Folding Time Scales Research Articles

Energy-level statistics in the fine conformational resolution of RNA folding dynamics.

This work is aimed at determining the energy-level statistics of the fine resolution of soft-mode dynamics warranting an adiabatically simplified structural relaxation of a folding biopolymer chain. The parameters defining the intrabasin structure relaxation are specified for RNA, so that each Watson-Crick base-pairing pattern may be treated as a quasiequilibrium ensemble of substates or torsional isomers within relevant folding time scales. The temperature-dependent threshold for energy dispersion associated with the fine structure of each superbasin is determined so as to warrant the adiabatic entrainment of the torsional dynamics.

Early events in the folding of an amphipathic peptide: A multinanosecond molecular dynamics study

Folding of the capped LQQLLQQLLQL peptide is investigated at the water–hexane interface by molecular dynamics simulations for 161.5 ns. Initially placed in the aqueous phase as a β-strand, the peptide rapidly adsorbs to the interface, where it adopts an amphipathic conformation. The marginal presence of nonamphipathic structures throughout the complete trajectory indicates that the corresponding conformations are strongly disfavored at the interface. It is further suggestive that folding in an interfacial environment proceeds through a pathway of successive amphipathic intermediates. The energetic and entropic penalties involved in the conformational changes along this pathway markedly increase the folding time scales of LQQLLQQLLQL, explaining why the α-helix, the hypothesized lowest free energy structure for a sequence with a hydrophobic periodicity of 3.6, has not been reached yet. The formation of a type I β-turn at the end of the simulation confirms the importance of such motifs as initiation sites allowing the peptide to coalesce towards a secondary structure. Proteins 1999;36:383–399. © 1999 Wiley-Liss, Inc.

Early events in the folding of an amphipathic peptide: A multinanosecond molecular dynamics study

Folding of the capped LQQLLQQLLQL peptide is investigated at the water-hexane interface by molecular dynamics simulations for 161.5 ns. Initially placed in the aqueous phase as a beta-strand, the peptide rapidly adsorbs to the interface, where it adopts an amphipathic conformation. The marginal presence of nonamphipathic structures throughout the complete trajectory indicates that the corresponding conformations are strongly disfavored at the interface. It is further suggestive that folding in an interfacial environment proceeds through a pathway of successive amphipathic intermediates. The energetic and entropic penalties involved in the conformational changes along this pathway markedly increase the folding time scales of LQQLLQQLLQL, explaining why the alpha-helix, the hypothesized lowest free energy structure for a sequence with a hydrophobic periodicity of 3.6, has not been reached yet. The formation of a type I beta-turn at the end of the simulation confirms the importance of such motifs as initiation sites allowing the peptide to coalesce towards a secondary structure. Proteins 1999;36:383-399.

Deciphering the timescales and mechanisms of protein folding using minimal off-lattice models

Considerable insights into the mechanisms and timescales of protein folding have been obtained from detailed studies of minimal off-lattice models. These models are coarse-grained representations of polypeptide chains. Many novel predictions of the mechanisms and timescales of the folding of proteins have been made using simulations of off-lattice models. The concepts derived from these simulations have been used to analyze the recent experiments and simulations of proteins and peptides.

The topomer-sampling model of protein folding.

Clearly, a protein cannot sample all of its conformations (e.g., approximately 3(100) approximately 10(48) for a 100 residue protein) on an in vivo folding timescale (<1 s). To investigate how the conformational dynamics of a protein can accommodate subsecond folding time scales, we introduce the concept of the native topomer, which is the set of all structures similar to the native structure (obtainable from the native structure through local backbone coordinate transformations that do not disrupt the covalent bonding of the peptide backbone). We have developed a computational procedure for estimating the number of distinct topomers required to span all conformations (compact and semicompact) for a polypeptide of a given length. For 100 residues, we find approximately 3 x 10(7) distinct topomers. Based on the distance calculated between different topomers, we estimate that a 100-residue polypeptide diffusively samples one topomer every approximately 3 ns. Hence, a 100-residue protein can find its native topomer by random sampling in just approximately 100 ms. These results suggest that subsecond folding of modest-sized, single-domain proteins can be accomplished by a two-stage process of (i) topomer diffusion: random, diffusive sampling of the 3 x 10(7) distinct topomers to find the native topomer ( approximately 0.1 s), followed by (ii) intratopomer ordering: nonrandom, local conformational rearrangements within the native topomer to settle into the precise native state.

