Subexponential-Time Framework for Optimal Embeddings of Graphs in Integer Lattices

We consider the problem of protein folding in the HP model on the 3D square lattice. This problem is combinatorially equivalent to folding a string of 0’s and 1’s so that the string forms a self-avoiding walk on the lattice and the number of adjacent pairs of 1’s is maximized. The previously best-known approximation algorithm for this problem has a guarantee of \(\frac{3}{8}=.375\) [HI95]. In this paper, we first present a new \(\frac{3}{8}\)-approximation algorithm for the 3D folding problem that improves on the absolute approximation guarantee of the previous algorithm. We then show a connection between the 3D folding problem and a basic combinatorial problem on binary strings, which may be of independent interest. Given a binary string in { a,b }*, we want to find a long subsequence of the string in which every sequence of consecutive a’s is followed by at least as many consecutive b’s. We show a non-trivial lower-bound on the existence of such subsequences. Using this result, we obtain an algorithm with a slightly improved approximation ratio of at least .37501 for the 3D folding problem. All of our algorithms run in linear time.

BackgroundPredicting protein structure from amino acid sequence is a prominent problem in computational biology. The long range interactions (or non-local interactions) are known as the main source of complexity for protein folding and dynamics and play the dominant role in the compact architecture. Some simple but exact model, such as HP model, captures the pain point for this difficult problem and has important implications to understand the mapping between protein sequence and structure.ResultsIn this paper, we formulate the biological problem into optimization model to study the hydrophobic-hydrophilic model on 3D square lattice. This is a combinatorial optimization problem and known as NP-hard. Particle swarm optimization is utilized as the heuristic framework to solve the hard problem. To avoid premature in computation, we incorporated the Tabu search strategy. In addition, a pulling strategy was designed to accelerate the convergence of algorithm based on the characteristic of native protein structure. Together a novel hybrid method combining particle swarm optimization, Tabu strategy, and pulling strategy can fold the amino acid sequences on 3D square lattice efficiently. Promising results are reported in several examples by comparing with existing methods. This allows us to use this tool to study the protein stability upon amino acid mutation on 3D lattice. In particular, we evaluate the effect of single amino acid mutation and double amino acids mutation via 3D HP lattice model and some useful insights are derived.ConclusionWe propose a novel hybrid method to combine several heuristic strategies to study HP model on 3D lattice. The results indicate that our hybrid method can predict protein structure more accurately and efficiently. Furthermore, it serves as a useful tools to probe the protein stability on 3D lattice and provides some biological insights.

Recent studies of the tube model of protein have indicated that the free energy landscape of proteins is presculpted by symmetry of the protein backbone and geometrical constraints played by the hydrogen bonds. In this study, we investigate the role of amino acid sequences in the folding of proteins. We consider two models that are differed by sequence specificity: the tube HP model with hydrophobic (H) and polar (P) sequences, and the tube Go model with native-centric contact potentials. Monte Carlo simulations are carried out for two sequences of length of 48 amino acids, whose ground states are a three-helix bundle and a GB1-like structure. The results show that folding inthe Go model is more cooperative than in the HP model. In the HP model the collapse transition and the folding transition are separated, whereas in the Go model the two transitions coincide.

The prediction of a minimum energy protein structure from its amino acid sequence represents an important and challenging problem in computational biology. In this paper, we propose a novel heuristic approach for protein structure prediction (PSP) based on the concept of optimal hydrophobic core formation. Using 2D HP model, a well-known set of sub-structures analogous to the secondary structures are obtained. Some sub-conformations are appropriately classified and then incorporated as prior knowledge. Unlike most of the popular PSP approaches which are stochastic in nature, the proposed method is deterministic. The effectiveness of the proposed algorithm is evaluated by well-known benchmark as well as non-benchmark sequences commonly used with 2D HP model. Maintaining similar accuracy as other core based and population based algorithms our method is significantly faster and reduces the computation time as it avoids blind search within the hydrophobic core (H-Core).

