Toy model for protein folding

Bilinear optimization of protein structure prediction: An exact approach via AB off-lattice model

Protein structure prediction (PSP) remains a central challenge in computational biology due to its inherent complexity and high dimensionality. While numerous heuristic approaches have appeared in the literature, their success varies. The AB off-lattice model, which characterizes proteins as sequences of A (hydrophobic) and B (hydrophilic) beads, presents a simplified perspective on PSP. This work presents a mathematical optimization-based methodology capitalizing on the off-lattice AB model. Dissecting the inherent non-linearities of the energy landscape of protein folding allowed for formulating the PSP as a bilinear optimization problem. This formulation was achieved by introducing auxiliary variables and constraints that encapsulate the nuanced relationship between the protein's conformational space and its energy landscape. The proposed bilinear model exhibited notable accuracy in pinpointing the global minimum energy conformations on a benchmark dataset presented by the Protein Data Bank (PDB). Compared to traditional heuristic-based methods, this bilinear approach yielded exact solutions, reducing the likelihood of local minima entrapment. This research highlights the potential of reframing the traditionally non-linear protein structure prediction problem into a bilinear optimization problem through the off-lattice AB model. Such a transformation offers a route toward methodologies that can determine the global solution, challenging current PSP paradigms. Exploration into hybrid models, merging bilinear optimization and heuristic components, might present an avenue for balancing accuracy with computational efficiency.

Folding Coarse-Grained Oligomer Models with PyRosetta

Non-biological foldamers are a promising class of macromolecules that share similarities to classical biopolymers such as proteins and nucleic acids. Currently, designing novel foldamers is a non-trivial process, often involving many iterations of trial synthesis and characterization until folded structures are observed. In this work, we aim to tackle these foldamer design challenges using computational modeling techniques. We developed CG PyRosetta, an extension to the popular protein folding python package, PyRosetta, which introduces coarse-grained (CG) residues into PyRosetta, enabling the folding of toy CG foldamer models. Although these models are simplified, they can help explore overarching physical hypotheses about how oligomers can form. Through systematic variation of CG parameters in these models, we can investigate various folding hypotheses at the CG scale to inform the design process of new foldamer chemistries. In this study, we demonstrate CG PyRosetta's ability to identify minimum energy structures with a diverse structural search over a range of simple models, as well as two hypothesis-driven parameter scans investigating the effects of side-chain size and internal backbone angle on secondary structures. We are able to identify several types of secondary structures from single- and double-helices to sheet-like and knot-like structures. We show how side-chain size and backbone bond angle both play an important role in the structure and energetics of these toy models. Optimal side-chain sizes promote favorable packing of side chains, while specific backbone bond angles influence the specific helix type found in folded structures.

Collective Variables for Free Energy Surface Tailoring: Understanding and Modifying Functionality in Systems Dominated by Rare Events

We introduce a method for elucidating and modifying the functionality of systems dominated by rare events that relies on the semiautomated tuning of their underlying free energy surface. The proposed approach seeks to construct collective variables (CVs) that encode the essential information regarding the rare events of the system of interest. The appropriate CVs are identified using harmonic linear discriminant analysis (HLDA), a machine-learning-based method that is trained solely on data collected from short ordinary simulations in the relevant metastable states of the system. Utilizing the interpretable form of the resulting CVs, the critical interaction potentials that determine the system's rare transitions are identified and purposely modified to tailor the free energy surface in a manner that alters functionality as desired. The applicability of the method is illustrated in the context of three different systems, thereby demonstrating that thermodynamic and kinetic properties can be tractably modified with little to no prior knowledge or intuition.

Supervised and unsupervised machine learning of structural phases of polymers adsorbed to nanowires

We identify configurational phases and structural transitions in a polymer nanotube composite by means of machine learning. We employ various unsupervised dimensionality reduction methods, conventional neural networks, as well as the confusion method, an unsupervised neural-network-based approach. We find neural networks are able to reliably recognize all configurational phases that have been found previously in experiment and simulation. Furthermore, we locate the boundaries between configurational phases in a way that removes human intuition or bias. This could be done before only by relying on preconceived, ad hoc order parameters.

