were estimated. During the folding process the system dwells in the bifurcation basin for a relatively short time before it proceeds to the native or latent state. We suggest that such a bifurcation may occur generally for proteins in which native and latent states have a sizable part of their structures in common. Moreover, there is the possibility of introducing changes in the system (e.g., mutations), which guide the system toward the native or misfolded state.

Temperature dependence and counter effect of the correlations of folding rate with chain length and with native topology

There is a controversy about the major determinants of the folding rate of small single-domain proteins. To shed light on this issue, we examined a possibility that the major determinants may change depending on temperature by conducting molecular dynamics simulations for 17 small single-domain proteins using an off-lattice Go-like model over a wide range of temperature. It was shown that the rank order of the folding rates is temperature dependent, which indicates that the major determinants are dependent on temperature. It was also found that as temperature is decreased, the correlation of the folding rate with the chain length becomes weakened, whereas that with the native topology becomes enhanced. Our simulation results, therefore, may provide a clue to reconcile the apparent controversy between the study by Plaxco based on experimental data and the previous theoretical and subsequent simulation studies: the former showed that the folding rate of two-state folders does not correlate with the chain length but correlates well with the native topology, whereas the latter showed that the folding rate does correlate with the chain length. We propose a possible scenario reconciling the controversy, explaining the reason why the correlation of the folding rate with the chain length became weakened and that with the native topology became enhanced with decreasing temperature.

Squeezed exponential kinetics to describe a nonglassy downhill folding as observed in a lattice protein model

We previously studied the so-called strange kinetics in the two-dimensional lattice HP model. To further study the strange kinetics, folding processes of a 27-mer cubic lattice protein model with Gō potential were investigated by simulating how the bundle of folding trajectories, consisting of a number of independent Monte Carlo simulations, evolves as the folding reaction proceeds, covering a wide range of temperature. Three realms of folding kinetics were observed depending on temperature. Although at temperatures where folding was two-state-like, the kinetics was conventional single exponential, we found that the time course data were well represented by a squeezed (or "shrunken") exponential function, exp [-(t/tau)beta] with beta > 1, at temperatures lower than the folding temperature, where folding was fastest and of a nonglassy downhill type. The squeezed exponential kinetics was found to pertain to the subdiffusion on the nonglassy downhill free energy surface and presents a marked contrast both to the single exponential kinetics and to the stretched exponential kinetics that was observed at lower temperatures where folding was also downhill but topological frustration came into effect. The observed temperature dependence of the folding kinetics suggests that some small single-domain proteins may follow the squeezed exponential kinetics at about the room temperature.

Close identity of a pressure-stabilized intermediate with a kinetic intermediate in protein folding

Atomic detailed structural study of a transiently existing folding intermediate is severely limited because of its short life. In ubiquitin, we found that a pressure-stabilized equilibrium conformer shares a common structural feature with the proline-trapped kinetic intermediate found in a pulse-labeling (1)H(2)H exchange NMR study [Briggs, M. S. & Roder, H. (1992) Proc. Natl. Acad. Sci. USA 89, 2017-2021]. The conformer is locally unfolded in the entire segment from residues 33 to 42 and in C-terminal residues 70-76. The close structural identity of an equilibrium intermediate stabilized under pressure with a transiently observed folding intermediate is likely to be general in terms of a folding funnel common to both experiments.

beta-Hairpins, alpha-helices, and the intermediates among the secondary structures in the energy landscape of a peptide from a distal beta-hairpin of SH3 domain

Energy landscape of a peptide, extracted from a distal beta-hairpin of src SH3 domain, in explicit water was obtained with the multicanonical molecular dynamics. A variety of beta-hairpins with various strand-strand hydrogen bonds were found in the energy landscape at 300 K. There was no energy barrier between random-coil and hairpins. Thus, the peptide conformation can easily change from the random-coil to the hairpins in the thermal fluctuations at 300 K. The landscape also included two clusters of alpha-helices, among which an energy barrier existed, and besides, these helix clusters were separated from the other conformations. Thus, the free-energy barrier exists among the helices and the other conformations. Intermediate clusters were found between the helix and the hairpin clusters. The current study showed that the isolated state of this peptide in water fluctuates among random-coil, beta-hairpin, and alpha-helix. In SH3 domain, which has a topology of mainly beta-protein, the whole-protein folding may proceed when the segment is folded in the beta-hairpin and the other parts of the protein are coupled with the beta-hairpin in an energetically or kinetically favorite way.