Predição de estruturas de proteínas (PSP) é um problema computacionalmente complexo. Modelos simplificados da molécula proteica (como o Modelo HP) e o uso de Algoritmos Evolutivos (AEs) estão entre as principais técnicas investigadas para PSP. Entretanto, a avaliação de uma estrutura representada pelo Modelo HP considera apenas o número de contatos hidrofóbicos, não possibilitando distinguir entre estruturas com o mesmo número de contatos hidrofóbicos. Neste trabalho, é apresentada uma nova formulação multiobjetivo para PSP em Modelo HP. Duas métricas são avaliadas: o número de contatos hidrofóbicos e a distância entre os aminoácidos hidrofóbicos, as quais são tratados pelo AE Multiobjetivo em Tabelas (AEMT). O algoritmo mostrou-se rápido e robusto.

Protein structure prediction is the problem of finding the functional conformation of a protein given only its amino acid sequence. The HP lattice model is an abstract formulation of this problem, which captures the fact that hydrophobicity is one of the major driving forces in the protein folding process. This model represents a hard combinatorial optimization problem and has been widely addressed through metaheuristics such as evolutionary algorithms. However, the conventional energy (evaluation) function of the HP model does not provide an adequate discrimination among potential solutions, which is an essential requirement for metaheuristics in order to perform an effective search. Therefore, alternative energy functions have been proposed in the literature to cope with this issue. In this study, we inquire into the effectiveness of several of such alternative approaches. We analyzed the degree of discrimination provided by each of the studied functions as well as their impact on the behavior of a basic memetic algorithm. The obtained results support the relevance of following this research direction. To our knowledge, this is the first work reported in this regard.

The HP model for protein structure prediction abstracts the fact that hydrophobicity is a dominant force in the protein folding process. This challenging combinatorial optimization problem has been widely addressed through metaheuristics. The evaluation function is a key component for the success of metaheuristics; the poor discrimination of the conventional evaluation function of the HP model has motivated the proposal of alternative formulations for this component. This comparative analysis inquires into the effectiveness of seven different evaluation functions for the HP model. The degree of discrimination provided by each of the studied functions, their capability to preserve a rank ordering among potential solutions which is consistent with the original objective of the HP model, as well as their effect on the performance of local search methods are analyzed. The obtained results indicate that studying alternative evaluation schemes for the HP model represents a highly valuable direction which merits more attention.

Proteins are one of the essential bio-molecules and control almost all functions in the living cell. They act to perform many of the biological functions in different forms, including catalytic activities in the cell (enzymes), transport proteins (hemoglobin), transmission and coordination between cells and organs of the body (hormones), molecules of the immune system (antibodies), and the transport of small molecules across the cell membranes and so on. Since the origin of life, the protein molecules have attained a specifically designed architecture by nature under the normal physiological conditions. Different functional proteins adopt compact three-dimensional shapes that are biologically active, known as the native conformations. Similarly, if any of the physiological conditions deviate, then the protein extends its conformation. The protein folding problem describes how the native conformation is predicted from a given amino acid sequence. The study of protein folding mechanisms has attracted researchers to conduct intensive research both in experimental and theoretical approaches over the past few years due to its tremendous scope and industrial applications. It is very straightforward to analyze the amino acid sequence of a given protein; however, the prediction of the three-dimensional structure followed by its folding mechanism is a challenging task. For almost 50 years, researchers have been trying to study the protein folding problem using many computational approaches. Many of the artificial intelligence algorithms are suitably used to study the protein folding mechanism. There is a broader scope in applying artificial neural networks (ANN) to study protein folding. As per the recent literature, ANN-based protein folding algorithms are a million times faster than any other algorithms. So in this chapter, the details of ANN-based algorithms and their implementation in the study of protein folding has been discussed by narrating the recent literature.

Protein folding: Web alert

In this paper, we present a novel approximation algorithm to solve the protein folding problem in HP model. Our algorithm is polynomial in terms of the length of the given HP string. The expected approximation ratio of our algorithm is for n ≥ 6, where n2 is the total number of H’s in a given HP string. The expected approximation ratio tends to reach 1 for large values of n. Hence our algorithm is expected to perform very well for larger HP strings.

