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Quantum annealing has shown promise for finding solutions to difficult optimization problems, including protein folding. Recently, we used the D-Wave Advantage quantum annealer to explore the folding problem in a coarse-grained lattice model, the HP model, in which amino acids are classified into two broad groups: hydrophobic (H) and polar (P). Using a set of 22 HP sequences with up to 64 amino acids, we demonstrated the fast and consistent identification of the correct HP model ground states using the D-Wave hybrid quantum-classical solver. An equally relevant biophysical challenge, called the protein design problem, is the inverse of the above, where the task is to predict protein sequences that fold to a given structure. Here, we approach the design problem by a two-step procedure, implemented and executed on a D-Wave machine. In the first step, we perform a pure sequence-space search by varying the type of amino acid at each sequence position, and seek sequences which minimize the HP-model energy of the target structure. After mapping this task onto an Ising spin glass representation, we employ a hybrid quantum-classical solver to deliver energy-optimal sequences for structures with 30-64 amino acids, with a 100% success rate. In the second step, we filter the optimized sequences from the first step according to their ability to fold to the intended structure. In addition, we try solving the sequence optimization problem using only the QPU, which confines us to sizes $\le$20, due to exponentially decreasing success rates. To shed light on the pure QPU results, we investigate the effects of control errors caused by an imperfect implementation of the intended Hamiltonian on the QPU, by numerically analyzing the Schr\"odinger equation. We find that the simulated success rates in the presence of control noise semi-quantitatively reproduce the modest pure QPU results for larger chains., Comment: 33 pages, 11 figures

Quantum annealing is a promising approach for obtaining good approximate solutions to difficult optimization problems. Folding a protein sequence into its minimum-energy structure represents such a problem. For testing new algorithms and technologies for this task, the minimal lattice-based HP model is well suited, as it represents a considerable challenge despite its simplicity. The HP model has favorable interactions between adjacent, not directly bound hydrophobic residues. Here, we develop a novel spin representation for lattice protein folding tailored for quantum annealing. With a distributed encoding onto the lattice, it differs from earlier attempts to fold lattice proteins on quantum annealers, which were based upon chain growth techniques. With our encoding, the Hamiltonian by design has the quadratic structure required for calculations on an Ising-type annealer, without having to introduce any auxiliary spin variables. This property greatly facilitates the study of long chains. The approach is robust to changes in the parameters required to constrain the spin system to chain-like configurations, and performs very well in terms of solution quality. The results are evaluated against existing exact results for HP chains with up to $N=30$ beads with 100% hit rate, thereby also outperforming classical simulated annealing. In addition, the method allows us to recover the lowest known energies for $N=48$ and $N=64$ HP chains, with similar hit rates. These results are obtained by the commonly used hybrid quantum-classical approach. For pure quantum annealing, our method successfully folds an $N=14$ HP chain. The calculations were performed on a D-Wave Advantage quantum annealer.

The formation of biomolecular condensates inside cells often involve intrinsically disordered proteins (IDPs), and several of these IDPs are also capable of forming droplet-like dense assemblies on their own, through liquid-liquid phase separation. When modelings thermodynamic phase changes, it is well-known that finite-size scaling analysis can be a valuable tool. However, to our knowledge, this approach has not been applied before to the computationally challenging problem of modeling sequence-dependent biomolecular phase separation. Here, we implement finite-size scaling methods to investigate the phase behavior of two 10-bead sequences in a continuous hydrophobic/polar protein model. Combined with reversible explicit-chain Monte Carlo simulations of these sequences, finite-size scaling analysis turns out to be both feasible and rewarding, despite relying on theoretical results for asymptotically large systems. While both sequences form dense clusters at low temperature, this analysis shows that only one of them undergoes liquid-liquid phase separation. Furthermore, the transition temperature at which droplet formation sets in, is observed to converge slowly with system size, so that even for our largest systems the transition is shifted by about 8%. Using finite-size scaling analysis, this shift can be estimated and corrected for., Comment: 18 pages, 8 figures, to appear in Physical Review E

We use Markov state models (MSMs) to analyze the dynamics of a $\beta$-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 have a stabilizing effect on the peptide, especially with BPTI crowders, which can be attributed to a reduced unfolding rate $k_\text{u}$, while the folding rate $k_\text{f}$ is left largely unchanged., Comment: 18 pages, 6 figures

