Lattice Protein Research Articles

Hybrid Ising Machine and Reinforcement Learning Approach to Folding Lattice Proteins

The hydrophobic polarity (HP) protein folding model is a simplified computation model utilized to study the folding of proteins into their three-dimensional structures. Understanding how proteins fold and misfold is fundamental to designing effective, safe drugs, allowing for targeted therapies and better patient outcomes. The amino acids of a protein can be categorized as either hydrophobic or polar, where the hydrophobic beads tend to cluster together in a properly folded protein to minimize exposure to the surrounding environment. Determining the lowest energy configurations of the HP model is a non-deterministic polynomial (NP) hard problem. One method is formulating the HP model as a quadratic unconstrained binary optimization (QUBO) problem which can be mapped to an Ising machine. In the Ising model, the lowest energy state of the system represents the correct lattice configuration of the folded protein. The Ising machine technique utilized of simulated annealing allows for future application of the work on an integrated silicon photonic chip. Alternatively, reinforcement learning designed with an energy-based reward function can optimize the HP model by adapting to the environment without the consequence of remaining in local minima like that of the Ising machine. A hybrid pipeline combining the Ising machine and reinforcement learning sequentially was designed to determine the two-dimensional lattice configurations of various protein chain lengths and sequences, combining the parallel optimization of the Ising machine on an integrated photonic platform and the adaptability of the reinforcement learning algorithm. Results found that this hybrid model improved the accuracy and ability of determining the lowest energy state in contrast to the Ising machine alone as well as improved the efficiency by reducing the number of iterations required when compared solely to reinforcement learning. Future work focuses on implementing a parallel pipeline approach as well as adapting to a three-dimensional model.

Protein superfolds are characterised as frustration-free topologies: A case study of pure parallel β-sheet topologies.

A protein superfold is a type of protein fold that is observed in at least three distinct, non-homologous protein families. Structural classification studies have revealed a limited number of prevalent superfolds alongside several infrequent occurring folds, and in α/β type superfolds, the C-terminal β-strand tends to favor the edge of the β-sheet, while the N-terminal β-strand is often found in the middle. The reasons behind these observations, whether they are due to evolutionary sampling bias or physical interactions, remain unclear. This article offers a physics-based explanation for these observations, specifically for pure parallel β-sheet topologies. Our investigation is grounded in several established structural rules that are based on physical interactions. We have identified "frustration-free topologies" which are topologies that can satisfy all the rules simultaneously. In contrast, topologies that cannot are termed "frustrated topologies." Our findings reveal that frustration-free topologies represent only a fraction of all theoretically possible patterns, these topologies strongly favor positioning the C-terminal β-strand at the edge of the β-sheet and the N-terminal β-strand in the middle, and there is significant overlap between frustration-free topologies and superfolds. We also used a lattice protein model to thoroughly investigate sequence-structure relationships. Our results show that frustration-free structures are highly designable, while frustrated structures are poorly designable. These findings suggest that superfolds are highly designable due to their lack of frustration, and the preference for positioning C-terminal β-strands at the edge of the β-sheet is a direct result of frustration-free topologies. These insights not only enhance our understanding of sequence-structure relationships but also have significant implications for de novo protein design.

Non-Poissonian bursts in the arrival of phenotypic variation can strongly affect the dynamics of adaptation.

Modelling the rate at which adaptive phenotypes appear in a population is a key to predicting evolutionary processes. Given random mutations, should this rate be modelled by a simple Poisson process, or is a more complex dynamics needed? Here we use analytic calculations and simulations of evolving populations on explicit genotype-phenotype maps to show that the introduction of novel phenotypes can be 'bursty' or overdispersed. In other words, a novel phenotype either appears multiple times in quick succession, or not at all for many generations. These bursts are fundamentally caused by statistical fluctuations and other structure in the map from genotypes to phenotypes. Their strength depends on population parameters, being highest for 'monomorphic' populations with low mutation rates. They can also be enhanced by additional inhomogeneities in the mapping from genotypes to phenotypes. We mainly investigate the effect of bursts using the well-studied genotype-phenotype map for RNA secondary structure, but find similar behaviour in a lattice protein model and in Richard Dawkins's biomorphs model of morphological development. Bursts can profoundly affect adaptive dynamics. Most notably, they imply that fitness differences play a smaller role in determining which phenotype fixes than would be the case for a Poisson process without bursts.