Effects of lattice constraints in coarse-grained protein models

We compare and contrast folding behavior in several coarse-grained protein models, both on- and off-lattice, in an attempt to uncover the effect of lattice constraints in these kinds of models. Using modern, extended ensemble Monte Carlo methods-Wang-Landau sampling, multicanonical sampling, replica-exchange Wang-Landau sampling, and replica-exchange multicanonical sampling, we investigate the thermodynamic and structural behavior of the protein Crambin within the context of the hydrophobic-polar, hydrophobic-"neutral"-polar (H0P), and semi-flexible H0P model frameworks. We uncover the folding process in all cases; all models undergo, at least, the two major structural transitions observed in nature-the coil-globule collapse and the folding transition. As the complexity of the model increases, these two major transitions begin to split into multi-step processes, wherein the lattice coarse-graining has a significant impact on the details of these processes. The results show that the level of structural coarse-graining is coupled to the level of interaction coarse-graining.

Sequence-based protein structure optimization using enhanced simulated annealing algorithm on a coarse-grained model

The understanding of protein structure is vital to determine biological function. We presented an enhanced simulated annealing (ESA) algorithm to investigate protein three-dimensional (3D) structure on a coarse-grained model. Inside the algorithm, we adjusted exploration equations to achieve good search intensity. To that end, our algorithm used (i) a multivariable disturbance operator for diversification of solution, (ii) a sign function to improve randomness of solution, and (iii) taking remainder operation performed on floating-point number to tackle out-of-range solution. By monitoring energy value throughout the simulation, the energy-optimal state can be found. The ESA algorithm was tested on artificial and real protein sequences with different lengths. The results show that our algorithm outperforms conventional simulated annealing algorithm and can compete with the reported algorithms before. Especially, our algorithm can obtain folding conformations with specific structural features. Further analysis shows that simulating trajectory of seeking the lowest energy can exhibit thermodynamic behavior of protein folding. Graphical Abstract.

Protein structure optimization using improved simulated annealing algorithm on a three-dimensional AB off-lattice model

This paper proposed an improved simulated annealing (ISA) algorithm for protein structure optimization based on a three-dimensional AB off-lattice model. In the algorithm, we provided a general formula used for producing initial solution, and designed a multivariable disturbance term, relating to the parameters of simulated annealing and a tuned constant, to generate neighborhood solution. To avoid missing optimal solution, storage operation was performed in searching process. We applied the algorithm to test artificial protein sequences from literature and constructed a benchmark dataset consisting of 10 real protein sequences from the Protein Data Bank (PDB). Otherwise, we generated C_α space-filling model to represent protein folding conformation. The results indicate our algorithm outperforms the five methods before in searching lower energies of artificial protein sequences. In the testing on real proteins, our method can achieve the energy conformations with C_α-RMSD less than 3.0 Å from the PDB structures. Moreover, C_α space-filling model may simulate dynamic change of protein folding conformation at atomic level.

An improved stochastic fractal search algorithm for 3D protein structure prediction

Protein structure prediction (PSP) is a significant area for biological information research, disease treatment, and drug development and so on. In this paper, three-dimensional structures of proteins are predicted based on the known amino acid sequences, and the structure prediction problem is transformed into a typical NP problem by an AB off-lattice model. This work applies a novel improved Stochastic Fractal Search algorithm (ISFS) to solve the problem. The Stochastic Fractal Search algorithm (SFS) is an effective evolutionary algorithm that performs well in exploring the search space but falls into local minimums sometimes. In order to avoid the weakness, Lvy flight and internal feedback information are introduced in ISFS. In the experimental process, simulations are conducted by ISFS algorithm on Fibonacci sequences and real peptide sequences. Experimental results prove that the ISFS performs more efficiently and robust in terms of finding the global minimum and avoiding getting stuck in local minimums.

