Thomas L. Hayden

Molecular conformations from distance matrices

Two algorithms are introduced that show exceptional promise in finding molecular conformations using distance geometry on nuclear magnetic resonance data. The first algorithm is a gradient version of the majorization algorithm from multidimensional scaling. The main contribution is a large decrease in CPU time. The second algorithm is an iterative algorithm between possible conformations obtained from the first algorithm and permissible data points near the configuration. These ideas are similar to alternating least squares or alternating projections on convex sets. The iterations significantly improve the conformation from the first algorithm when applied to the small peptide E. coli STh enterotoxin.

A chiasma-hormonal hypothesis relating Down's syndrome and maternal age.

THE incidence of the human congenital abnormality known as Downs syndrome (or mongolism) has been observed to increase with maternal age1. The trend of sharply increased incidence beyond the age of 30 has been well documented2, and various hypotheses have been formulated to explain it. These include cumulative ovum or uterine dysfunction (radiation3, chemicals4, disease5), acute effects of radiation and chemicals, hormonal effects6, the ‘production line’ hypothesis7,8 and changes in sexual behaviour9. We describe here our hypothesis that there is an interaction between the hormonally governed rate of meiosis and the timing of chiasma terminalisation. We propose that changing hormone levels during the menstrual cycle not only trigger resumption of meiosis in the ovum10, but also control the rate of meiosis through the availability of a limiting substance11. As hormone levels and the length of the cycle change with advancing age of the mother, meiosis slows down, and chiasma frequencies decline7,12. Meiotic chromosome bivalents are held together by their chiasmata against strong mutual repulsion during late dictyotene and diakinesis13, making smaller bivalents with fewer chiasmata, such as 21, the most vulnerable to premature separation during terminalisation14,15; premature separation can lead to trisomy 21, by far the most frequent cytological manifestation of Downs syndrome16.

The cone of distance matrices

Abstract The geometry of the cone of Euclidean distance matrices (EDMs) is analyzed using a new characterization of an EDM. The facial structure and the angle that EDMs of embedding dimension one make with the center ray are found. This result follows from a complete analysis of the critical points of the distance function in Frobenius norm from the matrix E consisting of zero diagonal and ones elsewhere to the EDMs of embedding dimension one.

Circum-Euclidean distance matrices and faces

Abstract We study the structure of circum-Euclidean distance matrices, those Euclidean distance matrices generated by points lying on a hypersphere. We show, for example, that such Euclidean distance matrices are characterized as having constant row sums and they constitute the interior of the cone of all Euclidean distance matrices. Also, we provide a formula for computing the radius of a representing configuration in the smallest embedding dimension r and show that rk D = r + 1. Finally we obtain a geometric characterization of the faces of this cone. Given a configuration of points and its Euclidean distance matrix D , any matrix in the minimal face containing D comes from a configuration that is a linear perturbation of the points that generate D .

Approximation by matrices positive semidefinite on a subspace

Abstract We obtain the best approximation to a matrix by matrices positive semidefinite on a subspace. As a by-product, we present two new characterizations of Euclidean distance matrices.

Distance matrices and regular figures

Abstract A regular figure (which includes all regular polygons) is a set of points on a hypersphere whose center coincides with their centroid. We characterize all regular figures as those whose points generate a Euclidean distance matrix (EDM) with eigenvector e , the vector of all ones. Restricting the classical maps of Schoenberg, Gower, and Critchley for all EDMs to the subcone of EDMs with eigenvector e yields new geometrical information about the generating points and a simple formula for the radius of the hypersphere.

Preconditioned Spectral Gradient Method

The spectral gradient method is a nonmonotone gradient method for large-scale unconstrained minimization. We strengthen the algorithm by modifications which globalize the method and present strategies to apply preconditioning techniques. The modified algorithm replaces a condition of uniform positive definitness of the preconditioning matrices, with mild conditions on the search directions. The result is a robust algorithm which is effective on very large problems. Encouraging numerical experiments are presented for a variety of standard test problems, for solving nonlinear Poisson-type equations, an also for finding molecular conformations by distance geometry.

Methods for constructing distance matrices and the inverse eigenvalue problem

Abstract Let D 1 ∈ R k×k and D 2 ∈ R l×l be two distance matrices. We provide necessary conditions on Z∈ R k×l in order that D= D 1 Z Z T D 2 ∈ R n×n be a distance matrix. We then show that it is always possible to border an n×n distance matrix, with certain scalar multiples of its Perron eigenvector, to construct an (n+1)×(n+1) distance matrix. We also give necessary and sufficient conditions for two principal distance matrix blocks D 1 and D 2 be used to form a distance matrix as above, where Z is a scalar multiple of a rank one matrix, formed from their Perron eigenvectors. Finally, we solve the inverse eigenvalue problem for distance matrices in certain special cases, including n=3,4,5,6 , any n for which there exists a Hadamard matrix, and some other cases.

Preconditioners for distance matrix algorithms

A recent gradient algorithm in nonlinear optimization uses a novel idea that avoids line searches. This so‐called spectral gradient algorithm works well when the spectrum of the Hessian of the function to be minimized has a small range or is clustered. In this article, we find a general preconditioning method for this algorithm. The preconditioning method is applied to the stress function, which arises in many applications of distance geometry, from statistics to finding molecular conformations. The Hessian of stress is shown to have a nice block structure. This structure yields a preconditioner which decreases the amount of computation needed to minimize stress by the spectral gradient algorithm.

Mean residence time in peripheral tissue

The published methods for determining the mean residence time for drugs in peripheral tissue are reviewed in terms of assumptions involved, advantages and disadvantages. A method for determining mean transit time in peripheral tissue is proposed; this may be a more useful indicator of the tissue retention properties for drug compounds.