Synergistic treatment of androgenetic alopecia with follicular co-delivery of minoxidil and cedrol in metal–organic frameworks stabilized by covalently cross-linked cyclodextrins

Androgenetic alopecia seriously affects the physical and mental health of patients. The main clinical therapeutic agent, minoxidil tincture, is challenged by solvent irritation and dose-dependent side effects. Our recent work has identified a biosafety natural product, cedrol, that is synergistic in combination with minoxidil, thereby improving medication safety by substantially reducing the clinical dose of minoxidil. In addition, ccross-linked CD-MOF were designed as carriers for hair follicle delivery, and γ-CD in the carriers was cross-linked by diphenyl carbonate with covalent bonds to protect the CD-MOF from rapid disintegration in an aqueous environment. This improved nanocarrier has a drug loading of 25%, whereas nanocarriers increased drug delivery to the hair follicles through ratchet effect, and increased human dermal papilla cells uptake of drugs via endocytosis pathways mainly mediated by lattice proteins, energy-dependent active transport, and lipid raft-dependent, thus improved cell viability, proliferation, and migration, followed by significantly enhancing the anti-androgenetic alopecia effect, with cedrol focusing on inhibiting 5α-reductase and activating Shh/Gli pathway, and minoxidil, which up-regulated VEGF, down-regulated TGF-β, and activated ERK/AKT pathway. This drug combination provides a new therapeutic strategy for androgenetic alopecia, while the newly developed cross-linked CD-MOF has been shown to serve as a promising follicular delivery vehicle.

Using quantum annealing to design lattice proteins

Using quantum annealing to design lattice proteins

Evolutionary dynamics of a lattice dimer: a toy model for stability vs. affinity trade-offs in proteins

Understanding how a stressor applied on a biological system shapes its evolution is key to achieving targeted evolutionary control. Here we present a toy model of two interacting lattice proteins to quantify the response to the selective pressure defined by the binding energy. We generate sequence data of proteins and study how the sequence and structural properties of dimers are affected by the applied selective pressure, both during the evolutionary process and in the stationary regime. In particular we show that internal contacts of native structures lose strength, while inter-structure contacts are strengthened due to the folding-binding competition. We discuss how dimerization is achieved through enhanced mutability on the interacting faces, and how the designability of each native structure changes upon introduction of the stressor.

Pharmacokinetics, absorption and transport mechanism for ginseng polysaccharides

BackgroundGinseng polysaccharide (GP) is one of the most abundant components in Panax ginseng. However, the absorption pathways and mechanisms of GPs have not been investigated systematically due to the challenges of their detection. MethodsThe fluorescein isothiocyanate derivative (FITC) was employed to label GP and ginseng acidic polysaccharide (GAP) to obtain target samples. HPLC-MS/MS assay was used to determine the pharmacokinetics of GP and GAP in rats. The Caco-2 cell model was used to investigate the uptake and transport mechanisms of GP and GAP in rats. ResultsOur results demonstrated that the absorption of GAP was more than that of GP in rats after gavage administration, while there was no significant difference between both after intravenous administration. In addition, we found that GAP and GP were more distributed in the kidney, liver and genitalia, suggesting that GAP and GP are highly targeted to the liver, kidney and genitalia. Importantly, we explored the uptake mechanism of GAP and GP. GAP and GP are endocytosed into the cell via lattice proteins or niche proteins. Both are transported lysosomally mediated to the endoplasmic reticulum (ER) and then enter the nucleus through the ER, thus completing the process of intracellular uptake and transportation. ConclusionOur results confirm that the uptake of GPs by small intestinal epithelial cells is primarily mediated via lattice proteins and the cytosolic cellar. The discovery of important pharmacokinetic properties and the uncovering of the absorption mechanism provide a research rationale for the research of GP formulation and clinical promotion.

