A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented

EFFECTS OF GROUP SELECTION WITH YELLOW BIRCH (BETULA ALLEGHANIENSIS) RETENTION ON THE UNDERSTORY AND SAPLING LAYER IN NORTHERN HARDWOOD FORESTS

Gap partitioning theory predicts that changes in microenvironment conditions found within a forest opening promote diversity in forest ecosystems. Under this theory we would expect to see variations in tree and understory diversity throughout and surrounding a forest opening. In order to test this theory, we examined manmade openings with legacy-tree retention in a northern hardwood forest located in the Upper Peninsula of Michigan. This work is part of an ongoing study that was started in 2003 with the creation of 49 openings centered on a reserve dominant or co-dominant yellow birch (Betula alleghaniensis Britt.). The primary objective of this research was to assess if opening size, plot location and/or plot transect azimuth had an influence on the dependent variables that we measured (herbaceous-layer species, cover groups, and sapling). Twenty reference sites were also selected from the surrounding forest and centered on a dominant or co-dominant yellow birch. At each site, the following variables were measured; herbaceous-layer species percent cover, cover groups (tree seedling (\u3c 50 cm), exposed soil, forest litter, exposed rock, woody shrubs, herbaceous plants, grass, sedge, rush, non-vascular plants, and woody debris), and sapling height (≥ 50 cm). We found that both opening size and plot location were influential on our measured variables to varying degrees. We found no evidence that transect azimuth was a significant predictor of any of the dependent variables. Opening size was significant when analyzing species diversity and evenness. Plot location was also significant when measuring species diversity as well as richness. Correlations with cover groups varied and some groups were not found to be associated with any of the opening measures (size, location, transect azimuth). Mean tallest tree sapling height was not found to be significantly different among opening sizes, but sapling height was significantly shorter in the references sites than any of the harvested openings. We also found that saplings under the legacy tree were the tallest on average when compared to the opening and the surrounding forest. Maples were by far the most abundance sapling species with sugar maple (Acer saccharum Marsh.) being the most common. Continued monitoring of sapling survival and growth will be important to gain a better understanding of tree diversity in openings with legacy- tree retention and have a better understanding of the future forest composition

Blind Prediction of Interfacial Water Positions in CAPRI

We report the first assessment of blind predictions of water positions at protein-protein interfaces, performed as part of the CAPRI (Critical Assessment of Predicted Interactions) community-wide experiment. Groups submitting docking predictions for the complex of the DNase domain of colicin E2 and Im2 immunity protein (CAPRI target 47), were invited to predict the positions of interfacial water molecules using the method of their choice. The predictions – 20 groups submitted a total of 195 models – were assessed by measuring the recall fraction of water-mediated protein contacts. Of the 176 high or medium quality docking models – a very good docking performance per se – only 44% had a recall fraction above 0.3, and a mere 6% above 0.5. The actual water positions were in general predicted to an accuracy level no better than 1.5 Å, and even in good models about half of the contacts represented false positives. This notwithstanding, three hotspot interface water positions were quite well predicted, and so was one of the water positions that is believed to stabilize the loop that confers specificity in these complexes. Overall the best interface water predictions was achieved by groups that also produced high quality docking models, indicating that accurate modelling of the protein portion is a determinant factor. The use of established molecular mechanics force fields, coupled to sampling and optimization procedures also seemed to confer an advantage. Insights gained from this analysis should help improve the prediction of protein-water interactions and their role in stabilizing protein complexes

An RBF-FD closest point method for solving PDEs on surfaces

Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret, J. Comput. Phys. 231(14):4662-4675, [2012]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.NSERC Canada (RGPIN 227823), Hong Kong Research Grant Council GRF Grant (HKBU 11528205), Hong Kong Baptist University FRG Grant

Docking by structural similarity at protein-protein interfaces

Rapid accumulation of experimental data on protein-protein complexes drives the paradigm shift in protein docking from ‘traditional,’ template free approaches to template based techniques. Homology docking algorithms based on sequence similarity between target and template complexes can account for up to 20% of known protein-protein interactions. When highly homologous templates for the target complex are not available, but the structure of the target monomers is known, docking by local structural alignment may provide an adequate solution. Such an algorithm was developed based on the structural comparison of monomers to co-crystallized interfaces. A library of the interfaces was generated from co-crystallized protein-protein complexes in PDB. The partial structure alignment algorithm was validated on the Dockground benchmark sets. The optimal performance of the partial (interface) structure alignment was achieved with the interface residues defined by 12Å distance across the interface. Overall, the partial structural alignment yielded more accurate models than the full structure alignment. Most templates identified by the partial structural alignment had low sequence identity to the target, which makes them hard to detect by sequence-based methods. The results indicate that the structure alignment techniques provide a much needed addition to the docking arsenal, with the combined structural alignment and template free docking success rate significantly surpassing that of the free docking alone

Protein Docking by the Interface Structure Similarity: How Much Structure Is Needed?

The increasing availability of co-crystallized protein-protein complexes provides an opportunity to use template-based modeling for protein-protein docking. Structure alignment techniques are useful in detection of remote target-template similarities. The size of the structure involved in the alignment is important for the success in modeling. This paper describes a systematic large-scale study to find the optimal definition/size of the interfaces for the structure alignment-based docking applications. The results showed that structural areas corresponding to the cutoff values <12 Å across the interface inadequately represent structural details of the interfaces. With the increase of the cutoff beyond 12 Å, the success rate for the benchmark set of 99 protein complexes, did not increase significantly for higher accuracy models, and decreased for lower-accuracy models. The 12 Å cutoff was optimal in our interface alignment-based docking, and a likely best choice for the large-scale (e.g., on the scale of the entire genome) applications to protein interaction networks. The results provide guidelines for the docking approaches, including high-throughput applications to modeled structures.This work was supported by National Institutes of Health grant R01 GM074255