How well do reduced models capture the dynamics in models of interacting neurons?

This paper introduces a class of stochastic models of interacting neurons with emergent dynamics similar to those seen in local cortical populations. Rigorous results on existence and uniqueness of nonequilibrium steady states are proved. These network models are then compared to very simple reduced models driven by the same mean excitatory and inhibitory currents. Discrepancies in firing rates between network and reduced models are investigated and explained by correlations in spiking, or partial synchronization, working in concert with “nonlinearities” in the time evolution of membrane potentials. The use of simple random walks and their first passage times to simulate fluctuations in neuronal membrane potentials and interspike times is also considered.

     

Fluorescence correlation spectroscopy: past, present, future

In recent years fluorescence correlation spectroscopy (FCS) has become a routine method for determining diffusion coefficients, chemical rate constants, molecular concentrations, fluorescence brightness, triplet state lifetimes, and other molecular parameters. FCS measures the spatial and temporal correlation of individual molecules with themselves and so provides a bridge between classical ensemble and contemporary single-molecule measurements. It also provides information on concentration and molecular number fluctuations for nonlinear reaction systems that complement single-molecule measurements. Typically implemented on a fluorescence microscope, FCS samples femtoliter volumes and so is especially useful for characterizing small dynamic systems such as biological cells. In addition to its practical utility, however, FCS provides a window on mesoscopic systems in which fluctuations from steady states not only provide the basis for the measurement but also can have important consequences for the behavior and evolution of the system. For example, a new and potentially interesting field for FCS studies could be the study of nonequilibrium steady states, especially in living cells.     

Protein aggregation determinants from a simplified model: cooperative folders resist aggregation

Two-chain aggregation simulations using minimalist models of proteins G, L, and mutants were used to investigate the fundamentals of protein aggregation. Mutations were selected to break up repeats of hydrophobic beads in the sequence while maintaining native topology and folding ability. Data are collected under conditions in which all chain types have similar folded populations and after equilibrating the separated chains to minimize competition between folding and aggregation. Folding cooperativity stands out as the best single-chain determinant under these conditions and for these simple models. It can be experimentally measured by the width of the unfolding transition during thermal denaturation and loosely related to population of intermediate-like states during folding. Additional measures of cooperativity and other properties such as radius of gyration fluctuations and patterning of hydrophobic residues are also examined. Initial contact system states with transition-state characteristics can be identified and are more expanded than average initial contact states. Two-chain minimalist model aggregates are considerably less structured than their native states and have minimal domain-swapping features.     

Long-range electrostatic effects on peptide folding.

The reversible folding/unfolding of a short peptide in solution is studied by molecular dynamics simulations. The effects of long-range electrostatic interactions are examined and found to be important both for the equilibrium between folded and unfolded states and the dynamics of the folding process. The neglect of long-range electrostatics leads to an increased population of unfolded states and increased structural fluctuations. When such interactions are taken into account, the peptide unfolds and folds to the experimentally determined structure several times during a 25 ns trajectory, with approximately equal populations of folded and unfolded states in the neighborhood of its proposed melting temperature. The effect of using spherical boundary conditions rather than periodic ones does not appear to have any major effect on the folding dynamics.     

Metastable, partially folded states in the productive folding and in the misfolding and amyloid aggregation of proteins

Understanding the energetic and structural basis of protein folding in a physiological context may represent an important step toward the elucidation of protein misfolding and aggregation events that take place in several pathological states. In particular, investigation of the structure and thermodynamic properties of partially folded intermediate states involved in productive folding or in misfolding/aggregation may provide insight into these processes and suggest novel approaches to prevent misfolding in living organisms. This goal, however, has remained elusive, because such intermediates are often transient and correspond to metastable states that are little populated under physiological conditions. Characterization of these states requires their stabilization by means of manipulation of the experimental conditions, involving changes in temperature, pH, or addition of different types of denaturants. In the past few years, hydrostatic pressure has been increasingly used as a thermodynamic variable in the study of both protein folding and misfolding/aggregation transitions. Compared with other chemical or physical denaturing agents, a unique feature of pressure is its ability to induce subtle changes in protein conformation, allowing the stabilization of partially folded states that are usually not significantly populated under more drastic conditions. Much of the recent work in this field has focused on the characterization of folding intermediates, because they seem to be involved in a variety of disease-causing protein misfolding and aggregation reactions. Here, we review recent examples of the use of hydrostatic pressure as a tool to gain insight into the forces and energetics governing the productive folding or the misfolding and amyloid aggregation of proteions.     

