Search

The rate of protein folding is governed by the transition state so that a detailed characterization of its structure is essential for understanding the folding process. In vitro experiments have provided a coarse-grained description of the folding transition state ensemble (TSE) of small proteins. Atomistic details could be obtained by molecular dynamics (MD) simulations but it is not straightforward to extract the TSE directly from the MD trajectories, even for small peptides. Here, the structures in the TSE are isolated by the cut-based free-energy profile (cFEP) using the network whose nodes and links are configurations sampled by MD and direct transitions among them, respectively. The cFEP is a barrier-preserving projection that does not require arbitrarily chosen progress variables. First, a simple two-dimensional free-energy surface is used to illustrate the successful determination of the TSE by the cFEP approach and to explain the difficulty in defining boundary conditions of the Markov state model for an entropically stabilized free-energy minimum. The cFEP is then used to extract the TSE of a β-sheet peptide with a complex free-energy surface containing multiple basins and an entropic region. In contrast, Markov state models with boundary conditions defined by projected variables and conventional histogram-based free-energy profiles are not able to identify the TSE of the β-sheet peptide. [ABSTRACT FROM AUTHOR]

Multiple independent molecular dynamics (MD) simulations are often carried out starting from a single protein structure or a set of conformations that do not correspond to a thermodynamic ensemble. Therefore, a significant statistical bias is usually present in the Markov state model generated by simply combining the whole MD sampling into a network whose nodes and links are clusters of snapshots and transitions between them, respectively. Here, we introduce a depth-first search algorithm to extract from the whole conformation space network the largest ergodic component, i.e., the subset of nodes of the network whose transition matrix corresponds to an ergodic Markov chain. For multiple short MD simulations of a globular protein (as in distributed computing), the steady state, i.e., stationary distribution determined using the largest ergodic component, yields more accurate free energy profiles and mean first passage times than the original network or the ergodic network obtained by imposing detailed balance by means of symmetrization of the transition counts. [ABSTRACT FROM AUTHOR]

It is difficult to investigate folding kinetics by conventional atomistic simulations of proteins. The replica exchange molecular dynamics (REMD) simulation technique enhances conformational sampling at the expenses of reduced kinetic information, which in REMD is directly available only for very short time scales. Here, we propose a procedure for obtaining kinetic data from REMD by making use of the equilibrium transitions network (ETN) sampled at the temperature of interest. This information is supplemented by mean folding times extracted from ETNs at higher REMD temperatures and scaled according to the Arrhenius equation. The procedure is applied to a three-stranded antiparallel β-sheet peptide which has a very heterogeneous denatured state with a broad entropic basin and several enthalpic traps. Despite the complexity of the system and the REMD exchange time of only 0.1 ns, the procedure is able to estimate folding times (ranging from about 0.1 μs at the melting temperature of 330 K to about 8 μs at 286 K) as well as transition times from individual non-native basins to the native state. [ABSTRACT FROM AUTHOR]

Protein folding is governed by a complex free energy surface whose entropic contributions are relevant because of the large number of degrees of freedom involved. Such complexity, in particular the conformational heterogeneity of the denatured state, is hidden in projections onto one or two order parameters (e.g. fraction of native contacts and/or radius of gyration), which usually results in relatively smooth surfaces. Recent approaches borrowed from network and graph theory have yielded quantitative unprojected representations of the free energy surfaces of a β-hairpin and a three-stranded β-sheet peptide using equilibrium folding-unfolding molecular dynamics simulations. Interestingly, the network and graph analyses of these structured peptides have revealed a very heterogeneous denatured state ensemble. It includes high-enthalpy, high-entropy conformations with fluctuating non-native secondary structure, as well as low-enthalpy, low-entropy traps. [Copyright &y& Elsevier]