Molecular Dynamics of a 15-Residue Poly(l-alanine) in Water: Helix Formation and Energetics

We present a molecular dynamics study of the α-helix formation in a system consisting of a 15-residue poly(l-alanine) and surrounding water molecules. By applying a relatively high temperature, we observed the α-helix formation several times during a 17-ns run, and reversible helix−coil transitions were also observed. The α-helix formations were usually initiated by the β-turn structures. A crank-shaft-like motion of the peptide was included in the folding process. In the formed α-helical domains, substantial 310-helix formations were found especially at the termini, as observed by the NMR study. The folding time scale at room temperature estimated from our simulation was found to lie in the range of 100 ns, which is in accord with the time scale of the T-jump experiments. The total energy of the whole system was lower in the α-helix state than in the random-coil state by 20.4 ± 4.8 kcal/mol, which is consistent with the experimental value obtained by calorimetry. This energy decrease in forming the α-hel...

Coarsely defined solution to the protein folding problem

In this work we introduce a least-action formulation of the protein folding problem casted within a coarse description of the peptide chain dynamics. Thus, a binary coding of local topological costraints associated to each structural motif is introduced to computationally mimick the long-time dynamics. Folding pathways are resolved as transitions between patterns of locally encoded structural signals. Our variational approach is aimed at identifying the most economic pathway with respect to the stepwise cost in conformational freedom. Our treatment allows us to account for the expediency of the process in proteins effectively capable of in vitro renaturation. We identify the dominant pathway by introducing a coarse version of Lagrangian microscopic dynamics. The coarse folding pathways are generated by a parallel search for structural patterns in a matrix of local topological constraints (LTM) of the chain. Each local topological constraint represents a coarse description of a local torsional state and each pattern is evaluated, translated and finally recorded as a contact matrix (CM), an operation which is subject to a renormalization feedback loop. The renormalization operation periodically introduces long-range correlations on the LTM according to the latest CM generated by translation. Local topological constraints may form consensus regions in portions of the chain which translate as secondary structure motifs or tertiary interactions. Nucleation steps and cooperative effects are accounted for by means of the renormalization operation which warrants the persistence of seeding patterns upon successive LTM evaluations. Relevant folding timescales beyond the realm of molecular-dynamics simulations become accessible through the coarsely codified representation of local torsional constraints. The validity of our approach is tested vis-a-vis experimentally probed folding pathways generating tertiary interactions in proteins which may recover their active structure under in vitro renaturation conditions. We focus on determining significant folding intermediates and the late kinetic bottlenecks which occur within the first 1/100 s of the renaturation process.

Microscopic dynamics from a coarsely defined solution to the protein folding problem