Dynamical Phase Transitions Reveal Amyloid-like States on Protein Folding Landscapes

Developing an understanding of protein misfolding processes presents a crucial challenge for unlocking the mysteries of human disease. In this article, we present our observations of β-sheet-rich misfolded states on a number of protein dynamical landscapes investigated through molecular dynamics simulation and Markov state models. We employ a nonequilibrium statistical mechanical theory to identify the glassy states in a protein’s dynamics, and we discuss the nonnative, β-sheet-rich states that play a distinct role in the slowest dynamics within seven protein folding systems. We highlight the fundamental similarity between these states and the amyloid structures responsible for many neurodegenerative diseases, and we discuss potential consequences for mechanisms of protein aggregation and intermolecular amyloid formation.     

Molecular crowders and cosolutes promote folding cooperativity of RNA under physiological ionic conditions

Folding mechanisms of functional RNAs under idealized in vitro conditions of dilute solution and high ionic strength have been well studied. Comparatively little is known, however, about mechanisms for folding of RNA in vivo where Mg²⁺ ion concentrations are low, K⁺ concentrations are modest, and concentrations of macromolecular crowders and low-molecular-weight cosolutes are high. Herein, we apply a combination of biophysical and structure mapping techniques to tRNA to elucidate thermodynamic and functional principles that govern RNA folding under in vivo–like conditions. We show by thermal denaturation and SHAPE studies that tRNA folding cooperativity increases in physiologically low concentrations of Mg²⁺ (0.5–2 mM) and K⁺ (140 mM) if the solution is supplemented with physiological amounts (∼20%) of a water-soluble neutral macromolecular crowding agent such as PEG or dextran. Low-molecular-weight cosolutes show varying effects on tRNA folding cooperativity, increasing or decreasing it based on the identity of the cosolute. For those additives that increase folding cooperativity, the gain is manifested in sharpened two-state-like folding transitions for full-length tRNA over its secondary structural elements. Temperature-dependent SHAPE experiments in the absence and presence of crowders and cosolutes reveal extent of cooperative folding of tRNA on a nucleotide basis and are consistent with the melting studies. Mechanistically, crowding agents appear to promote cooperativity by stabilizing tertiary structure, while those low molecular cosolutes that promote cooperativity stabilize tertiary structure and/or destabilize secondary structure. Cooperative folding of functional RNA under physiological-like conditions parallels the behavior of many proteins and has implications for cellular RNA folding kinetics and evolution.     

Entropy and barrier-controlled fluctuations determine conformational viscoelasticity of single biomolecules

Biological macromolecules have complex and nontrivial energy landscapes, endowing them with a unique conformational adaptability and diversity in function. Hence, understanding the processes of elasticity and dissipation at the nanoscale is important to molecular biology and emerging fields such as nanotechnology. Here we analyze single molecule fluctuations in an atomic force microscope, using a generic model of biopolymer viscoelasticity that includes local "internal" conformational dissipation. Comparing two biopolymers, dextran and cellulose (polysaccharides with and without local bistable transitions), demonstrates that signatures of simple conformational change are minima in both the elastic and internal friction constants around a characteristic force. A novel analysis of dynamics on a bistable energy landscape provides a simple explanation: an elasticity driven by the entropy, and friction by a barrier-controlled hopping time of populations between states, which is surprisingly distinct to the well-known relaxation time. This nonequilibrium microscopic analysis thus provides a means of quantifying new dynamical features of the energy landscape of the glucopyranose ring, revealing an unexpected underlying roughness and information on the shape of the barrier of the chair-boat transition in dextran. The results presented herein provide a basis toward probing the viscoelasticity of macromolecular conformational transitions on more complex energy landscapes, such as during protein folding.     