Folding lattice proteins with quantum annealing

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.

Extending fluorescence anisotropy to large complexes using reversibly switchable proteins

The formation of macromolecular complexes can be measured by detection of changes in rotational mobility using time-resolved fluorescence anisotropy. However, this method is limited to relatively small molecules (~0.1–30 kDa), excluding the majority of the human proteome and its complexes. We describe selective time-resolved anisotropy with reversibly switchable states (STARSS), which overcomes this limitation and extends the observable mass range by more than three orders of magnitude. STARSS is based on long-lived reversible molecular transitions of switchable fluorescent proteins to resolve the relatively slow rotational diffusivity of large complexes. We used STARSS to probe the rotational mobility of several molecular complexes in cells, including chromatin, the retroviral Gag lattice and activity-regulated cytoskeleton-associated protein oligomers. Because STARSS can probe arbitrarily large structures, it is generally applicable to the entire human proteome.

The cavity method to protein design problem

In this study, we propose an analytic statistical mechanics approach to solve a fundamental problem in biological physics called protein design. Protein design is an inverse problem of protein structure prediction, and its solution is the amino acid sequence that best stabilizes a given conformation. Despite recent rapid progress in protein design using deep learning, the challenge of exploring protein design principles remains. Contrary to previous computational physics studies, we used the cavity method, an extension of the mean-field approximation that becomes rigorous when the interaction network is a tree. We found that for small two-dimensional lattice hydrophobic-polar protein models, the design by the cavity method yields results almost equivalent to those from the Markov chain Monte Carlo method with lower computational cost.

Tetrahymena thermophila granule lattice protein 3 improves solubility of sexual stage malaria antigens expressed in Escherichia coli

The requirement for low cost manufacturing makes bacterial cells a logical platform for the production of recombinant subunit vaccines for malaria. However, protein solubility has been a major stumbling block with prokaryotic expression systems. Notable examples include the transmission blocking vaccine candidates, Pfs25 and Pfs48/45, which are almost entirely insoluble when expressed as recombinant proteins in Escherichia coli. Various solubility tags have been used with limited success in improving solubility, although recent studies with granule lattice protein 1 (Grl1p) from the ciliated protozoan, Tetrahymena thermophila, have shown promise. Here, we examine a related solubility tag, granule lattice protein 3 (Grl3p) from T. thermophila, and compare it to both Grl1p and the well-studied maltose binding protein (MBP) used to improve the solubility of multiple protein targets. We find that Grl3p performs comparably to Grl1p when linked to Pfs25 but significantly improves solubility when paired with Pfs48/45.

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.

Achieving a realistic native protein ensemble by HDX-MS and computational modeling

Amide hydrogen-deuterium exchange (HDX) measurements pioneered in the 1950s and developed extensively by Englander in the 1980s (1), still give us more information than we understand about protein motions. Some backbone amides exchange within milliseconds, whereas others may take months. How amides exchange in a folded protein and why some exchange so quickly and others so slowly has puzzled biophysicists for several generations. What exactly are the underlying physicochemical phenomena that result in amide exchange and how do the rates of exchange relate to structural dynamics? Early investigators, including Ken Dill and Hue Sun Chan, worked in statistical mechanical lattice models of proteins to try to answer this question (2,3).

Barriers and dynamical paths in alternating Gibbs sampling of restricted Boltzmann machines

Restricted Boltzmann machines (RBM) are bilayer neural networks used for the unsupervised learning of model distributions from data. The bipartite architecture of RBM naturally defines an elegant sampling procedure, called alternating Gibbs sampling (AGS), where the configurations of the latent-variable layer are sampled conditional to the data-variable layer and vice versa. We study here the performance of AGS on several analytically tractable models borrowed from statistical mechanics. We show that standard AGS is not more efficient than classical Metropolis-Hastings (MH) sampling of the effective energy landscape defined on the data layer. However, RBM can identify meaningful representations of training data in their latent space. Furthermore, using these representations and combining Gibbs sampling with the MH algorithm in the latent space can enhance the sampling performance of the RBM when the hidden units encode weakly dependent features of the data. We illustrate our findings on three datasets: Bars and Stripes and MNIST, well known in machine learning, and the so-called lattice proteins dataset, introduced in theoretical biology to study the sequence-to-structure mapping in proteins.