Polymer adsorption on curved surfaces

The conformational behavior of a coarse-grained finite polymer chain near an attractive spherical surface was investigated by means of multicanonical Monte Carlo computer simulations. In a detailed analysis of canonical equilibrium data over a wide range of sphere radius and temperature, we have constructed entire phase diagrams both for nongrafted and end-grafted polymers. For the identification of the conformational phases, we have calculated several energetic and structural observables such as gyration tensor based shape parameters and their fluctuations by canonical statistical analysis. Despite the simplicity of our model, it qualitatively represents in the considered parameter range real systems that are studied in experiments. The work discussed here could have experimental implications from protein-ligand interactions to designing nanosmart materials.

Two Perturbations for Geometry Optimization of Off-lattice Bead Protein Models

Referring to the optimization algorithm previously developed for atomic clusters, the present author develops an efficient method for geometry optimization of a coarse-grained protein model expressed with two kinds of beads (hydrophilic and hydrophobic ones). In the method, two types of geometrical perturbations, center-directed bead move and one bead rotation, are used to explore new configurations and local optimizations are performed after the perturbations. The center-directed bead move is used for hydrophobic beads and the one bead rotation is performed for both hydrophobic and hydrophilic beads. The optimization method was applied to protein models consisting of 13, 20, 21, and 34 beads. The present method produced the global minima of the 13-, 21-, and 34-bead models reported in the literature and updated the lowest energies of the protein models with 20 beads. These results indicate that the present method is efficient for searching for optimal structures of proteins.

Subphase transitions in first-order aggregation processes

In this paper, we investigate the properties of aggregation transitions in the context of generic coarse-grained homopolymer systems. By means of parallel replica-exchange Monte Carlo methods, we perform extensive simulations of systems consisting of up to 20 individual oligomer chains with five monomers each. Using the tools of the versatile microcanonical inflection-point analysis, we show that the aggregation transition is a first-order process consisting of a sequence of subtransitions between intermediate structural phases. We unravel the properties of these intermediate phases by collecting and analyzing their individual contributions towards the density of states of the system. The central theme of this systematic study revolves around translational entropy and its role in the striking phenomena of missing intermediate phases. We conclude with a brief discussion of the scaling properties of the transition temperature and the latent heat.

Control globular structure formation of a copolymer chain through inverse design

A copolymer chain in dilute solution can exhibit various globular structures with characteristic morphologies, which makes it a potentially useful candidate for artificial materials design. However, the chain has a huge conformation space and may not naturally form the globular structure we desire. An ideal way to control globular structure formation should be inverse design, i.e., starting from the target structure and finding out what kind of polymers can effectively generate it. To accomplish this, we propose an inverse design procedure, which is combined with Wang-Landau Monte Carlo to fully and precisely explore the huge conformation space of the chain. Starting from a desired target structure, all the geometrically possible sequences are exactly enumerated. Interestingly, reasonable interaction strengths are obtained and found to be not specified for only one sequence. Instead, they can be combined with many other sequences and also achieve a relatively high yield for target structure, although these sequences may be rather different. These results confirm the possibility of controlling globular structure formation of a copolymer chain through inverse design and pave the way for targeted materials design.

Knots as a Topological Order Parameter for Semiflexible Polymers

Using a combination of the multicanonical Monte Carlo algorithm and the replica-exchange method, we investigate the influence of bending stiffness on the conformational phases of a bead-stick homopolymer model and present the pseudophase diagram for the complete range of semiflexible polymers, from flexible to stiff. Although it is a simple model, we observe a rich variety of conformational phases, reminiscent of conformations observed for synthetic polymers or biopolymers. Depending on the bending stiffness, the model exhibits different pseudophases like bent, hairpin, or toroidal. In particular, we find thermodynamically stable knots and unusual transitions into these knotted phases with a clear phase coexistence, but almost constant mean total energy, and hence almost no latent heat.