Deconvoluting Protein (Un)folding Structural Ensembles Using X-Ray Scattering,Nuclear Magnetic Resonance Spectroscopy and Molecular Dynamics Simulation

The folding and unfolding of protein domains is an apparently cooperative process, but transient intermediates have been detected in some cases. Such (un)folding intermediates are challenging to investigate structurally as they are typically not long-lived and their role in the (un)folding reaction has often been questioned. One of the most well studied (un)folding pathways is that of Drosophila melanogaster Engrailed homeodomain (EnHD): this 61-residue protein forms a three helix bundle in the native state and folds via a helical intermediate. Here we used molecular dynamics simulations to derive sample conformations of EnHD in the native, intermediate, and unfolded states and selected the relevant structural clusters by comparing to small/wide angle X-ray scattering data at four different temperatures. The results are corroborated using residual dipolar couplings determined by NMR spectroscopy. Our results agree well with the previously proposed (un)folding pathway. However, they also suggest that the fully unfolded state is present at a low fraction throughout the investigated temperature interval, and that the (un)folding intermediate is highly populated at the thermal midpoint in line with the view that this intermediate can be regarded to be the denatured state under physiological conditions. Further, the combination of ensemble structural techniques with MD allows for determination of structures and populations of multiple interconverting structures in solution.     

The nature of folded states of globular proteins.

We suggest, using dynamical simulations of a simple heteropolymer modelling the alpha-carbon sequence in a protein, that generically the folded states of globular proteins correspond to statistically well-defined metastable states. This hypothesis, called the metastability hypothesis, states that there are several free energy minima separated by barriers of various heights such that the folded conformations of a polypeptide chain in each of the minima have similar structural characteristics but have different energies from one another. The calculated structural characteristics, such as bond angle and dihedral angle distribution functions, are assumed to arise from only those configurations belonging to a given minimum. The validity of this hypothesis is illustrated by simulations of a continuum model of a heteropolymer whose low temperature state is a well-defined beta-barrel structure. The simulations were done using a molecular dynamics algorithm (referred to as the "noisy" molecular dynamics method) containing both friction and noise terms. It is shown that for this model there are several distinct metastable minima in which the structural features are similar. Several new methods of analyzing fluctuations in structures belonging to two distinct minima are introduced. The most notable one is a dynamic measure of compactness that can in principle provide the time required for maximal compactness to be achieved. The analysis shows that for a given metastable state in which the protein has a well-defined folded structure the transition to a state of higher compactness occurs very slowly, lending credence to the notion that the system encounters a late barrier in the process of folding to the most compact structure. The examination of the fluctuations in the structures near the unfolding----folding transition temperature indicates that the transition state for the unfolding to folding process occurs closer to the folded state.     

Dynamical Phase Transitions Reveal Amyloid-like States on Protein Folding Landscapes

Developing an understanding of protein misfolding processes presents a crucial challenge for unlocking the mysteries of human disease. In this article, we present our observations of β-sheet-rich misfolded states on a number of protein dynamical landscapes investigated through molecular dynamics simulation and Markov state models. We employ a nonequilibrium statistical mechanical theory to identify the glassy states in a protein’s dynamics, and we discuss the nonnative, β-sheet-rich states that play a distinct role in the slowest dynamics within seven protein folding systems. We highlight the fundamental similarity between these states and the amyloid structures responsible for many neurodegenerative diseases, and we discuss potential consequences for mechanisms of protein aggregation and intermolecular amyloid formation.     