Thermodynamics of hydrophobic-polar model proteins on the face-centered cubic lattice.

The HP model, a coarse-grained protein representation with only hydrophobic (H) and polar (P) amino acids, has already been extensively studied on the simple cubic (SC) lattice. However, this geometry severely restricts possible bond angles, and a simple improvement is to instead use the face-centered cubic (fcc) lattice. In this paper, the density of states and ground state energies are calculated for several benchmark HP sequences on the fcc lattice using the replica-exchange Wang-Landau algorithm and a powerful set of Monte Carlo trial moves. Results from the fcc lattice proteins are directly compared with those obtained from a previous lattice protein folding study with a similar methodology on the SC lattice. A thermodynamic analysis shows comparable folding behavior between the two lattice geometries, but with a greater rate of hydrophobic-core formation persisting into lower temperatures on the fcc lattice.

Lattice protein design using Bayesian learning.

Protein design is the inverse approach of the three-dimensional (3D) structure prediction for elucidating the relationship between the 3D structures and amino acid sequences. In general, the computation of the protein design involves a double loop: A loop for amino acid sequence changes and a loop for an exhaustive conformational search for each amino acid sequence. Herein, we propose a novel statistical mechanical design method using Bayesian learning, which can design lattice proteins without the exhaustive conformational search. We consider a thermodynamic hypothesis of the evolution of proteins and apply it to the prior distribution of amino acid sequences. Furthermore, we take the water effect into account in view of the grand canonical picture. As a result, on applying the 2D lattice hydrophobic-polar (HP) model, our design method successfully finds an amino acid sequence for which the target conformation has a unique ground state. However, the performance was not as good for the 3D lattice HP models compared to the 2D models. The performance of the 3D model improves on using a 20-letter lattice proteins. Furthermore, we find a strong linearity between the chemical potential of water and the number of surface residues, thereby revealing the relationship between protein structure and the effect of water molecules. The advantage of our method is that it greatly reduces computation time, because it does not require long calculations for the partition function corresponding to an exhaustive conformational search. As our method uses a general form of Bayesian learning and statistical mechanics and is not limited to lattice proteins, the results presented here elucidate some heuristics used successfully in previous protein design methods.

Fluid flow in the sarcomere

A highly organized and densely packed lattice of molecular machinery within the sarcomeres of muscle cells powers contraction. Although many of the proteins that drive contraction have been studied extensively, the mechanical impact of fluid shearing within the lattice of molecular machinery has received minimal attention. It was recently proposed that fluid flow augments substrate transport in the sarcomere, however, this analysis used analytical models of fluid flow in the molecular machinery that could not capture its full complexity. By building a finite element model of the sarcomere, we estimate the explicit flow field, and contrast it with analytical models. Our results demonstrate that viscous drag forces on sliding filaments are surprisingly small in contrast to the forces generated by single myosin molecular motors. This model also indicates that the energetic cost of fluid flow through viscous shearing with lattice proteins is likely minimal. The model also highlights a steep velocity gradient between sliding filaments and demonstrates that the maximal radial fluid velocity occurs near the tips of the filaments. To our knowledge, this is the first computational analysis of fluid flow within the highly structured sarcomere.

Quantifying the Mutational Robustness of Protein-Coding Genes.