Confinement effects upon the separation of structural transitions in linear systems with restricted bond fluctuation ranges

By means of advanced parallel replica-exchange Monte Carlo methods we examine the influence of elasticity and confinement on the structural transitions of linear systems with restricted bonded interaction. For this purpose, we adopt a model for coarse-grained flexible polymers of finite length in the dilute regime. Hyperphase diagrams are constructed using energy-dependent canonical quantities to demonstrate the effects of the changes in the range of the confined interaction on the liquid and solid structural phases. With increasing bonded interaction range we observe the disappearance of the liquid phase and the fusion of the gas-liquid (or Θ) and the liquid-solid transitions. One of the most remarkable features, the liquid-gas transition, changes from second to first order if the confined interaction range exceeds a threshold that separates polymeric from nonpolymeric systems. The notoriously difficult sampling of the entropically suppressed conformations in the region of very strong first-order transitions is improved by using multiple Gaussian modified ensembles.

Protein structure prediction with local adjust tabu search algorithm

Background

Protein folding structure prediction is one of the most challenging problems in the bioinformatics domain. Because of the complexity of the realistic protein structure, the simplified structure model and the computational method should be adopted in the research. The AB off-lattice model is one of the simplification models, which only considers two classes of amino acids, hydrophobic (A) residues and hydrophilic (B) residues.

Results

The main work of this paper is to discuss how to optimize the lowest energy configurations in 2D off-lattice model and 3D off-lattice model by using Fibonacci sequences and real protein sequences. In order to avoid falling into local minimum and faster convergence to the global minimum, we introduce a novel method (SATS) to the protein structure problem, which combines simulated annealing algorithm and tabu search algorithm. Various strategies, such as the new encoding strategy, the adaptive neighborhood generation strategy and the local adjustment strategy, are adopted successfully for high-speed searching the optimal conformation corresponds to the lowest energy of the protein sequences. Experimental results show that some of the results obtained by the improved SATS are better than those reported in previous literatures, and we can sure that the lowest energy folding state for short Fibonacci sequences have been found.

Conclusions

Although the off-lattice models is not very realistic, they can reflect some important characteristics of the realistic protein. It can be found that 3D off-lattice model is more like native folding structure of the realistic protein than 2D off-lattice model. In addition, compared with some previous researches, the proposed hybrid algorithm can more effectively and more quickly search the spatial folding structure of a protein chain.

A balance-evolution artificial bee colony algorithm for protein structure optimization based on a three-dimensional AB off-lattice model

Protein structure prediction is a fundamental issue in the field of computational molecular biology. In this paper, the AB off-lattice model is adopted to transform the original protein structure prediction scheme into a numerical optimization problem. We present a balance-evolution artificial bee colony (BE-ABC) algorithm to address the problem, with the aim of finding the structure for a given protein sequence with the minimal free-energy value. This is achieved through the use of convergence information during the optimization process to adaptively manipulate the search intensity. Besides that, an overall degradation procedure is introduced as part of the BE-ABC algorithm to prevent premature convergence. Comprehensive simulation experiments based on the well-known artificial Fibonacci sequence set and several real sequences from the database of Protein Data Bank have been carried out to compare the performance of BE-ABC against other algorithms. Our numerical results show that the BE-ABC algorithm is able to outperform many state-of-the-art approaches and can be effectively employed for protein structure optimization.

Breakout character of islet amyloid polypeptide hydrophobic mutations at the onset of type-2 diabetes

Toxic fibrillar aggregates of islet amyloid polypeptide (IAPP) appear as the physical outcome of a peptidic phase transition signaling the onset of type-2 diabetes mellitus in different mammalian species. In particular, experimentally verified mutations on the amyloidogenic segment 20-29 in humans, cats, and rats are highly correlated with the molecular aggregation propensities. Through a microcanonical analysis of the aggregation of IAPP_{20-29} isoforms, we show that a minimalist one-bead hydrophobic-polar continuum model for protein interactions properly quantifies those propensities from free-energy barriers. Our results highlight the central role of sequence-dependent hydrophobic mutations on hot spots for stabilization, and thus for the engineering, of such biological peptides.

