Limitations of field-theory simulation for exploring phase separation: The role of repulsion in a lattice protein model.

Field-theory simulation by the complex Langevin method offers an alternative to conventional sampling techniques for exploring the forces driving biomolecular liquid-liquid phase separation. Such simulations have recently been used to study several polyampholyte systems. Here, we formulate a field theory corresponding to the hydrophobic/polar (HP) lattice protein model, with finite same-site repulsion and nearest-neighbor attraction between HH bead pairs. By direct comparison with particle-based Monte Carlo simulations, we show that complex Langevin sampling of the field theory reproduces the thermodynamic properties of the HP model only if the same-site repulsion is not too strong. Unfortunately, the repulsion has to be taken weaker than what is needed to prevent condensed droplets from assuming an artificially compact shape. Analysis of a minimal and analytically solvable toy model hints that the sampling problems caused by repulsive interaction may stem from loss of ergodicity.

Stability and Local Unfolding of SOD1 in the Presence of Protein Crowders.

Using NMR and Monte Carlo (MC) methods, we investigate the stability and dynamics of superoxide dismutase 1 (SOD1) in homogeneous crowding environments, where either bovine pancreatic trypsin inhibitor (BPTI) or the B1 domain of streptococcal protein G (PGB1) serves as a crowding agent. By NMR, we show that both crowders, and especially BPTI, cause a drastic loss in the overall stability of SOD1 in its apo monomeric form. Additionally, we determine chemical shift perturbations indicating that SOD1 interacts with the crowder proteins in a residue-specific manner that further depends on the identity of the crowding protein. Furthermore, the specificity of SOD1-crowder interactions is reciprocal: chemical shift perturbations on BPTI and PGB1 identify regions that interact preferentially with SOD1. By MC simulations, we investigate the local unfolding of SOD1 in the absence and presence of the crowders. We find that the crowders primarily interact with the long flexible loops of the folded SOD1 monomer. The basic mechanisms by which the SOD1 ß-barrel core unfolds remain unchanged when adding the crowders. In particular, both with and without the crowders, the second ß-sheet of the barrel is more dynamic and unfolding-prone than the first. Notably, the MC simulations (exploring the early stages of SOD1 unfolding) and the NMR experiments (under equilibrium conditions) identify largely the same set of PGB1 and BPTI residues as prone to form SOD1 contacts. Thus, contacts stabilizing the unfolded state of SOD1 in many cases appear to form early in the unfolding reaction.

Markov modeling of peptide folding in the presence of protein crowders.

We use Markov state models (MSMs) to analyze the dynamics of a ß-hairpin-forming peptide in Monte Carlo (MC) simulations with interacting protein crowders, for two different types of crowder proteins [bovine pancreatic trypsin inhibitor (BPTI) and GB1]. In these systems, at the temperature used, the peptide can be folded or unfolded and bound or unbound to crowder molecules. Four or five major free-energy minima can be identified. To estimate the dominant MC relaxation times of the peptide, we build MSMs using a range of different time resolutions or lag times. We show that stable relaxation-time estimates can be obtained from the MSM eigenfunctions through fits to autocorrelation data. The eigenfunctions remain sufficiently accurate to permit stable relaxation-time estimation down to small lag times, at which point simple estimates based on the corresponding eigenvalues have large systematic uncertainties. The presence of the crowders has a stabilizing effect on the peptide, especially with BPTI crowders, which can be attributed to a reduced unfolding rate ku, while the folding rate kf is left largely unchanged.

Peptide folding in the presence of interacting protein crowders.

Using Monte Carlo methods, we explore and compare the effects of two protein crowders, BPTI and GB1, on the folding thermodynamics of two peptides, the compact helical trp-cage and the ß-hairpin-forming GB1m3. The thermally highly stable crowder proteins are modeled using a fixed backbone and rotatable side-chains, whereas the peptides are free to fold and unfold. In the simulations, the crowder proteins tend to distort the trp-cage fold, while having a stabilizing effect on GB1m3. The extent of the effects on a given peptide depends on the crowder type. Due to a sticky patch on its surface, BPTI causes larger changes than GB1 in the melting properties of the peptides. The observed effects on the peptides stem largely from attractive and specific interactions with the crowder surfaces, and differ from those seen in reference simulations with purely steric crowder particles.

Equilibrium simulation of trp-cage in the presence of protein crowders.