Perspectives on the landscape and flux theory for describing emergent behaviors of the biological systems

We give a review on the landscape theory of the equilibrium biological systems and landscape-flux theory of the nonequilibrium biological systems as the global driving force. The emergences of the behaviors, the associated thermodynamics in terms of the entropy and free energy and dynamics in terms of the rate and paths have been quantitatively demonstrated. The hierarchical organization structures have been discussed. The biological applications ranging from protein folding, biomolecular recognition, specificity, biomolecular evolution and design for equilibrium systems as well as cell cycle, differentiation and development, cancer, neural networks and brain function, and evolution for nonequilibrium systems, cross-scale studies of genome structural dynamics and experimental quantifications/verifications of the landscape and flux are illustrated. Together, this gives an overall global physical and quantitative picture in terms of the landscape and flux for the behaviors, dynamics and functions of biological systems.

     

Direct Observation of Downhill Folding of λ-Repressor in a Microfluidic Mixer

The protein λ_6-85 has been implicated in barrierless folding by observations of kinetic relaxation after nanosecond T-jump. In this work we observed folding of this protein after dilution of a high denaturant in an ultrarapid microfluidic mixer at temperatures far below the thermal midpoint. The observations of total intensity and spectral shift of tryptophan fluorescence yielded distinctly different kinetics and activation energies. These results may be explained as diffusion on a low-barrier, one-dimensional, free-energy surface, with different probes having different sensitivities along the reaction coordinate. Additionally, we observed an extremely fast phase within the mixing time that was not observed by T-jump, suggesting that the ensemble of unfolded states populated at high denaturant is distinct from those accessible at high temperature.     

The effect of alumina nanoparticles on the thermal properties of PMMA: a molecular dynamics simulation

Metal oxides, as one of the most promising flame retardant additives, improve the fire retardant and the thermal stability properties of polymers. In the present study, molecular dynamics (MD) simulations based on the united atom model were applied to study the effect of alumina nanoparticles on the density, thermal conductivity, heat capacity, and thermal diffusivity of isotactic poly(methyl methacrylate) (is-PMMA). Thermal diffusivity of PMMA and PMMA/alumina nanocomposite were investigated through calculating thermal conductivity, density and heat capacity in the range of 300–700?K. Heat capacity can be calculated using fluctuations properties of energy. Thermal conductivity was calculated through the nonequilibrium molecular dynamics (NEMD) simulation by Fourier’s law approach. Our results show that the addition of alumina nanoparticles decreases the heat capacity and increases the glass transition temperature (T_g), thermal conductivity and thermal diffusivity of the PMMA. Therefore, the addition of alumina nanoparticles to PMMA improves the fire retardancy of the polymer. In addition, we illustrate the links between the intermolecular and bulk properties of PMMA in the presence of the alumina nanoparticles.     

Nonequilibrium fluctuations of mechanically stretched single red blood cells detected by optical tweezers

We study the thermal and out-of-equilibrium mechanical dynamics of single, living human red blood cells (RBCs) by combining two-probe passive and active microrheology techniques. Both experiments were performed quasisimultaneously on the same cell using two identical polystyrene probes, biochemically attached to the cell membrane. We obtained compelling evidence of nonequilibrium fluctuations in the RBCs under physiological condition and without the influence of any external chemicals. The spectral distributions of metabolically driven forces and viscoelastic response were evaluated in the relaxed and stretched states, intended to simulate the varying natural environment of the cells during blood circulation. We found that the internally generated forces are more pronounced in the stretched state, suggesting a stress-dependent RBC activity.     

Inhibited complete folding of consecutive human telomeric G-quadruplexes

Noncanonical DNA structures, termed G-quadruplexes, are present in human genomic DNA and are important elements in many DNA metabolic processes. Multiple sites in the human genome have G-rich DNA stretches able to support formation of several consecutive G-quadruplexes. One of those sites is the telomeric overhang region that has multiple repeats of TTAGGG and is tightly associated with both cancer and aging. We investigated the folding of consecutive G-quadruplexes in both potassium- and sodium-containing solutions using single-molecule FRET spectroscopy, circular dichroism, thermal melting and molecular dynamics simulations. Our observations show coexistence of partially and fully folded DNA, the latter consisting of consecutive G-quadruplexes. Following the folding process over hours in sodium-containing buffers revealed fast G-quadruplex folding but slow establishment of thermodynamic equilibrium. We find that full consecutive G-quadruplex formation is inhibited by the many DNA structures randomly nucleating on the DNA, some of which are off-path conformations that need to unfold to allow full folding. Our study allows describing consecutive G-quadruplex formation in both nonequilibrium and equilibrium conditions by a unified picture, where, due to the many possible DNA conformations, full folding with consecutive G-quadruplexes as beads on a string is not necessarily achieved.     

