Optimization methods for side-chain positioning and macromolecular docking

This dissertation proposes new optimization algorithms targeting protein-protein docking which is an important class of problems in computational structural biology. The ultimate goal of docking methods is to predict the 3-dimensional structure of a stable protein-protein complex. We study two specific problems encountered in predictive docking of proteins. The first problem is Side-Chain Positioning (SCP), a central component of homology modeling and computational protein docking methods. We formulate SCP as a Maximum Weighted Independent Set (MWIS) problem on an appropriately constructed graph. Our formulation also considers the significant special structure of proteins that SCP exhibits for docking. We develop an approximate algorithm that solves a relaxation of MWIS and employ randomized estimation heuristics to obtain high-quality feasible solutions to the problem. The algorithm is fully distributed and can be implemented on multi-processor architectures. Our computational results on a benchmark set of protein complexes show that the accuracy of our approximate MWIS-based algorithm predictions is comparable with the results achieved by a state-of-the-art method that finds an exact solution to SCP. The second problem we target in this work is protein docking refinement. We propose two different methods to solve the refinement problem. The first approach is based on a Monte Carlo-Minimization (MCM) search to optimize rigid-body and side-chain conformations for binding. In particular, we study the impact of optimally positioning the side-chains in the interface region between two proteins in the process of binding. We report computational results showing that incorporating side-chain flexibility in docking provides substantial improvement in the quality of docked predictions compared to the rigid-body approaches. Further, we demonstrate that the inclusion of unbound side-chain conformers in the side-chain search introduces significant improvement in the performance of the docking refinement protocols. In the second approach, we propose a novel stochastic optimization algorithm based on Subspace Semi-Definite programming-based Underestimation (SSDU), which aims to solve protein docking and protein structure prediction. SSDU is based on underestimating the binding energy function in a permissive subspace of the space of rigid-body motions. We apply Principal Component Analysis (PCA) to determine the permissive subspace and reduce the dimensionality of the conformational search space. We consider the general class of convex polynomial underestimators, and formulate the problem of finding such underestimators as a Semi-Definite Programming (SDP) problem. Using these underestimators, we perform a biased sampling in the vicinity of the conformational regions where the energy function is at its global minimum. Moreover, we develop an exploration procedure based on density-based clustering to detect the near-native regions even when there are many local minima residing far from each other. We also incorporate a Model Selection procedure into SSDU to pick a predictive conformation. Testing our algorithm over a benchmark of protein complexes indicates that SSDU substantially improves the quality of docking refinement compared with existing methods

Knowledge-based energy functions for computational studies of proteins

This chapter discusses theoretical framework and methods for developing knowledge-based potential functions essential for protein structure prediction, protein-protein interaction, and protein sequence design. We discuss in some details about the Miyazawa-Jernigan contact statistical potential, distance-dependent statistical potentials, as well as geometric statistical potentials. We also describe a geometric model for developing both linear and non-linear potential functions by optimization. Applications of knowledge-based potential functions in protein-decoy discrimination, in protein-protein interactions, and in protein design are then described. Several issues of knowledge-based potential functions are finally discussed.Comment: 57 pages, 6 figures. To be published in a book by Springe