While steric crowders tend to stabilize globular proteins, it has been found that protein crowders can have an either stabilizing or destabilizing effect, where a destabilization may arise from nonspecific attractive interactions between the test protein and the crowders. Here, we use Monte Carlo replica-exchange methods to explore the equilibrium behavior of the miniprotein trp-cage in the presence of protein crowders. Our results suggest that the surrounding crowders prevent trp-cage from adopting its global native fold, while giving rise to a stabilization of its main secondary-structure element, an α-helix. With the crowding agent used (bovine pancreatic trypsin inhibitor), the trp-cage-crowder interactions are found to be specific, involving a few key residues, most of which are prolines. The effects of these crowders are contrasted with those of hard-sphere crowders.

Thermodynamics of amyloid formation and the role of intersheet interactions.

The self-assembly of proteins into ß-sheet-rich amyloid fibrils has been observed to occur with sigmoidal kinetics, indicating that the system initially is trapped in a metastable state. Here, we use a minimal lattice-based model to explore the thermodynamic forces driving amyloid formation in a finite canonical (NVT) system. By means of generalized-ensemble Monte Carlo techniques and a semi-analytical method, the thermodynamic properties of this model are investigated for different sets of intersheet interaction parameters. When the interactions support lateral growth into multi-layered fibrillar structures, an evaporation/condensation transition is observed, between a supersaturated solution state and a thermodynamically distinct state where small and large fibril-like species exist in equilibrium. Intermediate-size aggregates are statistically suppressed. These properties do not hold if aggregate growth is one-dimensional.

Hybrid Monte Carlo with non-uniform step size.

The Hybrid Monte Carlo method offers a rigorous and potentially efficient approach to the simulation of dense systems, by combining numerical integration of Newton's equations of motion with a Metropolis accept-or-reject step. The Metropolis step corrects for sampling errors caused by the discretization of the equations of motion. The integration is usually performed using a uniform step size. Here, we present simulations of the Lennard-Jones system showing that the use of smaller time steps in the tails of each integration trajectory can reduce errors in energy. The acceptance rate is 10-15 percentage points higher in these runs, compared to simulations with the same trajectory length and the same number of integration steps but a uniform step size. We observe similar effects for the harmonic oscillator and a coarse-grained peptide model, indicating generality of the approach.

Local unfolding and aggregation mechanisms of SOD1: a Monte Carlo exploration.

Copper, zinc superoxide dismutase 1 (SOD1) is a ubiquitous homodimeric enzyme, whose misfolding and aggregation play a potentially key role in the neurodegenerative disease amyotrophic lateral sclerosis (ALS). SOD1 aggregation is thought to be preceded by dimer dissociation and metal loss, but the mechanisms by which the metal-free monomer aggregates remain incompletely understood. Here we use implicit solvent all-atom Monte Carlo (MC) methods to investigate the local unfolding dynamics of the ß-barrel-forming SOD1 monomer. Although event-to-event variations are large, on average, we find clear differences in dynamics among the eight strands forming the ß-barrel. Most dynamic is the eighth strand, ß8, which is located in the dimer interface of native SOD1. For the four strands in or near the dimer interface (ß1, ß2, ß7, and ß8), we perform aggregation simulations to assess the propensity of these chain segments to self-associate. We find that ß1 and ß2 readily self-associate to form intermolecular parallel ß-sheets, whereas ß8 shows a very low aggregation propensity.

Aggregate geometry in amyloid fibril nucleation.

We present and study a minimal structure-based model for the self-assembly of peptides into ordered ß-sheet-rich fibrils. The peptides are represented by unit-length sticks on a cubic lattice and interact by hydrogen bonding and hydrophobicity forces. Using Monte Carlo simulations with >10(5) peptides, we show that fibril formation occurs with sigmoidal kinetics in the model. To determine the mechanism of fibril nucleation, we compute the joint distribution in length and width of the aggregates at equilibrium, using an efficient cluster move and flat-histogram techniques. This analysis, based on simulations with 256 peptides in which aggregates form and dissolve reversibly, shows that the main free-energy barriers that a nascent fibril has to overcome are associated with changes in width.

Distinct phases of free α-synuclein--a Monte Carlo study.

The α-synuclein protein (αS), implicated in Parkinson's disease, shows conformational versatility. It aggregates into ß-sheet-rich fibrils, occurs in helical membrane-bound forms, is disordered as a free monomer, and has recently been suggested to have a folded helical tetramer as its main physiological form. Here, we use implicit solvent all-atom Monte Carlo methods to explore the conformational ensemble sampled by the free αS monomer. We analyze secondary structure propensities, size, and topological properties and compare with existing experimental data. Our study suggests that free αS has two distinct phases. One phase has the expected disordered character. The other phase also shows large conformational variability. However, in this phase, the ß-strand content is substantial, and the backbone fold shows statistical similarities with that in αS fibrils. Presence of this phase is consistent with data from low-temperature experiments. Conversion of disordered αS to this fibril-like form requires the crossing of a rather large apparent free-energy barrier.

