Virtually all docking methods include some local continuous minimization of an energy/scoring function in order to remove steric clashes and obtain more reliable energy values. In this paper, we describe an efficient rigid-body optimization algorithm that, compared to the most widely used algorithms, converges approximately an order of magnitude faster to conformations with equal or slightly lower energy. The space of rigid body transformations is a nonlinear manifold, namely, a space which locally resembles a Euclidean space. We use a canonical parametrization of the manifold, called the exponential parametrization, to map the Euclidean tangent space of the manifold onto the manifold itself. Thus, we locally transform the rigid body optimization to an optimization over a Euclidean space where basic optimization algorithms are applicable. Compared to commonly used methods, this formulation substantially reduces the dimension of the search space. As a result, it requires far fewer costly function and gradient evaluations and leads to a more efficient algorithm. We have selected the LBFGS quasi-Newton method for local optimization since it uses only gradient information to obtain second order information about the energy function and avoids the far more costly direct Hessian evaluations. Two applications, one in protein-protein docking, and the other in protein-small molecular interactions, as part of macromolecular docking protocols are presented. The code is available to the community under open source license, and with minimal effort can be incorporated into any molecular modeling package.

In this paper we extend a recently introduced rigid body minimization algorithm, defined on manifolds, to the problem of minimizing the energy of interacting flexible molecules. The goal is to integrate moving the ligand in six dimensional rotational/translational space with internal rotations around rotatable bonds within the two molecules. We show that adding rotational degrees of freedom to the rigid moves of the ligand results in an overall optimization search space that is a manifold to which our manifold optimization approach can be extended. The effectiveness of the method is shown for three different docking problems of increasing complexity. First we minimize the energy of fragment-size ligands with a single rotatable bond as part of a protein mapping method developed for the identification of binding hot spots. Second, we consider energy minimization for docking a flexible ligand to a rigid protein receptor, an approach frequently used in existing methods. In the third problem we account for flexibility in both the ligand and the receptor. Results show that minimization using the manifold optimization algorithm is substantially more efficient than minimization using a traditional all-atom optimization algorithm while producing solutions of comparable quality. In addition to the specific problems considered, the method is general enough to be used in a large class of applications such as docking multidomain proteins with flexible hinges. The code is available under open source license (at http://cluspro.bu.edu/Code/Code_Rigtree.tar), and with minimal effort can be incorporated into any molecular modeling package.

We study the impact of optimizing side-chain positions in the interface region between two proteins during the process of binding. Mathematically, the problem is similar to side-chain prediction, extensively explored in the process of protein structure prediction. The protein-protein docking application, however, has a number of characteristics that necessitate different algorithmic and implementation choices. In this work, we implement a distributed approximate algorithm that can be implemented on multi-processor architectures and enables trading off accuracy with running speed. We report computational results on benchmarks of enzyme-inhibitor and other types of complexes, establishing that the side-chain flexibility our algorithm introduces substantially improves the performance of docking protocols. Further, we establish that the inclusion of unbound side-chain conformers in the side-chain positioning problem is critical in these performance improvements.

Several emerging nano-technologies, including crossbar nano-architectures, have recently been studied as possible replacement or supplement to CMOS technology in the future. However, extreme process variation and high failure rates, mainly due to atomic device sizes, are major challenges for crossbar nano-architectures. This article presents variation- and defect-tolerant logic mapping on crossbar nano-architectures. Since variation/defect-aware mapping is an NP-hard problem, we introduce a set of Integer Linear Programming (ILP) formulations to effectively solve the problem in a reasonable time. The proposed ILP formulations can be used for both diode-based and FET-based crossbars. Experimental results on benchmark circuits show that our approach can reduce the critical-path delay 39% compared to the Simulated Annealing (SA) method. It can also successfully achieve 97% defect-free mapping with 40% defect density. It can tolerate process variations to meet timing constraints in 95% of the cases, compared to only 77% achieved by SA.

Our work is motivated by energy minimization in the space of rigid affine transformations of macromolecules, an essential step in computational protein-protein docking. We introduce a novel representation of rigid body motion that leads to a natural formulation of the energy minimization problem as an optimization on the SO(3)×frakturR3 manifold, rather than the commonly used SE(3). The new representation avoids the complications associated with optimization on the SE(3) manifold and provides additional flexibilities for optimization not available in that formulation. The approach is applicable to general rigid body minimization problems. Our computational results for a local optimization algorithm developed based on the new approach show that it is about an order of magnitude faster than a state of art local minimization algorithms for computational protein-protein docking.

The fast Fourier transform (FFT) sampling algorithm has been used with success in application to protein-protein docking and for protein mapping, the latter docking a variety of small organic molecules for the identification of binding hot spots on the target protein. Here we explore the local rather than global usage of the FFT sampling approach in docking applications. If the global FFT based search yields a near-native cluster of docked structures for a protein complex, then focused resampling of the cluster generally leads to a substantial increase in the number of conformations close to the native structure. In protein mapping, focused resampling of the selected hot spot regions generally reveals further hot spots that, while not as strong as the primary hot spots, also contribute to ligand binding. The detection of additional ligand binding regions is shown by the improved overlap between hot spots and bound ligands.

Fast Fourier transform (FFT) based approaches have been successful in application to modeling of relatively rigid protein-protein complexes. Recently, we have been able to adapt the FFT methodology to treatment of flexible protein-peptide interactions. Here, we report our latest attempt to expand the capabilities of the FFT approach to treatment of flexible protein-ligand interactions in application to the D3R PL-2016-1 challenge. Based on the D3R assessment, our FFT approach in conjunction with Monte Carlo minimization off-grid refinement was among the top performing methods in the challenge. The potential advantage of our method is its ability to globally sample the protein-ligand interaction landscape, which will be explored in further applications.

Our work is motivated by energy minimization of biological macromolecules, an essential step in computational docking. By allowing some ligand flexibility, we generalize a recently introduced novel representation of rigid body minimization as an optimization on the SO(3)×double-struckR3 manifold, rather than on the commonly used Special Euclidean group SE(3). We show that the resulting flexible docking can also be formulated as an optimization on a Lie group that is the direct product of simpler Lie groups for which geodesics and exponential maps can be easily obtained. Our computational results for a local optimization algorithm developed based on this formulation show that it is about an order of magnitude faster than the state-of-the-art local minimization algorithms for computational protein-small molecule docking.