Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories

Single-molecule FRET and molecular diffusion analysis characterize stable oligomers of amyloid-β 42 of extremely low population

Soluble oligomers produced during protein aggregation have been thought to be toxic species causing various diseases. Characterization of these oligomers is difficult because oligomers are a heterogeneous mixture, which is not readily separable, and may appear transiently during aggregation. Single-molecule spectroscopy can provide valuable information by detecting individual oligomers, but there have been various problems in determining the size and concentration of oligomers. In this work, we develop and use a method that analyzes single-molecule fluorescence burst data of freely diffusing molecules in solution based on molecular diffusion theory and maximum likelihood method. We demonstrate that the photon count rate, diffusion time, population, and Förster resonance energy transfer (FRET) efficiency can be accurately determined from simulated data and the experimental data of a known oligomerization system, the tetramerization domain of p53. We used this method to characterize the oligomers of the 42-residue amyloid-β (Aβ42) peptide. Combining peptide incubation in a plate reader and single-molecule free-diffusion experiments allows for the detection of stable oligomers appearing at various stages of aggregation. We find that the average size of these oligomers is 70-mer and their overall population is very low, less than 1 nM, in the early and middle stages of aggregation of 1 µM Aβ42 peptide. Based on their average size and long diffusion time, we predict the oligomers have a highly elongated rod-like shape.

Microscopic origin of diffusive dynamics in the context of transition path time distributions for protein folding and unfolding

Experimentally measured transition path time distributions are usually analyzed theoretically in terms of a diffusion equation over a free energy barrier. It is though well understood that the free energy profile separating the folded and unfolded states of a protein is characterized as a transition through many stable micro-states which exist between the folded and unfolded states. Why is it then justified to model the transition path dynamics in terms of a diffusion equation, namely the Smoluchowski equation (SE)? In principle, van Kampen has shown that a nearest neighbor Markov chain of thermal jumps between neighboring microstates will lead in a continuum limit to the SE, such that the friction coefficient is proportional to the mean residence time in each micro-state. However, the practical question of how many microstates are needed to justify modeling the transition path dynamics in terms of an SE has not been addressed. This is a central topic of this paper where we compare numerical results for transition paths based on the diffusion equation on the one hand and the nearest neighbor Markov jump model on the other. Comparison of the transition path time distributions shows that one needs at least a few dozen microstates to obtain reasonable agreement between the two approaches. Using the Markov nearest neighbor model one also obtains good agreement with the experimentally measured transition path time distributions for a DNA hairpin and PrP protein. As found previously when using the diffusion equation, the Markov chain model used here also reproduces the experimentally measured long time tail and confirms that the transition path barrier height is ∼3k_BT. This study indicates that in the future, when attempting to model experimentally measured transition path time distributions, one should perhaps prefer a nearest neighbor Markov model which is well defined also for rough energy landscapes. Such studies can also shed light on the minimal number of microstates needed to unravel the experimental data.

The Effect of Time Resolution on Apparent Transition Path Times Observed in Single-Molecule Studies of Biomolecules

Single-molecule experiments have now achieved a time resolution allowing observation of transition paths, the brief trajectory segments where the molecule undergoing an unfolding or folding transition enters the energetically or entropically unfavorable barrier region from the folded/unfolded side and exits to the unfolded/folded side, thereby completing the transition. This resolution, however, is yet insufficient to identify the precise entrance/exit events that mark the beginning and the end of a transition path: the nature of the diffusive dynamics is such that a molecular trajectory will recross the boundary between the barrier region and the folded/unfolded state, multiple times, at a time scale much shorter than that of the typical experimental resolution. Here we use theory and Brownian dynamics simulations to show that, as a result of such recrossings, the apparent transition path times are generally longer than the true ones. We quantify this effect using a simple model where the observed dynamics is a moving average of the true dynamics and discuss experimental implications of our results.

Effects of Hydrodynamic Backflow on the Transmission Coefficient of a Barrier-Crossing Brownian Particle

The slow power law decay of the velocity autocorrelation function of a particle moving stochastically in a condensed-phase fluid is widely attributed to the momentum that fluid molecules displaced by the particle transfer back to it during the course of its motion. The forces created by this backflow effect are known as Basset forces, and they have been found in recent analytical work and numerical simulations to be implicated in a number of interesting dynamical phenomena, including boosted particle mobility in tilted washboard potentials. Motivated by these findings, the present paper is an investigation of the role of backflow in thermally activated barrier crossing, the governing process in essentially all condensed-phase chemical reactions. More specifically, it is an exact analytical calculation, carried out within the framework of the reactive-flux formalism, of the transmission coefficient κ(t) of a Brownian particle that crosses an inverted parabola under the influence of a colored noise process originating in the Basset force and a Markovian time-local friction. The calculation establishes that κ(t) is significantly enhanced over its backflow-free limit.