Ground-state properties of a polymer chain in an attractive sphere

We analyze the structural formation of a polymer chain inside of an attractive sphere depending on the attraction strength. Our model is composed of a coarse-grained polymer and an attractive sphere potential. Within this frame, multicanonical Monte Carlo simulations are employed to identify the global minimum energies for a polymer chain interacting with the attractive inner wall of the sphere. Different compact structures are found with varying attraction strengths, among which are spherical, three/two, or monolayer. The conformational properties of these structures are presented.

Structural behavior of a polymer chain inside an attractive sphere

We analyze the structural behavior of a single polymer chain inside an attractive sphere. Our model is composed of a coarse-grained polymer and an attractive-sphere potential. By means of extensive multicanonical Monte Carlo simulations, it is shown that the system exhibits a rich phase diagram ranging from highly ordered compact to extended random coil structures and from desorbed to partially or even completely adsorbed conformations. These findings are identified with different structural observables.

STUDIES OF FINDING LOW ENERGY CONFIGURATIONS IN OFF-LATTICE PROTEIN MODELS

We studied two three-dimensional off-lattice protein models with two species of monomers, hydrophobic and hydrophilic. Low energy configurations in both models were optimized using the energy landscape paving (ELP) method and subsequent gradient descent. The numerical results show that the proposed methods are very promising for finding the ground states of proteins. For all sequences with lengths 13 ≤ n ≤ 55, the algorithm finds states with lower energy than previously proposed putative ground states.

3D protein structure prediction with genetic tabu search algorithm

Background

Protein structure prediction (PSP) has important applications in different fields, such as drug design, disease prediction, and so on. In protein structure prediction, there are two important issues. The first one is the design of the structure model and the second one is the design of the optimization technology. Because of the complexity of the realistic protein structure, the structure model adopted in this paper is a simplified model, which is called off-lattice AB model. After the structure model is assumed, optimization technology is needed for searching the best conformation of a protein sequence based on the assumed structure model. However, PSP is an NP-hard problem even if the simplest model is assumed. Thus, many algorithms have been developed to solve the global optimization problem. In this paper, a hybrid algorithm, which combines genetic algorithm (GA) and tabu search (TS) algorithm, is developed to complete this task.

Results

In order to develop an efficient optimization algorithm, several improved strategies are developed for the proposed genetic tabu search algorithm. The combined use of these strategies can improve the efficiency of the algorithm. In these strategies, tabu search introduced into the crossover and mutation operators can improve the local search capability, the adoption of variable population size strategy can maintain the diversity of the population, and the ranking selection strategy can improve the possibility of an individual with low energy value entering into next generation. Experiments are performed with Fibonacci sequences and real protein sequences. Experimental results show that the lowest energy obtained by the proposed GATS algorithm is lower than that obtained by previous methods.

Conclusions

The hybrid algorithm has the advantages from both genetic algorithm and tabu search algorithm. It makes use of the advantage of multiple search points in genetic algorithm, and can overcome poor hill-climbing capability in the conventional genetic algorithm by using the flexible memory functions of TS. Compared with some previous algorithms, GATS algorithm has better performance in global optimization and can predict 3D protein structure more effectively.

Conformational phase diagram for polymers adsorbed on ultrathin nanowires

We study the conformational behavior of a polymer adsorbed at an attractive stringlike nanowire and construct the complete structural phase diagram in dependence of the binding strength and effective thickness of the nanowire. For this purpose, Monte Carlo optimization techniques are employed to identify lowest-energy structures for a coarse-grained model of a polymer in contact with the nanowire. Among the representative conformations in the different phases are, for example, compact droplets attached to the wire and also nanotubelike monolayer films wrapping it in a very ordered way. We here systematically analyze low-energy shapes and structural order parameters to elucidate the transitions between the structural phases.

