Non-Markovian properties and multiscale hidden Markovian network buried in single molecule time series

Nonlinear measurements of kinetics and generalized dynamical modes. I. Extracting the one-dimensional Green's function from a time series

Often, a single correlation function is used to measure the kinetics of a complex system. In contrast, a large set of k-vector modes and their correlation functions are commonly defined for motion in free space. This set can be transformed to the van Hove correlation function, which is the Green's function for molecular diffusion. Here, these ideas are generalized to other observables. A set of correlation functions of nonlinear functions of an observable is used to extract the corresponding Green's function. Although this paper focuses on nonlinear correlation functions of an equilibrium time series, the results are directly connected to other types of nonlinear kinetics, including perturbation-response experiments with strong fields. Generalized modes are defined as the orthogonal polynomials associated with the equilibrium distribution. A matrix of mode-correlation functions can be transformed to the complete, single-time-interval (1D) Green's function. Diagonalizing this matrix finds the eigendecays. To understand the advantages and limitation of this approach, Green's functions are calculated for a number of models of complex dynamics within a Gaussian probability distribution. Examples of non-diffusive motion, rate heterogeneity, and range heterogeneity are examined. General arguments are made that a full set of nonlinear 1D measurements is necessary to extract all the information available in a time series. However, when a process is neither dynamically Gaussian nor Markovian, they are not sufficient. In those cases, additional multidimensional measurements are needed.

Single-Molecule Förster Resonance Energy Transfer Measurement Reveals the Dynamic Partially Ordered Structure of the Epidermal Growth Factor Receptor C-Tail Domain

Intrinsically disordered proteins (IDPs) or regions (IDRs) are thought to exhibit unique functionalities without forming ordered structures. However, these molecular mechanisms are not easily elucidated, partly because of the difficultly of measuring structural information. In this study, we applied the alternative laser excitation (ALEX) method and circular dichroism (CD) spectroscopy to investigate the structure of the C-terminal tail (CTT) domain of the human epidermal growth factor receptor (EGFR). The single-molecule distributions of Förster resonance energy transfer (FRET) obtained by ALEX under solution conditions modified by the addition of potassium chloride (KCl), urea, or guanidinium chloride (GdmCl) allowed us to separately examine the influences of charge interactions and secondary structure formation. The CD spectrum analyses indicated the types of included secondary structure. The results suggested that the structure of the CTT is influenced by secondary structure formation, which is a principally antiparallel β-sheet, rather than by charge interactions and that phosphorylation of the major Grb2-binding sites partially denatures that secondary structure. Our findings suggest that the EGFR CTT might regulate ligand binding kinetics by local β-sheet formation or by the disruption associated with phosphorylation states.

Single-molecule fluorescence-based analysis of protein conformation, interaction, and oligomerization in cellular systems

Single-molecule imaging (SMI) of proteins in operation has a history of intensive investigations over 20 years and is now widely used in various fields of biology and biotechnology. We review the recent advances in SMI of fluorescently-tagged proteins in structural biology, focusing on technical applicability of SMI to the measurements in living cells. Basic technologies and recent applications of SMI in structural biology are introduced. Distinct from other methods in structural biology, SMI directly observes single molecules and single-molecule events one-by-one, thus, explicitly analyzing the distribution of protein structures and the history of protein dynamics. It also allows one to detect single events of protein interaction. One unique feature of SMI is that it is applicable in complicated and heterogeneous environments, including living cells. The numbers, location, movements, interaction, oligomerization, and conformation of single-protein molecules have been determined using SMI in cellular systems.

Error-based extraction of states and energy landscapes from experimental single-molecule time-series

Characterization of states, the essential components of the underlying energy landscapes, is one of the most intriguing subjects in single-molecule (SM) experiments due to the existence of noise inherent to the measurements. Here we present a method to extract the underlying state sequences from experimental SM time-series. Taking into account empirical error and the finite sampling of the time-series, the method extracts a steady-state network which provides an approximation of the underlying effective free energy landscape. The core of the method is the application of rate-distortion theory from information theory, allowing the individual data points to be assigned to multiple states simultaneously. We demonstrate the method's proficiency in its application to simulated trajectories as well as to experimental SM fluorescence resonance energy transfer (FRET) trajectories obtained from isolated agonist binding domains of the AMPA receptor, an ionotropic glutamate receptor that is prevalent in the central nervous system.

Single-molecule FRET studies of HIV TAR-DNA hairpin unfolding dynamics

We directly measure the dynamics of the HIV trans-activation response (TAR)-DNA hairpin with multiple loops using single-molecule Förster resonance energy transfer (smFRET) methods. Multiple FRET states are identified that correspond to intermediate melting states of the hairpin. The stability of each intermediate state is calculated from the smFRET data. The results indicate that hairpin unfolding obeys a "fraying and peeling" mechanism, and evidence for the collapse of the ends of the hairpin during folding is observed. These results suggest a possible biological function for hairpin loops serving as additional fraying centers to increase unfolding rates in otherwise stable systems. The experimental and analytical approaches developed in this article provide useful tools for studying the mechanism of multistate DNA hairpin dynamics and of other general systems with multiple parallel pathways of chemical reactions.