Configurational subdiffusion of peptides: a network study

Fractional Extended Diffusion Theory to capture anomalous relaxation from biased/accelerated molecular simulations

Biased and accelerated molecular simulations (BAMS) are widely used tools to observe relevant molecular phenomena occurring on time scales inaccessible to standard molecular dynamics, but evaluation of the physical time scales involved in the processes is not directly possible from them. For this reason, the problem of recovering dynamics from such kinds of simulations is the object of very active research due to the relevant theoretical and practical implications of dynamics on the properties of both natural and synthetic molecular systems. In a recent paper [A. Rapallo et al., J. Comput. Chem. 42, 586-599 (2021)], it has been shown how the coupling of BAMS (which destroys the dynamics but allows to calculate average properties) with Extended Diffusion Theory (EDT) (which requires input appropriate equilibrium averages calculated over the BAMS trajectories) allows to effectively use the Smoluchowski equation to calculate the orientational time correlation function of the head-tail unit vector defined over a peptide in water solution. Orientational relaxation of this vector is the result of the coupling of internal molecular motions with overall molecular rotation, and it was very well described by correlation functions expressed in terms of weighted sums of suitable time-exponentially decaying functions, in agreement with a Brownian diffusive regime. However, situations occur where exponentially decaying functions are no longer appropriate to capture the actual dynamical behavior, which exhibits persistent long time correlations, compatible with the so called subdiffusive regimes. In this paper, a generalization of EDT will be given, exploiting a fractional Smoluchowski equation (FEDT) to capture the non-exponential character observed in the relaxation of intramolecular distances and molecular radius of gyration, whose dynamics depend on internal molecular motions only. The calculation methods, proper to EDT, are adapted to implement the generalization of the theory, and the resulting algorithm confirms FEDT as a tool of practical value in recovering dynamics from BAMS, to be used in general situations, involving both regular and anomalous diffusion regimes.

Role of ergodicity, aging, and Gaussianity in resolving the origins of biomolecule subdiffusion

The internal motions of biomolecules are essential to their function. Although biological macromolecules conventionally show subdiffusive dynamics, only recently has subdiffusion been associated with non-ergodicity. These findings have stimulated new questions in biophysics and statistical mechanics. Is non-ergodic subdiffusion a general strategy shared by biomolecules? What underlying mechanisms are responsible for it? Here, we performed extensive molecular dynamics (MD) simulations to characterize the internal dynamics of six different biomolecules, ranging from single or double-stranded DNA, a single domain protein (KRAS), two globular proteins (PGK and SHP2), to an intrinsically disordered protein (SNAP-25). We found that the subdiffusive behavior of these biomolecules falls into two classes. The internal motion of the first three cases is ergodic subdiffusion and can be interpreted by fractional Brownian motion (FBM), while the latter three cases involve non-ergodic subdiffusion and can be modeled by mixed origins of continuous-time random walk (CTRW) and FBM.

Non-ergodicity of a globular protein extending beyond its functional timescale

Internal motions of folded proteins have been assumed to be ergodic, i.e., that the dynamics of a single protein molecule averaged over a very long time resembles that of an ensemble. Here, by performing single-molecule fluorescence resonance energy transfer (smFRET) experiments and molecular dynamics (MD) simulations of a multi-domain globular protein, cytoplasmic protein-tyrosine phosphatase (SHP2), we demonstrate that the functional inter-domain motion is observationally non-ergodic over the time spans 10⁻¹² to 10⁻⁷ s and 10⁻¹ to 10² s. The difference between observational non-ergodicity and simple non-convergence is discussed. In comparison, a single-strand DNA of similar size behaves ergodically with an energy landscape resembling a one-dimensional linear chain. The observed non-ergodicity results from the hierarchical connectivity of the high-dimensional energy landscape of the protein molecule. As the characteristic time for the protein to conduct its dephosphorylation function is ∼10 s, our findings suggest that, due to the non-ergodicity, individual, seemingly identical protein molecules can be dynamically and functionally different.

Internal motions of folded proteins have been assumed to be ergodic, i.e., that the dynamics of a single protein molecule averaged over a very long time resembles that of an ensemble.

Origin of subdiffusions in proteins: Insight from peptide systems

Subdiffusive kinetics are popular in proteins and peptides as observed in experiments and simulations. For protein systems with diverse interactions, are there multiple mechanisms to produce the common subdiffusion behavior? To approach this problem, long trajectories of two model peptides are simulated to study the mechanism of subdiffusion and the relations with their interactions. The free-energy profiles and the subdiffusive kinetics are observed for these two peptides. A hierarchical plateau analysis is employed to extract the features of the landscape from the mean square of displacement. The mechanism of subdiffusions can be postulated by comparing the exponents by simulations with those based on various models. The results indicate that the mechanisms of these two peptides are different and are related to the characteristics of their energy landscapes. The subdiffusion of the flexible peptide is mainly caused by depth distribution of traps on the energy landscape, while the subdiffusion of the helical peptide is attributed to the fractal topology of local minima on the landscape. The emergence of these different mechanisms reflects different kinetic scenarios in peptide systems though the peptides behave in a similar way of diffusion. To confirm these ideas, the transition networks between various conformations of these peptides are generated. Based on the network description, the controlled kinetics based only on the topology of the networks are calculated and compared with the results based on simulations. For the flexible peptide, the feature of controlled diffusion is distinct from that of simulation, and for the helical peptide, two kinds of kinetics have a similar exponent of subdiffusion. These results further exemplify the importance of the landscape topology in the kinetics of structural proteins and the effect of depth distribution of traps for the subdiffusion of disordered peptides.