The self-assembly of proteins into $\beta$-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., Comment: 22 pages

The design of hybrid peptide-solid interfaces for nanotechnological applications such as biomolecular nanoarrays requires a deep understanding of the basic mechanisms of peptide binding and assembly at solid substrates. Here we show by means of experimental and computational analyses that the adsorption properties of mutated synthetic peptides at semiconductors exhibit a clear sequence-dependent adhesion specificity. Our simulations of a novel hybrid peptide-substrate model reveal the correspondence between proline mutation and binding affinity to a clean silicon substrate. After synthesizing theoretically suggested amino-acid sequences with different binding behavior, we confirm the relevance of the selective mutations upon adhesion in our subsequent atomic force microscopy experiments., Comment: 14 pages, 4 figures

The escape process from the native valley for proteins subjected to a constant stretching force is examined using a model for a Beta-barrel. For a wide range of forces, the unfolding dynamics can be treated as one-dimensional diffusion, parametrized in terms of the end-to-end distance. In particular, the escape times can be evaluated as first passage times for a Brownian particle moving on the protein free-energy landscape, using the Smoluchowski equation. At strong forces, the unfolding process can be viewed as a diffusive drift away from the native state, while at weak forces thermal activation is the relevant mechanism. An escape-time analysis within this approach reveals a crossover from an exponential to an inverse Gaussian escape-time distribution upon passing from weak to strong forces. Moreover, a single expression valid at weak and strong forces can be devised both for the average unfolding time as well as for the corresponding variance. The analysis offers a possible explanation of recent experimental findings for ddFLN4 and ubiquitin., Comment: 6 pages, 4 figures, submitted for pubblication to Physical Review Letters

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.

We investigate the mechanical unfolding of the tenth type III domain from fibronectin, FnIII10, 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 the 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 the 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., Comment: 15 pages, 9 figures

Recent experiments have identified peptides with adhesion affinity for GaAs and Si surfaces. Here we use all-atom Monte Carlo (MC) simulations with implicit solvent to investigate the behavior in aqueous solution of four such peptides, all with 12 residues. At room temperature, we find that all the four peptides are largely unstructured, which is consistent with experimental data. At the same time, we find that one of the peptides is structurally different and more flexible, compared to the others. This finding points at structural differences as a possible explanation for differences in adhesion properties between these peptides. By also analyzing designed mutants of two of the peptides, an experimental test of this hypothesis is proposed., Comment: 21 pages, 5 figures

A simplified interaction potential for protein folding studies at the atomic level is discussed and tested on a set of peptides with about 20 residues each. The test set contains both alpha-helical (Trp cage, Fs) and beta-sheet (GB1p, GB1m2, GB1m3, Betanova, LLM) peptides. The model, which is entirely sequence-based, is able to fold these different peptides for one and the same choice of model parameters. Furthermore, the melting behavior of the peptides is in good quantitative agreement with experimental data. Apparent folded populations obtained using different observables are compared, and are found to be very different for some of the peptides (e.g., Betanova). In other cases (in particular, GB1m2 and GB1m3), the different estimates agree reasonably well, indicating a more two-state-like melting behavior., Comment: 26 pages, 10 figures

The 16-22 amino acid fragment of the beta-amyloid peptide associated with the Alzheimer's disease, Abeta, is capable of forming amyloid fibrils. Here we study the aggregation mechanism of Abeta(16-22) peptides by unbiased thermodynamic simulations at the atomic level for systems of one, three and six Abeta(16-22) peptides. We find that the isolated Abeta(16-22) peptide is mainly a random coil in the sense that both the alpha-helix and beta-strand contents are low, whereas the three- and six-chain systems form aggregated structures with a high beta-sheet content. Furthermore, in agreement with experiments on Abeta(16-22) fibrils, we find that large parallel beta-sheets are unlikely to form. For the six-chain system, the aggregated structures can have many different shapes, but certain particularly stable shapes can be identified., Comment: 19 pages, 7 figures (to appear in Biophys. J.)

Using a simple hydrophobic/polar protein model, we perform a Monte Carlo study of the thermodynamics and kinetics of binding to a target structure for two closely related sequences, one of which has a unique folded state while the other is unstructured. We obtain significant differences in their binding behavior. The stable sequence has rigid docking as its preferred binding mode, while the unstructured chain tends to first attach to the target and then fold. The free-energy profiles associated with these two binding modes are compared., Comment: 17 pages, 7 figures (to appear in J. Chem. Phys.)