In this work we introduce a least-action formulation of the protein folding problem casted within a coarse description of the soft mode dynamics of the peptide chain. Ultimately, we show that this coarse variational approach can be lifted to yield the microscopic long-time torsional dynamics responsible for the actual folding process. As a first step, a binary coding of local topological constraints associated to each structural motif is introduced to coarsely mimic the long-time dynamics. Folding pathways are initially resolved as transitions between patterns of locally encoded structural signals. Our variational approach is aimed at identifying the most economic pathway with respect to the stepwise cost in conformational freedom. Our treatment allows us to account for the expediency of the process in proteins effectively capable of in vitro renaturation. We identify the dominant pathway by introducing a coarse version of Lagrangian microscopic dynamics. The coarse folding pathways are generated by a parallel search for structural patterns in a matrix of local topological constraints (LTM) of the chain. Each local topological constraint represents a coarse description of a local torsional state and each pattern is evaluated, translated, and finally recorded as a contact matrix (CM), an operation that is subject to a renormalization feedback loop. The renormalization operation periodically introduces long-range correlations on the LTM according to the latest CM generated by translation. Local topological constraints may form consensus regions in portions of the chain that translate as secondary structure motifs or tertiary interactions. Nucleation steps and cooperative effects are accounted for by means of the renormalization operation, which warrants the persistence of seeding patterns upon successive LTM evaluations. Relevant folding time scales beyond the realm of molecular dynamics simulations become accessible through the coarsely codified representation of local torsional constraints. The validity of our approach is tested vis à vis experimentally probed folding pathways generating tertiary interactions in proteins that may recover their active structure under in vitro renaturation conditions. We focus on determining significant folding intermediates and the late kinetic bottlenecks that occur within the first 1100 s of the renaturation process. After the computational accessibility of this coarse solution of the folding problem becomes apparent, we show how to lift our variational problem to microscopic dynamics of the peptide chain. The consistency of our approach is revealed by an actual generation of Newton’s laws at the microscopic level through an inverse projection of the coarse dynamics originally generated through the pattern recognition computation.

Brownian dynamics simulations of protein folding: access to milliseconds time scale and beyond.

Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion-collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion-collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments.

Comment on “Chain Length Scaling of Protein Folding Time”

In a recent Letter, Gutin, Abkevich, and Shakhnovich (GAS) reported on a series of dynamical Monte Carlo simulations on lattice models of proteins. Based on these highly simplified models, they found that four different potential energies lead to four different folding time scales tau_f, where tau_f scales with chain length as N^lambda (see, also, Refs. [2-4]), with lambda varying from 2.7 to 6.0. However, due to the lack of microscopic models of protein folding dynamics, the interpretation and origin of the data have remained somewhat speculative. It is the purpose of this Comment to point out that the application of a simple "mesoscopic" model (cond-mat/9512019, PRL 77, 2324, 1996) of protein folding provides a full account of the data presented in their paper. Moreover, we find a major qualitative disagreement with the argumentative interpretation of GAS. Including, the origin of the dynamics, and size of the critical folding nucleus.

The Lagrangian Structure of Long-Time Torsional Dynamics Leading to RNA Folding

Within the context of biopolymer renaturation in vitro, a principle of maximization in the economy of the folding process has been previously formulated as the principle of sequential or stepwise minimization of conformational entropy loss (SMEL). When specialized to the RNA folding context, this principle leads to a predictive folding algorithm under the assumption that an “adiabatic approximation” is valid. This approximation requires that conformational microstates be lumped up into base-pairing patterns (BPPs) which are treated as quasiequilibrium states, while folding pathways are coarsely represented as sequences of BPP transitions. In this work, we develop a semiempirical microscopic treatment aimed at validating the adiabatic approximation and its underlying SMEL principle. We start by coarse-graining the conformation torsional space X = 3N-torus, with N = length of the chain, representing it as the lattice (Z2)3N, where Z2 = integers modulo 2. This is done so that each point in the lattice represents a complete set of local torsional isomeric states coarsely specifying the chain conformation. Then, a coarse Lagrangian governing the long-time dynamics of chain torsions is identified as the variational counterpart of the SMEL principle. To prove this statement, the Lagrangian computation of the coarse Shannon information entropy σ associated to the specific partition of X into BPPs is performed at different times and contrasted with the adiabatic computation, revealing (a) the subordination of torsional microstate dynamics to BPP transitions within time scales relevant to folding and (b) the coincidence of both plots in the range of folding time scales.

Molecular dynamics simulations of synthetic peptide folding.