Accelerating atomic-level protein simulations by flat-histogram techniques.

Flat-histogram techniques provide a powerful approach to the simulation of first-order-like phase transitions and are potentially very useful for protein studies. Here, we test this approach by implicit solvent all-atom Monte Carlo (MC) simulations of peptide aggregation, for a 7-residue fragment (GIINFEQ) [corrected] of the Cu/Zn superoxide dismutase 1 protein (SOD1). In simulations with 8 chains, we observe two distinct aggregated/non-aggregated phases. At the midpoint temperature, these phases coexist, separated by a free-energy barrier of height 2.7 k(B)T. We show that this system can be successfully studied by carefully implemented flat-histogram techniques. The frequency of barrier crossing, which is low in conventional canonical simulations, can be increased by turning to a two-step procedure based on the Wang-Landau and multicanonical algorithms.

Monte Carlo study of the formation and conformational properties of dimers of Aß42 variants.

Small soluble oligomers, and dimers in particular, of the amyloid ß-peptide (Aß) are believed to play an important pathological role in Alzheimer's disease. Here, we investigate the spontaneous dimerization of Aß42, with 42 residues, by implicit solvent all-atom Monte Carlo simulations, for the wild-type peptide and the mutants F20E, E22G and E22G/I31E. The observed dimers of these variants share many overall conformational characteristics but differ in several aspects at a detailed level. In all four cases, the most common type of secondary structure is intramolecular antiparallel ß-sheets. Parallel, in-register ß-sheet structure, as in models for Aß fibrils, is rare. The primary force driving the formation of dimers is hydrophobic attraction. The conformational differences that we do see involve turns centered in the 20-30 region. The probability of finding turns centered in the 25-30 region, where there is a loop in Aß fibrils, is found to increase upon dimerization and to correlate with experimentally measured rates of fibril formation for the different Aß42 variants. Our findings hint at reorganization of this part of the molecule as a potentially critical step in Aß aggregation.

Comparing the folding free-energy landscapes of Abeta42 variants with different aggregation properties.

The properties of the amyloid-beta peptide that lead to aggregation associated with Alzheimer's disease are not fully understood. This study aims at identifying conformational differences among four variants of full-length Abeta42 that are known to display very different aggregation properties. By extensive all-atom Monte Carlo simulations, we find that a variety of beta-sheet structures with distinct turns are readily accessible for full-length Abeta42. In the simulations, wild type (WT) Abeta42 preferentially populates two major classes of conformations, either extended with high beta-sheet content or more compact with lower beta-sheet content. The three mutations studied alter the balance between these classes. Strong mutational effects are observed in a region centered at residues 23-26, where WT Abeta42 tends to form a turn. The aggregation-accelerating E22G mutation associated with early onset of Alzheimer's disease makes this turn region conformationally more diverse, whereas the aggregation-decelerating F20E mutation has the reverse effect, and the E22G/I31E mutation reduces the turn population. Comparing results for the four Abeta42 variants, we identify specific conformational properties of residues 23-26 that might play a key role in aggregation.

Changing the mechanical unfolding pathway of FnIII10 by tuning the pulling strength.

We investigate the mechanical unfolding of the tenth type III domain from fibronectin (FnIII(10)) both at constant force and at constant pulling velocity, by all-atom Monte Carlo simulations. We observe both apparent two-state unfolding and several unfolding pathways involving one of three major, mutually exclusive intermediate states. All three major intermediates lack two of seven native beta-strands, and share a quite similar extension. The unfolding behavior is found to depend strongly on the pulling conditions. In particular, we observe large variations in the relative frequencies of occurrence for the intermediates. At low constant force or low constant velocity, all three major intermediates occur with a significant frequency. At high constant force or high constant velocity, one of them, with the N- and C-terminal beta-strands detached, dominates over the other two. Using the extended Jarzynski equality, we also estimate the equilibrium free-energy landscape, calculated as a function of chain extension. The application of a constant pulling force leads to a free-energy profile with three major local minima. Two of these correspond to the native and fully unfolded states, respectively, whereas the third one can be associated with the major unfolding intermediates.

Formation and growth of oligomers: a Monte Carlo study of an amyloid tau fragment.