We present a Monte Carlo study of a model protein with 54 amino acids that folds directly to its native three-helix-bundle state without forming any well-defined intermediate state. The free-energy barrier separating the native and unfolded states of this protein is found to be weak, even at the folding temperature. Nevertheless, we find that melting curves to a good approximation can be described in terms of a simple two-state system, and that the relaxation behavior is close to single exponential. The motion along individual reaction coordinates is roughly diffusive on timescales beyond the reconfiguration time for an individual helix. A simple estimate based on diffusion in a square-well potential predicts the relaxation time within a factor of two., Comment: 22 pages, 5 figures

An atomic protein model with a minimalistic potential is developed and then tested on an alpha-helix and a beta-hairpin, using exactly the same parameters for both peptides. We find that melting curves for these sequences to a good approximation can be described by a simple two-state model, with parameters that are in reasonable quantitative agreement with experimental data. Despite the apparent two-state character of the melting curves, the energy distributions are found to lack a clear bimodal shape, which is discussed in some detail. We also perform a Monte Carlo-based kinetic study and find, in accord with experimental data, that the alpha-helix forms faster than the beta-hairpin., Comment: 18 pages, 4 figures

ZSPA-1 is an engineered protein that binds to its parent, the three-helix-bundle Z domain of staphylococcal protein A. Uncomplexed ZSPA-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., Comment: 12 pages, 5 figures

We study the folding thermodynamics of a beta-hairpin and two three-stranded beta-sheet peptides using a simplified sequence-based all-atom model, in which folding is driven mainly by backbone hydrogen bonding and effective hydrophobic attraction. The native populations obtained for these three sequences are in good agreement with experimental data. We also show that the apparent native population depends on which observable is studied; the hydrophobicity energy and the number of native hydrogen bonds give different results. The magnitude of this dependence matches well with the results obtained in two different experiments on the beta-hairpin., Comment: 17 pages, 7 figures, to appear in Proteins

The hydrophobic/polar HP model on the square lattice has been widely used to investigate basics of protein folding. In the cases where all designing sequences (sequences with unique ground states) were enumerated without restrictions on the number of contacts, the upper limit on the chain length N has been 18-20 because of the rapid exponential growth of the numbers of conformations and sequences. We show how a few optimizations push this limit by about 5 units. Based on these calculations, we study the statistical distribution of hydrophobicity along designing sequences. We find that the average number of hydrophobic and polar clumps along the chains is larger for designing sequences than for random ones, which is in agreement with earlier findings for N up to 18 and with results for real enzymes. We also show that this deviation from randomness disappears if the calculations are restricted to maximally compact structures., Comment: 18 pages, 4 figures

A reduced protein model with five to six atoms per amino acid and five amino acid types is developed and tested on a three-helix-bundle protein, a 46-amino acid fragment from staphylococcal protein A. The model does not rely on the widely used Go approximation where non-native interactions are ignored. We find that the collapse transition is considerably more abrupt for the protein A sequence than for random sequences with the same composition. The chain collapse is found to be at least as fast as helix formation. Energy minimization restricted to the thermodynamically favored topology gives a structure that has a root-mean-square deviation of 1.8 A from the native structure. The sequence-dependent part of our potential is pairwise additive. Our calculations suggest that fine-tuning this potential by parameter optimization is of limited use., Comment: 18 pages, 8 figures

We study the thermodynamic behavior of a model protein with 54 amino acids that is designed to form a three-helix bundle in its native state. The model contains three types of amino acids and five to six atoms per amino acid, and has the Ramachandran torsion angles as its only degrees of freedom. The force field is based on hydrogen bonds and effective hydrophobicity forces. We study how the character of the collapse transition depends on the strengths of these forces. For a suitable choice of these two parameters, it is found that the collapse transition is first-order-like and coincides with the folding transition. Also shown is that the corresponding one- and two-helix segments make less stable secondary structure than the three-helix sequence., Comment: 14 pages, 8 figures