Constructive Approximation and Learning by Greedy Algorithms

This thesis develops several kernel-based greedy algorithms for different machine learning problems and analyzes their theoretical and empirical properties. Greedy approaches have been extensively used in the past for tackling problems in combinatorial optimization where finding even a feasible solution can be a computationally hard problem (i.e., not solvable in polynomial time). A key feature of greedy algorithms is that a solution is constructed recursively from the smallest constituent parts. In each step of the constructive process a component is added to the partial solution from the previous step and, thus, the size of the optimization problem is reduced. The selected components are given by optimization problems that are simpler and easier to solve than the original problem. As such schemes are typically fast at constructing a solution they can be very effective on complex optimization problems where finding an optimal/good solution has a high computational cost. Moreover, greedy solutions are rather intuitive and the schemes themselves are simple to design and easy to implement. There is a large class of problems for which greedy schemes generate an optimal solution or a good approximation of the optimum. In the first part of the thesis, we develop two deterministic greedy algorithms for optimization problems in which a solution is given by a set of functions mapping an instance space to the space of reals. The first of the two approaches facilitates data understanding through interactive visualization by providing means for experts to incorporate their domain knowledge into otherwise static kernel principal component analysis. This is achieved by greedily constructing embedding directions that maximize the variance at data points (unexplained by the previously constructed embedding directions) while adhering to specified domain knowledge constraints. The second deterministic greedy approach is a supervised feature construction method capable of addressing the problem of kernel choice. The goal of the approach is to construct a feature representation for which a set of linear hypotheses is of sufficient capacity — large enough to contain a satisfactory solution to the considered problem and small enough to allow good generalization from a small number of training examples. The approach mimics functional gradient descent and constructs features by fitting squared error residuals. We show that the constructive process is consistent and provide conditions under which it converges to the optimal solution. In the second part of the thesis, we investigate two problems for which deterministic greedy schemes can fail to find an optimal solution or a good approximation of the optimum. This happens as a result of making a sequence of choices which take into account only the immediate reward without considering the consequences onto future decisions. To address this shortcoming of deterministic greedy schemes, we propose two efficient randomized greedy algorithms which are guaranteed to find effective solutions to the corresponding problems. In the first of the two approaches, we provide a mean to scale kernel methods to problems with millions of instances. An approach, frequently used in practice, for this type of problems is the Nyström method for low-rank approximation of kernel matrices. A crucial step in this method is the choice of landmarks which determine the quality of the approximation. We tackle this problem with a randomized greedy algorithm based on the K-means++ cluster seeding scheme and provide a theoretical and empirical study of its effectiveness. In the second problem for which a deterministic strategy can fail to find a good solution, the goal is to find a set of objects from a structured space that are likely to exhibit an unknown target property. This discrete optimization problem is of significant interest to cyclic discovery processes such as de novo drug design. We propose to address it with an adaptive Metropolis–Hastings approach that samples candidates from the posterior distribution of structures conditioned on them having the target property. The proposed constructive scheme defines a consistent random process and our empirical evaluation demonstrates its effectiveness across several different application domains

Protein-Protein Docking Using Long Range Nuclear Magnetic Resonance Constraints

One of the main methods for experimentally determining protein structure is nuclear magnetic resonance (NMR) spectroscopy. The advantage of using NMR compared to other methods is that the molecule may be studied in its natural state and environment. However, NMR is limited in its facility to analyze multi-domain molecules because of the scarcity of inter-atomic NMR constraints between the domains. In those cases it might be possible to dock the domains based on long range NMR constraints that are related to the molecule's overall structure. We present two computational methods for rigid docking based on long range NMR constraints. The first docking method is based on the overall alignment tensor of the complex. The docking algorithm is based on the minimization of the difference between the predicted and experimental alignment tensor. In order to efficiently dock the complex we introduce a new, computationally efficient method called PATI for predicting the molecular alignment tensor based on the three-dimensional structure of the molecule. The increase in speed compared to the currently best-known method (PALES) is achieved by re-expressing the problem as one of numerical integration, rather than a simple uniform sampling (as in the PALES method), and by using a convex hull rather than a detailed representation of the surface of a molecule. Using PATI, we derive a method called PATIDOCK for efficiently docking a two-domain complex based solely on the novel idea of using the difference between the experimental alignment tensor and the predicted alignment tensor computed by PATI. We show that the alignment tensor fundamentally contains enough information to accurately dock a two-domain complex, and that we can very quickly dock the two domains by pre-computing the right set of data. A second new docking method is based on a similar concept but using the rotational diffusion tensor. We derive a minimization algorithm for this docking method by separating the problem into two simpler minimization problems and approximating our energy function by a quadratic equation. These methods provide two new efficient procedures for protein docking computations