Because the time scale of protein folding is much greater than that of the widely used simulations of native structures, a detailed report of molecular dynamics simulations of folding has not been available. In this study, we included the average solvent effect in the potential functions to simplify the calculation of the solvent effect and carried out long molecular dynamics simulations of the alanine-based synthetic peptides at 274 K. From either an extended or a randomly generated conformation, the simulations approached a helix-coil equilibrium in about 3 ns. The multiple minima problem did not prevent helix folding. The calculated helical ratio of Ac-AAQ-AAAAQAAAAQAAY-NH2 was 47%, in good agreement with the circular dichroism measurement (about 50%). A helical segment with frayed ends was the most stable conformation, but the hydrophobic interaction favored the compact, distorted helix-turn-helix conformations. The transition between the two types of conformations occurred in a much larger time scale than helix propagation. The transient hydrogen bonds between the glutamine side chain and the backbone carbonyl group could reduce the free energy barrier of helix folding and unfolding. The substitution of a single alanine residue in the middle of the peptide with valine or glycine decreased the average helical ratio significantly, in agreement with experimental observations.

The time dependence of chemical modification reveals slow steps in the folding of a group I ribozyme

L-21 ScaI ribozyme is a linear form of the self-splicing intron from the precursor of the Tetrahymena thermophila LSU intron. The time scales for tertiary folding of L-21 ScaI were investigated in two ways after bringing it from a partially denatured state at 60 degrees C and 1 mM Mg2+ to a renatured state of 15 degrees C and 10 mM Mg2+. First, formation of a catalytically active structure was monitored by measuring the kinetics of the reaction: p*CUCUA3 + G<==>p*CUCU + GA3. This reaction mimics the first step of splicing. After 1 min of folding time, the catalytic rate is roughly 10% of the rate attained after 8 h of folding. This indicates that much of the structure refolds quickly. Also, at least two time scales of folding are observed, separated by a lag time of about 30 min. To define the regions folding on various time scales, all the guanosines of L-21 ScaI were probed with kethoxal at 15 degrees C while folding was in progress. Based on folding time scales, the guanosines can be placed into at least four classes. These are guanosines that (1) are already protected at 60 degrees C in 1 mM Mg2+ or which fold immediately, (2) fold during the lag time, (3) continue to fold after 1 h, and (4) never fold. These results give insight into the folding pathway of a group I ribozyme at nucleotide resolution. This provides useful information on the regions whose foldings are important for catalytic function of the ribozyme. The method may also provide a general way to suggest regions of an RNA that may interact with each other to form tertiary structure.

Kinetics and thermodynamics of folding in model proteins.

Monte Carlo simulations on a class of lattice models are used to probe the thermodynamics and kinetics of protein folding. We find two transition temperatures: one at T theta, when chains collapse from a coil to a compact phase, and the other at Tf (< T theta), when chains adopt a conformation corresponding to their native state. The kinetics are probed by several correlation functions and are interpreted in terms of the underlying energy landscape. The transition from the coil to the native state occurs in three distinct stages. The initial stage corresponds to a random collapse of the protein chain. At intermediate times tau c, during which much of the native structure is acquired, there are multiple pathways. For longer times tau r (>> tau c) the decay is exponential, suggestive of a late transition state. The folding time scale (approximately tau r) varies greatly depending on the model. Implications of our results for in vitro folding of proteins are discussed.

Vertical dispersion of inertial waves in the upper ocean

A linear model of the vertical dispersion of near‐inertial waves is developed. A porosity distribution near the bottom of the computational domain minimizes bottom reflections and simulates an ocean of infinite depth. The model is used to show that the vertical dispersion of near‐inertial waves in the upper ocean may, under certain conditions, contribute significantly to the observed rapid decay of inertial oscillations in the surface layer. The kinetic energy of inertial oscillations at mid‐latitudes decays with an e folding time scale of 10 days or less, when the parameter λ(km)/N(cph)d(m) is less than or of the order of unity, where λ is the wavelength of the wind‐generated near‐inertial waves, N is the Väisälä frequency in the upper pycnocline, and d is the surface layer thickness. At the top of the pycnocline the model predicts a velocity maximum, which develops as energy propagates downward, out of the surface layer. However, when the upper pycnocline is sufficiently peaked, a resonant frequency interference effect is predicted. This effect modulates the dissipation of surface layer inertial oscillations, and their magnitude after a storm need not decay monotonically. We also make qualitative comparisons with deep‐ocean current meter observations taken during the Mixed Layer Experiment (MILE) and with shallow water (105 m) observations taken in the Baltic Sea.