The conformation space of a 20 residue antiparallel β-sheet peptide, sampled by molecular dynamics simulations, is mapped to a network. Snapshots saved along the trajectory are grouped according to secondary structure into nodes of the network and the transitions between them are links. The conformation space network describes the significant free energy minima and their dynamic connectivity without requiring arbitrarily chosen reaction coordinates. As previously found for the Internet and the World-Wide Web as well as for social and biological networks, the conformation space network is scale-free and contains highly connected hubs like the native state which is the most populated free energy basin. Furthermore, the native basin exhibits a hierarchical organization, which is not found for a random heteropolymer lacking a predominant free-energy minimum. The network topology is used to identify conformations in the folding transition state (TS) ensemble, and provides a basis for understanding the heterogeneity of the TS and denatured state ensemble as well as the existence of multiple pathways. [Copyright &y& Elsevier]

Examines the molecular dynamics of protein folding in the transition state ensemble (TSE). Structures of putative TSE; Formation of side-chain interaction in two-state protein; Properties of the TSE.

Deals with folding simulations of a three-stranded antiparallel beta-sheet peptide. Reversible folding; Folding mechanism; Energy surface at 360 K.

The conformational space of a 20-residue three-stranded antiparallel β-sheet peptide (double hairpin) was sampled by equilibrium folding/unfolding molecular dynamics simulations for a total of 20 µs. The resulting one-dimensional free-energy profiles (FEPs) provide a detailed description of the free-energy basins and barriers for the folding reaction. The similarity of the FEPs obtained using the probability of folding before unfolding ( pfold) or the mean first passage time supports the robustness of the procedure. The folded state and the most populated free-energy basins in the denatured state are described by the one-dimensional FEPs, which avoid the overlap of states present in the usual one- or two-dimensional projections. Within the denatured state, a basin with fluctuating helical conformations and a heterogeneous entropic state are populated near the melting temperature at about 11% and 33%, respectively. Folding pathways from the helical basin or enthalpic traps (with only one of the two hairpins formed) reach the native state through the entropic state, which is on-pathway and is separated by a low barrier from the folded state. A simplified equilibrium kinetic network based on the FEPs shows the complexity of the folding reaction and indicates, as augmented by additional analyses, that the basins in the denatured state are connected primarily by the native state. The overall folding kinetics shows single-exponential behavior because barriers between the non-native basins and the folded state have similar heights. [ABSTRACT FROM AUTHOR]

The kinetics of biomolecular isomerization processes, such as protein folding, is governed by a free-energy surface of high dimensionality and complexity. As an alternative to projections into one or two dimensions, the free-energy surface can be mapped into a weighted network where nodes and links are configurations and direct transitions among them, respectively. In this work, the free-energy basins and barriers of the alanine dipeptide are determined quantitatively using an algorithm to partition the network into clusters (i.e., states) according to the equilibrium transitions sampled by molecular dynamics. The network-based approach allows for the analysis of the thermodynamics and kinetics of biomolecule isomerization without reliance on arbitrarily chosen order parameters. Moreover, it is shown on low-dimensional models, which can be treated analytically, as well as for the alanine dipeptide, that the broad-tailed weight distribution observed in their networks originates from free-energy basins with mainly enthalpic character. [ABSTRACT FROM AUTHOR]

The assumption that similar structures have similar folding probabilities (pfold) leads naturally to a procedure to evaluate pfold for every snapshot saved along an equilibrium folding-unfolding trajectory of a structured peptide or protein. The procedure utilizes a structurally homogeneous clustering and does not require any additional simulation. It can be used to detect multiple folding pathways as shown for a three-stranded antiparallel β-sheet peptide investigated by implicit solvent molecular dynamics simulations. [ABSTRACT FROM AUTHOR]

