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.

When a foreign gene meets its native counterpart: computational biophysics analysis of two PgiC loci in the grass Festuca ovina.

Duplicative horizontal gene transfer may bring two previously separated homologous genes together, which may raise questions about the interplay between the gene products. One such gene pair is the "native" PgiC1 and "foreign" PgiC2 in the perennial grass Festuca ovina. Both PgiC1 and PgiC2 encode cytosolic phosphoglucose isomerase, a dimeric enzyme whose proper binding is functionally essential. Here, we use biophysical simulations to explore the inter-monomer binding of the two homodimers and the heterodimer that can be produced by PgiC1 and PgiC2 in F. ovina. Using simulated native-state ensembles, we examine the structural properties and binding tightness of the dimers. In addition, we investigate their ability to withstand dissociation when pulled by a force. Our results suggest that the inter-monomer binding is tighter in the PgiC2 than the PgiC1 homodimer, which could explain the more frequent occurrence of the foreign PgiC2 homodimer in dry habitats. We further find that the PgiC1 and PgiC2 monomers are compatible with heterodimer formation; the computed binding tightness is comparable to that of the PgiC1 homodimer. Enhanced homodimer stability and capability of heterodimer formation with PgiC1 are properties of PgiC2 that may contribute to the retaining of the otherwise redundant PgiC2 gene.

Aggregation and coacervation with Monte Carlo simulations.

Protein aggregation into oligomeric species has been linked to a number of neurodegenerative diseases. The ability to assemble into ordered fibril-like forms under certain conditions is now regarded as a very common property of polypeptide chains. A different form of assembly is known for some proteins in which protein molecules sequester into liquid-like droplets. Understanding the biophysical mechanisms behind these phenomena is of great relevance not only for deeper insights into the associated disorders but also for any protein-based therapeutics. Here we review a few examples of the use of Markov chain Monte Carlo simulations in the study of these two forms of protein assembly.

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.

Folding of Top7 in unbiased all-atom Monte Carlo simulations.

For computational studies of protein folding, proteins with both helical and ß-sheet secondary structure elements are very challenging, as they expose subtle biases of the physical models. Here, we present reproducible folding of a 92 residue α/ß protein (residues 3-94 of Top7, PDB ID: 1QYS) in computer simulations starting from random initial conformations using a transferable physical model which has been previously shown to describe the folding and thermodynamic properties of about 20 other smaller proteins of different folds. Top7 is a de novo designed protein with two α-helices and a five stranded ß-sheet. Experimentally, it is known to be unusually stable for its size, and its folding transition distinctly deviates from the two-state behavior commonly seen in natural single domain proteins. In our all-atom implicit solvent parallel tempering Monte Carlo simulations, Top7 shows a rapid transition to a group of states with high native-like secondary structure, and a much slower subsequent transition to the native state with a root mean square deviation of about 3.5 Å from the experimentally determined structure. Consistent with experiments, we find Top7 to be thermally extremely stable, although the simulations also find a large number of very stable non-native states with high native-like secondary structure.

Role of Prion Disease-Linked Mutations in the Intrinsically Disordered N-Terminal Domain of the Prion Protein.

Prion diseases are fatal neurodegenerative disorders in mammals and other animal species. In humans, about 15% of these maladies are caused by pathogenic mutations (PMs) in the gene encoding for the prion protein (PrP(C)). Seven PMs are located in the naturally unfolded PrP(C) N-terminal domain, which constitutes about half of the protein. Intriguingly and in sharp contrast to other PMs clustered in the folded domain, N-terminal PMs barely affect the conversion to the pathogenic (scrapie, or PrP(Sc)) isoform of PrP(C). Here, we hypothesize that the neurotoxicity of these PMs arises from changes in structural determinants of the N-terminal domain, affecting the protein binding with its cellular partners and/or the cotranslational translocation during the PrP(C) biosynthesis. We test this idea by predicting the conformational ensemble of the wild-type (WT) and mutated mouse PrP(C) N-terminal domain, whose sequence is almost identical to that of the human one and for which the largest number of in vivo data is available. The conformational properties of the WT are consistent with those inferred experimentally. Importantly, the PMs turn out to affect in a subtle manner the intramolecular contacts in the putative N-terminal domain binding sites for Cu(2+) ions, sulphated glycosaminoglycans, and other known PrP(C) cellular partners. The PMs also alter the local structural features of the transmembrane domain and adjacent stop transfer effector, which act together to regulate the protein topology. These results corroborate the hypothesis that N-terminal PMs affect the PrP(C) binding to functional interactors and/or the translocation.