Optimization in bioinformatics

In this work, we present novel optimization approaches for important bioinformatical problems. The rst part deals mainly with the local optimization of molecular structures and its applications to molecular docking, while the second part discusses discrete global optimization. In the rst part, we present a novel algorithm to an old task: nd the next local optimum into a given direction on a molecular potential energy function (line search). We show that replacing a standard line search method with the new algorithm reduced the number of function/gradient evaluations in our test runs down to 47.7% (down to 85% on average) . Then, we include this method into our novel approach for locally optimizing exible ligands in the presence of their receptors, which we describe in detail, avoiding the singularity problem of orientational parameters. We extend this approach to a full ligand-receptor docking program using a Lamarckian genetic algorithm. Our validation runs show that we gained an up to tenfold speedup in comparison to other tested methods. Then, we further incorporate side chain exibility of the receptor into our approach and introduce limited backbone exibility by interpolating between known extremal conformations using spherical linear extrapolation. Our results show that this approach is very promising for exible ligand-receptor docking. However, the drawback is that we need known extremal backbone conformations for the interpolation. In the last section of the rst part, we allow a loop region to be fully exible. We present a new method to nd all possible conformations using the Go-Scheraga ring closure equations and interval arithmetic. Our results show that this algorithm reliably nds alternative conformations and is able to identify promising loop/ligand complexes of the studied example. In the second part of this work, we describe the bond order assignment problem for molecular structures. We present our novel linear 0-1-programming formulation for the very efficient computation of all optimal and suboptimal bond order assignments and show that our approach does not only outperform the original heuristic approach of Wang et al. but also commonly used software for determining bond orders on our test set considering all optimal results. This test set consists of 761 thoroughly prepared drug like molecules that were originally used for the validation of the Merck Molecular Force Field. Then, we present our lter method for feature subset selection that is based on mutual information and uses second order information. We show our mathematically well motivated criterion and, in contrast to other methods, solve the resulting optimization problem exactly by quadratic 0-1-programming. In the validation runs, our method could achieve in 18 out of 21 test scenarios the best classification accuracies. In the last section, we give our integer linear programming formulation for the detection of deregulated subgraphs in regulatory networks using expression proles. Our approach identies the subnetwork of a certain size of the regulatory network with the highest sum of node scores. To demonstrate the capabilities of our algorithm, we analyzed expression proles from nonmalignant primary mammary epithelial cells derived from BRCA1 mutation carriers and epithelial cells without BRCA1 mutation. Our results suggest that oxidative stress plays an important role in epithelial cells with BRCA1 mutations that may contribute to the later development of breast cancer. The application of our algorithm to already published data can yield new insights. As expression data and network data are still growing, methods as our algorithm will be valuable to detect deregulated subgraphs in different conditions and help contribute to a better understanding of diseases.In der vorliegenden Arbeit präsentieren wir neue Optimierungsansätze für wichtige Probleme der Bioinformatik. Der erste Teil behandelt vorwiegend die lokale Optimierung von Molekülen und die Anwendung beim molekularen Docking. Der zweite Teil diskutiert diskrete globale Optimierung. Im ersten Teil präsentieren wir einen neuartigen Algorithmus für ein altes Problem: finde das nächste lokale Optimum in einer gegebenen Richtung auf einer Energiefunktion (Liniensuche, "line search"). Wir zeigen, dass die Ersetzung einer Standardliniensuche mit unserer neuen Methode die Anzahl der Funktions- und Gradientauswertungen