Adsorption of a hydrophobic-polar-model heteropolymer in an attractive nanotube

The adsorption of an off-lattice hydrophobic-polar-model heteropolymer in an attractive hydrophobic nanotube is presented by means of a multicanonical Monte Carlo simulation. In the model, the Lennard-Jones potential is assumed as an interaction potential between the effective monomers and the nanotube wall. The global minimum-energy configurations and the values of radius of gyration and end-to-end distance of these configurations are compared for both pure hydrophobic and polar heteropolymer models and with interaction term included model. The low-energy conformations for heteropolymers are extended when they interact with the nanotube which the conformational extensions are detected with some structural parameters.

Thermodynamics of tubelike flexible polymers

In this work, we discuss the general phase behavior of short tubelike flexible polymers. The geometric thickness constraint is implemented through the concept of the global radius of curvature. We use sophisticated Monte Carlo sampling methods to simulate small bead-stick polymer models with Lennard-Jones interaction among nonbonded monomers. We analyze energetic fluctuations and structural quantities to classify conformational pseudophases. We find that the tube thickness influences the thermodynamic behavior of simple tubelike polymers significantly, i.e., for a given temperature, the formation of secondary structures strongly depends on the tube thickness.

Structure optimization of the two-dimensional off-lattice hydrophobic-hydrophilic model

A two-dimensional off-lattice protein model with two species of monomers, hydrophobic and hydrophilic, was studied. Low-energy configurations in the model were optimized using the improved energy landscape paving (ELP+) method. In ELP+, the energy landscape paving (ELP) was first applied to search for the low-energy states. After the ELP led to the basins of the local energy minima, the additional degree-of-freedom of bond length was introduced, and the gradient descent method was then used to search for lower energy states near the local minima. Numerical results show that the proposed methods are quite effective for finding the ground states of proteins. A comparison between ELP+ and other methods is made.

Stochastic dynamics of model proteins on a directed graph

A method for reconstructing the potential energy landscape of simple polypeptidic chains is described. We show how to obtain a faithful representation of the energy landscape in terms of a suitable directed graph. Topological and dynamical indicators of the graph are shown to yield an effective estimate of the time scales associated with both folding and equilibration processes. This conclusion is drawn by comparing molecular dynamics simulations at constant temperature with the dynamics on the graph, defined as a temperature-dependent Markov process. The main advantage of the graph representation is that its dynamics can be naturally renormalized by collecting nodes into "hubs" while redefining their connectivity. We show that the dynamical properties at large time scales are preserved by the renormalization procedure. Moreover, we obtain clear indications that the heteropolymers exhibit common topological properties, at variance with the homopolymer, whose peculiar graph structure stems from its spatial homogeneity. In order to distinguish between "fast" and "slow" folders, one has to look at the kinetic properties of the corresponding directed graphs. In particular, we find that the average time needed to the fast folder for reaching its native configuration is two orders of magnitude smaller than its equilibration time while for the bad folder these time scales are comparable.

Enhanced sampling in generalized ensemble with large gap of sampling parameter: case study in temperature space random walk

We present an efficient sampling method for computing a partition function and accelerating configuration sampling. The method performs a random walk in the lambda space, with lambda being any thermodynamic variable that characterizes a canonical ensemble such as the reciprocal temperature beta or any variable that the Hamiltonian depends on. The partition function is determined by minimizing the difference of the thermal conjugates of lambda (the energy in the case of lambda = beta), defined as the difference between the value from the dynamically updated derivatives of the partition function and the value directly measured from simulation. Higher-order derivatives of the partition function are included to enhance the Brownian motion in the lambda space. The method is much less sensitive to the system size, and to the size of lambda window than other methods. On the two dimensional Ising model, it is shown that the method asymptotically converges the partition function, and the error of the logarithm of the partition function is much smaller than the algorithm using the Wang-Landau recursive scheme. The method is also applied to off-lattice model proteins, the AB models, in which cases many low energy states are found in different models.