We develop a new elementary move for simulations of polymer chains in torsion angle space. The method is flexible and easy to implement. Tentative updates are drawn from a (conformation-dependent) Gaussian distribution that favors approximately local deformations of the chain. The degree of bias is controlled by a parameter b. The method is tested on a reduced model protein with 54 amino acids and the Ramachandran torsion angles as its only degrees of freedom, for different b. Without excessive fine tuning, we find that the effective step size can be increased by a factor of three compared to the unbiased b=0 case. The method may be useful for kinetic studies, too., Comment: 14 pages, 4 figures

We study the thermodynamic behavior of a model protein with 54 amino acids that forms a three-helix bundle in its native state. The model contains three types of amino acids and five to six atoms per amino acid and has the Ramachandran torsional angles $\phi_i$, $\psi_i$ as its degrees of freedom. The force field is based on hydrogen bonds and effective hydrophobicity forces. For a suitable choice of the relative strength of these interactions, we find that the three-helix-bundle protein undergoes an abrupt folding transition from an expanded state to the native state. Also shown is that the corresponding one- and two-helix segments are less stable than the three-helix sequence., Comment: 15 pages, 7 figures

We study the statistical properties of hydrophobic/polar model sequences with unique native states on the square lattice. It is shown that this ensemble of sequences differs from random sequences in significant ways in terms of both the distribution of hydrophobicity along the chains and total hydrophobicity. Whenever statistically feasible, the analogous calculations are performed for a set of real enzymes, too., Comment: 17 pages, 5 figures, to appear in Biophys. J

We study the phase behavior of single homopolymers in a simple hydrophobic/hydrophilic off-lattice model with sequence independent local interactions. The specific heat is, not unexpectedly, found to exhibit a pronounced peak well below the collapse temperature, signalling a possible low-temperature phase transition. The system size dependence at this maximum is investigated both with and without the local interactions, using chains with up to 50 monomers. The size dependence is found to be weak. The specific heat itself seems not to diverge. The homopolymer results are compared with those for two non-uniform sequences. Our calculations are performed using the methods of simulated and parallel tempering. The performances of these algorithms are discussed, based on careful tests for a small system., Comment: 13 pages LaTeX, 6 Postscript figures, References added

Background: Designing amino acid sequences that are stable in a given target structure amounts to maximizing a conditional probability. A straightforward approach to accomplish this is a nested Monte Carlo where the conformation space is explored over and over again for different fixed sequences, which requires excessive computational demand. Several approximate attempts to remedy this situation, based on energy minimization for fixed structure or high-$T$ expansions, have been proposed. These methods are fast but often not accurate since folding occurs at low $T$. Results: We develop a multisequence Monte Carlo procedure, where both sequence and conformation space are simultaneously probed with efficient prescriptions for pruning sequence space. The method is explored on hydrophobic/polar models. We first discuss short lattice chains, in order to compare with exact data and with other methods. The method is then successfully applied to lattice chains with up to 50 monomers, and to off-lattice 20-mers. Conclusions: The multisequence Monte Carlo method offers a new approach to sequence design in coarse-grained models. It is much more efficient than previous Monte Carlo methods, and is, as it stands, applicable to a fairly wide range of two-letter models., Comment: 23 pages, 7 figures

A new method for sequence optimization in protein models is presented. The approach, which has inherited its basic philosophy from recent work by Deutsch and Kurosky [Phys. Rev. Lett. 76, 323 (1996)] by maximizing conditional probabilities rather than minimizing energy functions, is based upon a novel and very efficient multisequence Monte Carlo scheme. By construction, the method ensures that the designed sequences represent good folders thermodynamically. A bootstrap procedure for the sequence space search is devised making very large chains feasible. The algorithm is successfully explored on the two-dimensional HP model with chain lengths N=16, 18 and 32., Comment: 7 pages LaTeX, 4 Postscript figures; minor changes

We study a simple heteropolymer model containing sequence-independent local interactions on both square and triangular lattices. Sticking to a two-letter code, we investigate the model for varying strength $\kappa$ of the local interactions; $\kappa=0$ corresponds to the well-known HP model [K.F. Lau and K.A. Dill, Macromolecules 22, 3986 (1989)]. By exhaustive enumerations for short chains, we obtain all structures which act as a unique and pronounced energy minimum for at least one sequence. We find that the number of such designable structures depends strongly on $\kappa$. Also, we find that the number of designable structures can differ widely for the two lattices at a given $\kappa$. This is the case, for example, at $\kappa=0$, which implies that the HP model exhibits different behavior on the two lattices. Our findings clearly show that sequence-independent local properties of the chains can play an important role in the formation of unique minimum energy structures., Comment: 10 pages LaTeX, 3 Postscript figures. Figure and references added