in unseren Testläufen auf bis zu 47.7% reduzierte (85% im Mittel). Danach nehmen wir diese Methode in unseren neuen Ansatz zur lokalen Optimierung von flexiblen Liganden im Beisein ihres Rezeptors auf, den wir im Detail beschreiben. Unser Verfahren vermeidet das Singularitätsproblem von Orientierungsparametern. Wir erweitern diese Methode zu einem vollständigen Liganden-Rezeptor-Dockingprogramm, indem wir einen Lamarck'schen genetischen Algorithmus einsetzen. Unsere Validierungsläufe zeigen, dass wir im Vergleich zu anderen getesteten Methoden einen bis zu zehnfachen Geschwindigkeitszuwachs erreichen. Danach arbeiten wir in unseren Ansatz Seitenketten- und begrenzte Backbone exibilität ein, indem wir zwischen bekannten Extremkonformationen mittels sphärischer linearer Extrapolation interpolieren. Unsere Resultate zeigen, dass unsere Methode sehr viel versprechend für flexibles Liganden-Rezeptor-Docking ist. Dennoch hat dieser Ansatz den Nachteil, dass man bekannte Extremkonformationen des Backbones für die Interpolation benötigt. Im letzten Abschnitt des ersten Teils behandeln wir eine Loopregion voll flexibel. Wir zeigen eine neue Methode, die die Go-Scheraga Ringschlussgleichungen und Intervalarithmetik nutzt, um alle möglichen Konformationen zu nden. Unsere Resultate zeigen, dass dieser Algorithmus zuverlässig in der Lage ist, alternative Konformationen zu nden. Er identiziert sehr vielversprechende Loop-Ligandenkomplexe unseres Testbeispiels. Im zweiten Teil dieser Arbeit beschreiben wir das Bindungsordnungszuweisungsproblem von Molekülen. Wir präsentieren unsere neuartige Formulierung, die auf linearer 0-1-Programmierung basiert. Dieser Ansatz ist in der Lage sehr effizient alle optimalen und suboptimalen Bindngsordnungszuweisungen zu berechnen. Unsere Methode ist nicht nur besser als der ursprüngliche Ansatz von Wang et al., sondern auch weitverbreiteter Software zur Bindungszuordnung auf unserem Testdatensatz überlegen. Dieser Datensatz besteht aus 761 sorgfältig präparierten, arzneimittelähnlichen Molekülen, die ursprünglich zur Validierung des Merck-Kraftfeldes eingesetzt wurden. Danach präsentieren wir unsere Filtermethode zur "Feature Subset Selection", die auf "Mutual Information" basiert und Informationen zweiter Ordnung nutzt. Wir geben unser mathematisch motiviertes Kriterium an und lösen das resultierende Optimierungsproblem global optimal im Gegensatz zu anderen Ansätzen. In unseren Validierungsläufen konnte unsere Methode in 18 von 21 Testszenarien die beste Klassizierungsrate erreichen. Im letzten Abschnitt geben wir unsere, auf linearer 0-1-Programmierung basierende Formulierung zur Berechnung von deregulierten Untergraphen in regulatorischen Netzwerken an. Die Basisdaten für diese Methode sind Expressionsprole. Unser Ansatz identiziert die Unternetze einer gewissen Größe mit der höchsten Summe der Knotenscores. Wir analysierten Expressionsprole von nicht bösartigen Brustepithelzellen von BRCA1 Mutationsträgern und Epithelzellen ohne BRCA1 Mutation, um die Fähigkeiten unseres Algorithmuses zu demonstrieren. Unsere Resultate legen nahe, dass oxidativer Stress eine wichtige Rolle bei Epithelzellen mit BRCA1 Mutation spielt, der zur späteren Entwicklung von Brustkrebs beitragen könnte. Die Anwendung unseres Ansatzes auf bereits publizierte Daten kann zu neuen Erkenntnissen führen. Da sowohl Expressions- wie auch Netzwerkdaten ständig anwachsen, sind es Methoden wie unser Algorithmus die wertvoll sein werden, um deregulierte Subgraphen in verschiedenen Situationen zu entdecken. Damit trägt unser Ansatz zu einem besseren Verständnis von Krankheiten und deren Verlauf bei

Quadratic Binary Programming Models in Computational Biology

In this paper we formulate four problems in computational molecular biology as 0-1 quadratic programs. These problems are all NP-hard and the current solution methods used in practice consist of heuristics or approximation algorithms tailored to each problem. Using test problems from scientific databases, we address the question, “Can a general-purpose solver obtain good answers in reasonable time?” In addition, we use the latest heuristics as incumbent solutions to address the question, “Can a general-purpose solver confirm optimality or find an improved solution in reasonable time?” Our computational experiments compare four different reformulation methods: three forms of linearization and one form of quadratic convexification