Repeat proteins are built of modules, each of which constitutes a structural motif. We have investigated whether fragments of a designed consensus armadillo repeat protein (ArmRP) recognize each other. We examined a split ArmRP consisting of an N-capping repeat (denoted Y), three internal repeats (M), and a C-capping repeat (A). We demonstrate that the C-terminal MA fragment adopts a fold similar to the corresponding part of the entire protein. In contrast, the N-terminal YM2 fragment constitutes a molten globule. The two fragments form a 1:1 YM2:MA complex with a nanomolar dissociation constant essentially identical to the crystal structure of the continuous YM3A protein. Molecular dynamics simulations show that the complex is structurally stable over a 1 μs timescale and reveal the importance of hydrophobic contacts across the interface. We propose that the existence of a stable complex recapitulates possible intermediates in the early evolution of these repeat proteins. [ABSTRACT FROM AUTHOR]

We propose a network-based method for determining basins and barriers of complex free energy surfaces (e.g., the protein folding landscape) from the time series of a single intramolecular distance. First, a network of transitions is constructed by clustering the points of the time series according to the short-time distribution of the signal. The transition network, which reflects the short-time kinetics, is then used for the iterative determination of individual basins by the minimum-cut-based free energy profile, a barrier-preserving one-dimensional projection of the free energy surface. The method is tested using the time series of a single Cβ−Cβdistance extracted from equilibrium molecular dynamics (MD) simulations of a structured peptide (20 residue three-stranded antiparallel β-sheet). Although the information of only one distance is employed to describe a system with 645 degrees of freedom, both the native state and the unfolding barrier of about 10 kJ/mol are determined with remarkable accuracy. Moreover, non-native conformers are identified by comparing long-time distributions of the same distance. To examine the applicability to single-molecule Förster resonance energy transfer (FRET) experiments, a time series of donor and acceptor photons is generated using the MD trajectory. The native state of the β-sheet peptide is determined accurately from the emulated FRET signal. Applied to real single-molecule FRET measurements on a monomeric variant of λ-repressor, the network-based method correctly identifies the folded and unfolded populations, which are clearly separated in the minimum-cut-based free energy profile. [ABSTRACT FROM AUTHOR]

Multiple 4-μs molecular dynamics (MD) simulations are used to study the folding process of the cross-linked α-helical peptide Ac-EACAR5EAAAR10EAACR15Q-NH2(EAAAR peptide). The folding kinetics are single exponential at 330 K, while they are complex at 281 K with a clear deviation from single-exponential behavior, in agreement with time-resolved infrared (IR) spectroscopy measurements. Network analysis of the conformation space sampled by the MD simulations reveals four main folding channels which start from conformations with partially formed helical structure and non-native salt-bridges in a kinetically partitioned unfolded state. The independent folding pathways explain the comparable quality of models based on stretched exponential and multiexponential fitting of the kinetic traces at low temperature. The rearrangement of bulky side chains, and in particular their reorientation with respect to the cross-linker, makes the EAAAR peptide a slower folder at 281 K than a similar peptide devoid of the three glutamate side chains. On the basis of this simulation result, extracted from a total MD sampling of 1.0 ms, a mutant with additional bulky side chains (three methionines replacing alanines at positions 2, 7, and 12) is suggested to fold slower than the EAAAR peptide. This prediction is confirmed by time-resolved IR spectroscopy. [ABSTRACT FROM AUTHOR]

In the recently proposed distributed computing approach to protein folding a very large number of short independent simulations is performed. Using this method, folding events on a time scale orders of magnitude shorter than the experimental one have been reported. However, it has also been observed that the folding process is not an elementary kinetic step and that the presence of initial lag phases can bias short simulations toward atypical pathways. We study here a 20-residue three-stranded antiparallel β-sheet peptide whose equilibrium properties can be characterized by atomistic molecular dynamics simulations. We found that the folding rate of this peptide is estimated correctly by the distributed computing approach when trajectories > ≈ 1/100 of the equilibrium folding time are considered. We also found that the fastest folding events occur through high-energy pathways, which are unlikely under equilibrium conditions. These very fast folding pathways do not relax within the equilibrium denatured state that is stabilized by the transient presence of both native and nonnative interactions, and they are characterized by the nearly simultaneous formation of the two β-hairpins and a very small number of non-native contacts. [ABSTRACT FROM AUTHOR]