Small oligomers formed early in the process of amyloid fibril formation may be the major toxic species in Alzheimer's disease. We investigate the early stages of amyloid aggregation for the tau fragment AcPHF6 (Ac-VQIVYK-NH2) using an implicit solvent all-atom model and extensive Monte Carlo simulations of 12, 24, and 36 chains. A variety of small metastable aggregates form and dissolve until an aggregate of a critical size and conformation arises. However, the stable oligomers, which are beta-sheet-rich and feature many hydrophobic contacts, are not always growth-ready. The simulations indicate instead that these supercritical oligomers spend a lengthy period in equilibrium in which considerable reorganization takes place accompanied by exchange of chains with the solution. Growth competence of the stable oligomers correlates with the alignment of the strands in the beta-sheets. The larger aggregates seen in our simulations are all composed of two twisted beta-sheets, packed against each other with hydrophobic side chains at the sheet-sheet interface. These beta-sandwiches show similarities with the proposed steric zipper structure for PHF6 fibrils but have a mixed parallel/antiparallel beta-strand organization as opposed to the parallel organization found in experiments on fibrils. Interestingly, we find that the fraction of parallel beta-sheet structure increases with aggregate size. We speculate that the reorganization of the beta-sheets into parallel ones is an important rate-limiting step in the formation of PHF6 fibrils.

Spontaneous beta-barrel formation: an all-atom Monte Carlo study of Abeta16-22 oligomerization.

Using all-atom Monte Carlo simulations with implicit water, combined with a cluster size analysis, we study the aggregation of Abeta(16) (-22), a peptide capable of forming amyloid fibrils. We consider a system of six initially randomly oriented Abeta(16) (-22) peptides, and investigate the thermodynamics and structural properties of aggregates formed by this system. The system is unaggregated without ordered secondary structure at high temperature, and forms beta-sheet rich aggregates at low temperature. At the crossover between these two regimes, we find that clusters of all sizes occur, whereas the beta-strand content is low. In one of several runs, we observe the spontaneous formation of a beta-barrel with six antiparallel strands. The beta-barrel stands out as the by far most long-lived aggregate seen in our simulations.

Thermal versus mechanical unfolding of ubiquitin.

The authors studied the temperature-induced unfolding of ubiquitin by all-atom Monte Carlo simulations. The unfolding behavior is compared with that seen in previous simulations of the mechanical unfolding of this protein, based on the same model. In mechanical unfolding, secondary-structure elements were found to break in a quite well-defined order. In thermal unfolding, the authors saw somewhat larger event-to-event fluctuations, but the unfolding pathway was still far from random. Two long-lived secondary-structure elements could be identified in the simulations. These two elements have been found experimentally to be the thermally most stable ones. Interestingly, one of these long-lived elements, the first beta-hairpin, was found to break early in the mechanical unfolding simulations. Their combined simulation results thus enable the authors to predict in detail important differences between the thermal and mechanical unfolding behaviors of ubiquitin.

PROFASI: A Monte Carlo simulation package for protein folding and aggregation.

We present a flexible and efficient program package written in C++, PROFASI, for simulating protein folding and aggregation. The systems are modeled using an all-atom description of the protein chains with only torsional degrees of freedom, and implicit water. The program package has a modular structure that makes the interaction potential easy to modify. The currently implemented potential is able to fold several peptides with about 20 residues, and has also been used to study aggregation and force-induced unfolding. The simulation methods implemented in PROFASI are Monte Carlo-based and include a semilocal move and simulated tempering. Adding new updates is easy. The code runs fast in both single- and multi-chain applications, as is illustrated by several examples.

Dissecting the mechanical unfolding of ubiquitin.

The unfolding behavior of ubiquitin under the influence of a stretching force recently was investigated experimentally by single-molecule constant-force methods. Many observed unfolding traces had a simple two-state character, whereas others showed clear evidence of intermediate states. Here, we use Monte Carlo simulations to investigate the force-induced unfolding of ubiquitin at the atomic level. In agreement with experimental data, we find that the unfolding process can occur either in a single step or through intermediate states. In addition to this randomness, we find that many quantities, such as the frequency of occurrence of intermediates, show a clear systematic dependence on the strength of the applied force. Despite this diversity, one common feature can be identified in the simulated unfolding events, which is the order in which the secondary-structure elements break. This order is the same in two- and three-state events and at the different forces studied. The observed order remains to be verified experimentally but appears physically reasonable.

Sequence-based study of two related proteins with different folding behaviors.

Z(SPA-1) is an engineered protein that binds to its parent, the three-helix-bundle Z domain of staphylococcal protein A. Uncomplexed Z(SPA-1) shows a reduced helix content and a melting behavior that is less cooperative, compared with the wild-type Z domain. Here we show that the difference in folding behavior between these two sequences can be partly understood in terms of an off-lattice model with 5-6 atoms per amino acid and a minimalistic potential, in which folding is driven by backbone hydrogen bonding and effective hydrophobic attraction.