Predicting Atomic Details of the Unfolding Pathway for YibK,a Knotted Protein from the SPOUT Superfamily

Abstract

Several protein structures have been reported to contain intricate knots of the polypeptide backbone but the mechanism of the (un)folding process of knotted proteins remains unknown. The members of the SPOUT superfamily of RNA methyltransferases are some of the most intensely studied systems for investigation of the knot formation and function. YibK (whose biochemical function remains unknown) is the representative protein of the SPOUT superfamily. This protein exhibits a deep trefoil knot at the C-terminus.

We conducted an extensive computational analysis of the unfolding process for the monomeric form of YibK. In order to predict the (un)folding pathway of YibK, we have calculated the order of secondary structure disassembly using UNFOLD, and performed thermal unfolding simulations using classical Molecular Dynamics (MD), as well as simulations employing reduced representation of the peptide chain using either MD with the UNRES method or the Monte Carlo (MC) unfolding with the REFINER method.

Results obtained from all methods used in this work are in qualitative agreement. We found that YibK unfolds through four intermediate states. The trefoil knot in YibK disappears at the end of the unfolding process, long after the protein loses its native topology. We observed that the C-terminus leaves the knotting loop folded into a hairpin-like structure, in agreement with the results of coarse-grained simulation reported earlier. We propose that the folding pathway of YibK corresponds to the reversed sequence of events observed in the unfolding pathway elucidated in this study. Thus, we predict that the knot formation is the slowest part of the YibK folding process.     

Optimal Control of Transitions between Nonequilibrium Steady States

Biological systems fundamentally exist out of equilibrium in order to preserve organized structures and processes. Many changing cellular conditions can be represented as transitions between nonequilibrium steady states, and organisms have an interest in optimizing such transitions. Using the Hatano-Sasa Y-value, we extend a recently developed geometrical framework for determining optimal protocols so that it can be applied to systems driven from nonequilibrium steady states. We calculate and numerically verify optimal protocols for a colloidal particle dragged through solution by a translating optical trap with two controllable parameters. We offer experimental predictions, specifically that optimal protocols are significantly less costly than naive ones. Optimal protocols similar to these may ultimately point to design principles for biological energy transduction systems and guide the design of artificial molecular machines.     

Protein flexibility and conformational states of Leishmania antigen eIF-4A: identification of a novel plausible protein adjuvant using comparative genomics and molecular modeling

Recent homology modeling studies have identified specific residues (epitope) of the Leishmania RNA helicase protein (LmeIF) that stimulates production of IL-12 cytokine. However, question remains concerning how LmeIF’s N-terminal moiety initiates adjuvant effects. Extensive molecular modeling combining the normal mode analysis (NMA) and molecular dynamics simulations, in the present study, has demonstrated that the LmeIF structure may exist in two different forms corresponding to the extended and collapsed (closed) states of the entire structure. The computational results showed that the two domains of the LmeIF structure tend to undergo large fluctuations in a concerted fashion and have strong effect on the solvent accessible surface of the epitope situated on the N-terminal structure. The conformational freedom of the C-terminal domains may explain why the entire LmeIF protein is not as active as the N-terminal moiety. Thereafter, a comparative genome analysis with subsequent homology modeling and molecular electrostatic potential (MEP) techniques allowed us to predict a novel and plausible RNA helicase (LI-helicase) from the Listeria source with adjuvant property as observed for the Leishmania eIF-4A protein. The structural folding and MEP maps revealed similar topologies of the epitope of both LmeIF and LI-helicase proteins and striking identity in the local disposition of the charged groups.

An animated Interactive 3D Complement (I3DC) is available in Proteopedia at http://proteopedia.org/w/Journal:JBSD:7