Scott Grandison

Computation of small-angle scattering profiles with three-dimensional Zernike polynomials.

Small-angle X-ray scattering (SAXS) methods are extensively used for characterizing macromolecular structure and dynamics in solution. The computation of theoretical scattering profiles from three-dimensional models is crucial in order to test structural hypotheses. Here, a new approach is presented to efficiently compute SAXS profiles that are based on three-dimensional Zernike polynomial expansions. Comparison with existing methods and experimental data shows that the Zernike method can be used to effectively validate three-dimensional models against experimental data. For molecules with large cavities or complicated surfaces, the Zernike method more accurately accounts for the solvent contributions. The program is available as open-source software at http://sastbx.als.lbl.gov.

The Application of 3D Zernike Moments for the Description of “Model-Free” Molecular Structure, Functional Motion, and Structural Reliability

Protein structures are not static entities consisting of equally well-determined atomic coordinates. Proteins undergo continuous motion, and as catalytic machines, these movements can be of high relevance for understanding function. In addition to this strong biological motivation for considering shape changes is the necessity to correctly capture different levels of detail and error in protein structures. Some parts of a structural model are often poorly defined, and the atomic displacement parameters provide an excellent means to characterize the confidence in an atoms spatial coordinates. A mathematical framework for studying these shape changes, and handling positional variance is therefore of high importance. We present an approach for capturing various protein structure properties in a concise mathematical framework that allows us to compare features in a highly efficient manner. We demonstrate how three-dimensional Zernike moments can be employed to describe functions, not only on the surface of a protein but throughout the entire molecule. A number of proof-of-principle examples are given which demonstrate how this approach may be used in practice for the representation of movement and uncertainty.

Monotonicity of some modified Bessel function products

For non-negative real order, the product of modified Bessel functions of first and second kind is shown to be strictly decreasing for positive real arguments. After recalling some established results, only elementary methods are required to complete the proof.

A rapid boundary integral equation technique for protein electrostatics

A new boundary integral formulation is proposed for the solution of electrostatic field problems involving piecewise uniform dielectric continua. Direct Coulomb contributions to the total potential are treated exactly and Greens theorem is applied only to the residual reaction field generated by surface polarisation charge induced at dielectric boundaries. The implementation shows significantly improved numerical stability over alternative schemes involving the total field or its surface normal derivatives. Although strictly respecting the electrostatic boundary conditions, the partitioned scheme does introduce a jump artefact at the interface. Comparison against analytic results in canonical geometries, however, demonstrates that simple interpolation near the boundary is a cheap and effective way to circumvent this characteristic in typical applications. The new scheme is tested in a naive model to successfully predict the ground state orientation of biomolecular aggregates comprising the soybean storage protein, glycinin.

Ligand Electron Density Shape Recognition Using 3D Zernike Descriptors

We present a novel approach to crystallographic ligand density interpretation based on Zernike shape descriptors. Electron density for a bound ligand is expanded in an orthogonal polynomial series (3D Zernike polynomials) and the coefficients from this expansion are employed to construct rotation-invariant descriptors. These descriptors can be compared highly efficiently against large databases of descriptors computed from other molecules. In this manuscript we describe this process and show initial results from an electron density interpretation study on a dataset containing over a hundred OMIT maps. We could identify the correct ligand as the first hit in about 30 % of the cases, within the top five in a further 30 % of the cases, and giving rise to an 80 % probability of getting the correct ligand within the top ten matches. In all but a few examples, the top hit was highly similar to the correct ligand in both shape and chemistry. Further extensions and intrinsic limitations of the method are discussed.

Biomechanics of anther opening

Many influential papers are published each year on the fundamental role that genetics plays in plant growth anddevelopment.However, despite the beautifully written and illustratedOnGrowth and Form by D’Arcy Wentworth Thompson (1917) being published nearly 100 years ago, it could be argued that the role of physics during developmental processes is still a somewhat neglected field. Yet genetic processes do not happen in isolation, and it is the complex interplay between genetics, physics and the environment that is the true backdrop againstwhich these processes occur (Niklas, 1992). It is therefore especially welcome to read ‘A biomechanical model of anther opening reveals the roles of dehydration and secondary thickening’ by Nelson et al. in this issue of New Phytologist (pp. 1030–1037), where an elegantly simple mathematical formulation of anther opening illustrates how these three factors act together during the vital process. Not only does this paper highlight the importance of biophysical processes, it further highlights the importance and power of theoretical approaches to biological problems.