We use large-scale mutagenesis data and computer simulations to quantify the mutational robustness of protein-coding genes by taking into account constraints arising from protein function and the genetic code. Analyses of the distribution of amino acid substitutions from 18 mutagenesis studies revealed an average of 45% of neutral variants; while mutagenesis data of 12 proteins artificially designed under no other constraints but stability, reach an average of 60%. Simulations using a lattice protein model allow us to contrast these estimates to the expected mutational robustness of protein families by generating unbiased samples of foldable sequences, which we find to have 30% of neutral variants. In agreement with mutagenesis data of designed proteins, the model shows that maximally robust protein families might access up to twice the amount of neutral variants observed in the unbiased samples (i.e. 60%). A biophysical model of protein-ligand binding suggests that constraints associated to molecular function have only a moderate impact on robustness of approximately 5 to 10% of neutral variants; and that the direction of this effect depends on the relation between functional performance and thermodynamic stability. Although the genetic code constraints the access of a gene's nucleotide sequence to only 30% of the full distribution of amino acid mutations, it provides an extra 15 to 20% of neutral variants to the estimations above, such that the expected, observed, and maximal robustness of protein-coding genes are approximately 50, 65, and 75%, respectively. We discuss our results in the light of three main hypothesis put forward to explain the existence of mutationally robust genes.

Contributions of topological polar-polar contacts to achieve better folding stability of 2D/3D HP lattice proteins: An <i>in silico</i> approach

<abstract> <p>Many of the simplistic hydrophobic-polar lattice models, such as Dill's model (called <bold>Model 1</bold> herein), are aimed to fold structures through hydrophobic-hydrophobic interactions mimicking the well-known hydrophobic collapse present in protein structures. In this work, we studied 11 designed hydrophobic-polar sequences, S<sub>1</sub>-S<sub>8</sub> folded in 2D-square lattice, and S<sub>9</sub>-S<sub>11</sub> folded in 3D-cubic lattice. And to better fold these structures we have developed <bold>Model 2</bold> as an approximation to convex function aimed to weight hydrophobic-hydrophobic but also polar-polar contacts as an augmented version of <bold>Model 1</bold>. In this partitioned approach hydrophobic-hydrophobic ponderation was tuned as <italic>α</italic>-1 and polar-polar ponderation as <italic>α</italic>. This model is centered in preserving required hydrophobic substructure, and at the same time including polar-polar interactions, otherwise absent, to reach a better folding score now also acquiring the polar-polar substructure. In all tested cases the folding trials were better achieved with <bold>Model 2</bold>, using <italic>α</italic> values of 0.05, 0.1, 0.2 and 0.3 depending of sequence size, even finding optimal scores not reached with <bold>Model 1</bold>. An important result is that the better folding score, required the lower <italic>α</italic> weighting. And when <italic>α</italic> values above 0.3 are employed, no matter the nature of the hydrophobic-polar sequence, banning of hydrophobic-hydrophobic contacts started, thus yielding misfolding of sequences. Therefore, the value of <italic>α</italic> to correctly fold structures is the result of a careful weighting among hydrophobic-hydrophobic and polar-polar contacts.</p> </abstract>

Evolution Rapidly Optimizes Stability and Aggregation in Lattice Proteins Despite Pervasive Landscape Valleys and Mazes.

The "fitness" landscapes of genetic sequences are characterized by high dimensionality and "ruggedness" due to sign epistasis. Ascending from low to high fitness on such landscapes can be difficult because adaptive trajectories get stuck at low-fitness local peaks. Compounding matters, recent theoretical arguments have proposed that extremely long, winding adaptive paths may be required to reach even local peaks: a "maze-like" landscape topography. The extent to which peaks and mazes shape the mode and tempo of evolution is poorly understood, due to empirical limitations and the abstractness of many landscape models. We explore the prevalence, scale, and evolutionary consequences of landscape mazes in a biophysically grounded computational model of protein evolution that captures the "frustration" between "stability" and aggregation propensity. Our stability-aggregation landscape exhibits extensive sign epistasis and local peaks galore. Although this frequently obstructs adaptive ascent to high fitness and virtually eliminates reproducibility of evolutionary outcomes, many adaptive paths do successfully complete the ascent from low to high fitness, with hydrophobicity a critical mediator of success. These successful paths exhibit maze-like properties on a global landscape scale, in which taking an indirect path helps to avoid low-fitness local peaks. This delicate balance of "hard but possible" adaptation could occur more broadly in other biological settings where competing interactions and frustration are important.