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.

An effective all-atom potential for proteins.

We describe and test an implicit solvent all-atom potential for simulations of protein folding and aggregation. The potential is developed through studies of structural and thermodynamic properties of 17 peptides with diverse secondary structure. Results obtained using the final form of the potential are presented for all these peptides. The same model, with unchanged parameters, is furthermore applied to a heterodimeric coiled-coil system, a mixed alpha/beta protein and a three-helix-bundle protein, with very good results. The computational efficiency of the potential makes it possible to investigate the free-energy landscape of these 49-67-residue systems with high statistical accuracy, using only modest computational resources by today's standards.PACS Codes: 87.14.E-, 87.15.A-, 87.15.Cc.

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.

Caching of a chameleon segment facilitates folding of a protein with end-to-end beta-sheet.

We report results from all-atom simulations of a 49-residue C-terminal fragment of TOP7 in implicit solvent. Using parallel tempering simulations with high statistics, we probe the thermodynamic properties of the protein over a large range of temperatures and evaluate its free energy landscape at room temperature. Our results confirm that the protein folds by a caching mechanism that relies on a chameleon segment. This mechanism differs from the one seen in high-temperature unfolding simulations. Finally, we discuss a possible mechanism for dimerization of the protein.

Simulation of Top7-CFr: a transient helix extension guides folding.

Protein structures often feature beta-sheets in which adjacent beta-strands have large sequence separation. How the folding process orchestrates the formation and correct arrangement of these strands is not comprehensively understood. Particularly challenging are proteins in which beta-strands at the N and C termini are neighbors in a beta-sheet. The N-terminal beta-strand is synthesized early on, but it can not bind to the C terminus before the chain is fully synthesized. During this time, there is a danger that the beta-strand at the N terminus interacts with nearby molecules, leading to potentially harmful aggregates of incompletely folded proteins. Simulations of the C-terminal fragment of Top7 show that this risk of misfolding and aggregation can be avoided by a "caching" mechanism that relies on the "chameleon" behavior of certain segments.

Structural reorganisation and potential toxicity of oligomeric species formed during the assembly of amyloid fibrils.

Increasing evidence indicates that oligomeric protein assemblies may represent the molecular species responsible for cytotoxicity in a range of neurological disorders including Alzheimer and Parkinson diseases. We use all-atom computer simulations to reveal that the process of oligomerization can be divided into two steps. The first is characterised by a hydrophobic coalescence resulting in the formation of molten oligomers in which hydrophobic residues are sequestered away from the solvent. In the second step, the oligomers undergo a process of reorganisation driven by interchain hydrogen bonding interactions that induce the formation of beta sheet rich assemblies in which hydrophobic groups can become exposed. Our results show that the process of aggregation into either ordered or amorphous species is largely determined by a competition between the hydrophobicity of the amino acid sequence and the tendency of polypeptide chains to form arrays of hydrogen bonds. We discuss how the increase in solvent-exposed hydrophobic surface resulting from such a competition offers an explanation for recent observations concerning the cytotoxicity of oligomeric species formed prior to mature amyloid fibrils.

Improving an all-atom force field.

Experimentally well-characterized proteins that are small enough to be computationally tractable provide useful information for refining existing all-atom force fields. This is used by us for reparametrizing a recently developed all-atom force field. Relying on high statistics parallel tempering simulations of a designed 20 residue beta-sheet peptide, we propose incremental changes that improve the force field's range of applicability.

Folding of a miniprotein with mixed fold.

Using the 28 residue betabetaalpha protein FSD-EY as a target system, we examine correction terms for the ECEPP/3 force field. We find an increased probability of formation of the native state at low temperatures resulting from a reduced propensity to form alpha helices and increased formation of beta sheets. Our analysis of the observed folding events suggests that the C-terminal helix of FSD-EY is much more stable than the N-terminal beta hairpin and forms first. The hydrophobic groups of the helix provide a template which promotes the formation of the beta hairpin that is never observed to form without the helix.

Folding of proteins with diverse folds.

Using parallel tempering simulations with high statistics, we investigate the folding and thermodynamic properties of three small proteins with distinct native folds: the all-helical 1RIJ, the all-sheet beta3s, and BBA5, which has a mixed helix-sheet fold. In all three cases, simulations with our energy function find the native structures as global minima in free energy at experimentally relevant temperatures. However, the folding process strongly differs for the three molecules, indicating that the folding mechanism is correlated with the form of the native structure.