The thermodynamic behavior of a three-dimensional off-lattice model for protein folding is probed. The model has only two types of residues, hydrophobic and hydrophilic. In absence of local interactions, native structure formation does not occur for the temperatures considered. By including sequence independent local interactions, which qualitatively reproduce local properties of functional proteins, the dominance of a native state for many sequences is observed. As in lattice model approaches, folding takes place by gradual compactification, followed by a sequence dependent folding transition. Our results differ from lattice approaches in that bimodal energy distributions are not observed and that high folding temperatures are accompanied by relatively low temperatures for the peak of the specific heat. Also, in contrast to earlier studies using lattice models, our results convincingly demonstrate that one does not need more than two types of residues to generate sequences with good thermodynamic folding properties in three dimensions., Comment: 18 pages, 11 Postscript figures

Folding properties of a two-dimensional toy protein model containing only two amino-acid types, hydrophobic and hydrophilic, respectively, are analyzed. An efficient Monte Carlo procedure is employed to ensure that the ground states are found. The thermodynamic properties are found to be strongly sequence dependent in contrast to the kinetic ones. Hence, criteria for good folders are defined entirely in terms of thermodynamic fluctuations. With these criteria sequence patterns that fold well are isolated. For 300 chains with 20 randomly chosen binary residues approximately 10% meet these criteria. Also, an analysis is performed by means of statistical and artificial neural network methods from which it is concluded that the folding properties can be predicted to a certain degree given the binary numbers characterizing the sequences., Comment: 15 pages, 8 Postscript figures. Minor changes

This paper is withdrawn., Comment: Withdrawn

We develop a simple optimization procedure for assigning binary values to the amino acids. The binary values are determined by a maximization of the degree of pattern conservation in groups of closely related protein sequences. The maximization is carried out at fixed composition. For compositions approximately corresponding to an equipartition of the residues, the optimal encoding is found to be strongly correlated with hydrophobicity. The stability of the procedure is demonstrated. Our calculations are based upon sequences in the SWISS-PROT database., Comment: 9 pages, 4 Postscript figures. References and figure added

The question of whether proteins originate from random sequences of amino acids is addressed. A statistical analysis is performed in terms of blocked and random walk values formed by binary hydrophobic assignments of the amino acids along the protein chains. Theoretical expectations of these variables from random distributions of hydrophobicities are compared with those obtained from functional proteins. The results, which are based upon proteins in the SWISS-PROT data base, convincingly show that the amino acid sequences in proteins differ from what is expected from random sequences in a statistical significant way. By performing Fourier transforms on the random walks one obtains additional evidence for non-randomness of the distributions. We have also analyzed results from a synthetic model containing only two amino-acid types, hydrophobic and hydrophilic. With reasonable criteria on good folding properties in terms of thermodynamical and kinetic behavior, sequences that fold well are isolated. Performing the same statistical analysis on the sequences that fold well indicates similar deviations from randomness as for the functional proteins. The deviations from randomness can be interpreted as originating from anticorrelations in terms of an Ising spin model for the hydrophobicities. Our results, which differ from previous investigations using other methods, might have impact on how permissive with respect to sequence specificity the protein folding process is -- only sequences with non-random hydrophobicity distributions fold well. Other distributions give rise to energy landscapes with poor folding properties and hence did not survive the evolution., Comment: 16 pages, 8 Postscript figures. Minor changes, references added

We study the thermodynamic behavior of a simple off-lattice model for protein folding. The model is two-dimensional and has two different ``amino acids''. Using numerical simulations of all chains containing eight or ten monomers, we examine the sequence dependence at a fixed temperature. It is shown that only a few of the chains exist in unique folded state at this temperature, and the energy level spectra of chains with different types of behavior are compared. Furthermore, we use this model as a testbed for two improved Monte Carlo algorithms. Both algorithms are based on letting some parameter of the model become a dynamical variable; one of the algorithms uses a fluctuating temperature and the other a fluctuating monomer sequence. We find that by these algorithms one gains large factors in efficiency in comparison with conventional methods., Comment: 17 pages, 9 Postscript figures. Combined with chem-ph/9505004

See chem-ph/9505003., Comment: Withdrawn, combined with chem-ph/9505003