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The transmembrane (TM) domains of many integral membrane proteins are composed of alpha-helix bundles. Structure determination at high resolution (<4 A) of TM domains is still exceedingly difficult experimentally. Hence, some TM-protein structures have only been solved at intermediate (5-10 A) or low (>10 A) resolutions using, for example, cryo-electron microscopy (cryo-EM). These structures reveal the packing arrangement of the TM domain, but cannot be used to determine the positions of individual amino acids. The observation that typically, the lipid-exposed faces of TM proteins are evolutionarily more variable and less charged than their core provides a simple rule for orienting their constituent helices. Based on this rule, we developed score functions and automated methods for orienting TM helices, for which locations and tilt angles have been determined using, e.g., cryo-EM data. The method was parameterized with the aim of retrieving the native structure of bacteriorhodopsin among near- and far-from-native templates. It was then tested on proteins that differ from bacteriorhodopsin in their sequences, architectures, and functions, such as the acetylcholine receptor and rhodopsin. The predicted structures were within 1.5-3.5 A from the native state in all cases. We conclude that the computational method can be used in conjunction with cryo-EM data to obtain approximate model structures of TM domains of proteins for which a sufficiently heterogeneous set of homologs is available. We also show that in those proteins in which relatively short loops connect neighboring helices, the scoring functions can discriminate between near- and far-from-native conformations even without the constraints imposed on helix locations and tilt angles that are derived from cryo-EM.

Nonparametric density estimation has broad applications in computational finance especially in cases where high frequency data are available. However, the technique is often intractable, given the run times necessary to evaluate a density. We present a new and efficient algorithm based on multipole techniques. Given the n kernels that estimate the density, current methods take O(n) time directly to sum the kernels to perform a single density query. In an on-line algorithm where points are continually added to the density, the cumulative O(n 2 ) running time for n queries makes it very costly, if not impractical, to compute the density for large n. Our new Multipole-accelerated On-line Density Estimation (MODE) algorithm is general in that it can be applied to any kernel (in arbitrary dimensions) that admits a Taylor series expansion. The running time for a density query reduces to O(log n) or even constant time, depending on the kernel chosen, and, hence, the cumulative running time is reduced to O(n log n) or O(n), respectively. Our results show that the MODE algorithm provides dramatic advantages over the direct approach to density evaluation. For example, we show using a modest computing platform that on-line density updates and queries for 1 million points and two dimensions take 8 days to compute using the direct approach versus 40 seconds with the MODE approach.

We identify a structural feature of transmembrane helical proteins that restricts their conformational space and suggests a new way of understanding the construction and stability of their native states. We show that five kinds of well-known specific favorable interhelical interactions (hydrogen bonds, aromatic interactions, salt bridges, and two interactions from packing motifs) precisely determine the packing of the transmembrane helices in 15 diverse proteins. To show this, we iteratively reassemble the helix bundle of each protein using only these interactions, generic interaction geometries, and individual helix backbone conformations. On average, the representative set of rebuilt structures best satisfying the constraints imposed by the five types of interhelical interactions has an average Calpha root-mean-square deviation from the native of 1.03 A. Implications for protein folding, structure and motion prediction, modeling, and design are discussed.

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