Given an input consisting of an n-vertex convex polygon with k hole vertices or an n-vertex planar straight line graph (PSLG) with k holes and/or reflex vertices inside the convex hull, the parameterized minimum number convex partition (MNCP) problem asks for a partition into a minimum number of convex pieces. We give a fixed-parameter tractable algorithm for this problem that runs in the following time complexities: – linear time if k is constant, – time polynomial in n if $k=O(\frac{{\rm log}n}{{\rm log log}n})$, or, to be exact, in O(nk$^{\rm 6{\it k}-5}$ 216k) time.

Protein folding and de novo protein design for biotechnological applications

ProteinShop Gives Researchers a Hands-On Tool for Manipulating, Visualizing Protein Structures. The Human Genome Project and other biological research efforts are creating an avalanche of new data about the chemical makeup and genetic codes of living organisms. But in order to make sense of this raw data, researchers need software tools which let them explore and model data in a more intuitive fashion. With this in mind, researchers at Lawrence Berkeley National Laboratory and the University of California, Davis, have developed ProteinShop, a visualization and modeling program which allows researchers to manipulate protein structures with pinpoint control, guided in large part by their own biological and experimental instincts. Biologists have spent the last half century trying to unravel the ''protein folding problem,'' which refers to the way chains of amino acids physically fold themselves into three-dimensional proteins. This final shape, which resembles a crumpled ribbon or piece of origami, is what determines how the protein functions and translates genetic information. Understanding and modeling this geometrically complex formation is no easy matter. ProteinShop takes a given sequence of amino acids and uses visualization guides to help generate predictions about the secondary structures, identifying alpha helices and flat beta strands, and the coil regions that bind them. Once secondary structures are in place, researchers can twist and turn these pre-configurations until they come up with a number of possible tertiary structure conformations. In turn, these are fed into a computationally intensive optimization procedure that tries to find the final, three-dimensional protein structure. Most importantly, ProteinShop allows users to add human knowledge and intuition to the protein structure prediction process, thus bypassing bad configurations that would otherwise be fruitless for optimization. This saves compute cycles and accelerates the entire process, so that more and larger problems can be attempted. Currently, the program designers are working to make ProteinShop more applicable and adaptable to different protein folding methodologies. If users could manipulate structures from a biological point of view, and then put them back in the queue for more optimization, the process of experimentation and discovery in protein research could be greatly enhanced. The group is also investigating the use of stereoscopic rendering and three-dimensional input devices to remove the limitations of a two-dimensional interface. Clearly, protein-folding research will have far-reaching ramifications. It could lead to new insights about diseases ranging from Alzheimer's to Cystic fibrosis, which scientists believe are caused by protein folding gone wrong. A better understanding of protein structures could also lead to the engineering of altogether new proteins, and shed light on how drugs bind proteins to alter their structure and function. Above all, ProteinShop is an important tool that will help scientists unravel one of the most challenging problems that theoretical and computational chemistry has to offer.

One of the central challenges in structural molecular biology today is the protein folding problem, i.e. the acquisition of the 3D structure of a protein from its linear sequence of amino-acids. Different computational approaches to study protein folding and protein unfolding have recently become common tools available to the researcher. However, due to the lack of appropriate infrastructures, it is very difficult to directly compare simulations performed by different groups, with different methods, in different experimental conditions or for different proteins. Thus, we set out to create a public data repository with the goal of addressing the problem of comparison, analysis and sharing of information and data on protein folding and protein unfolding simulations. The P-found system for protein folding and protein unfolding simulations is presented. At the moment, the data repository allows uploading of molecular dynamics (MD) protein folding and unfolding simulations, calculates and stores several time series with the variation over time of pre-defined molecular properties, and allows searching and downloading of these data. In the near future, simulations performed by other than MD methods may be uploaded, and data mining techniques for analysis and comparison of multiple simulations will be implemented. The home page for the P-found system is accessible at http://www.p-found.org

We use two simple models and the energy landscape perspective to study protein folding kinetics. A major challenge has been to use the landscape perspective to interpret experimental data, which requires ensemble averaging over the microscopic trajectories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experiments show that two-state relaxation times have "chevron plot" dependences on denaturant and non-Arrhenius dependences on temperature. We find that HP and HP+ models can account for these behaviors. The HP model often gives bumpy landscapes with many kinetic traps and multiple-exponential behavior, whereas the HP+ model gives more smooth funnels and two-state behavior. Multiple-exponential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often involves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process.