Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

In silico detection of control signals: mRNA 3′-end-processing sequences in diverse species

We have investigated mRNA 3′-end-processing signals in each of six eukaryotic species (yeast, rice, arabidopsis, fruitfly, mouse, and human) through the analysis of more than 20,000 3′-expressed sequence tags. The use and conservation of the canonical AAUAAA element vary widely among the six species and are especially weak in plants and yeast. Even in the animal species, the AAUAAA signal does not appear to be as universal as indicated by previous studies. The abundance of single-base variants of AAUAAA correlates with their measured processing efficiencies. As found previously, the plant polyadenylation signals are more similar to those of yeast than to those of animals, with both common content and arrangement of the signal elements. In all species examined, the complete polyadenylation signal appears to consist of an aggregate of multiple elements. In light of these and previous results, we present a broadened concept of 3′-end-processing signals in which no single exact sequence element is universally required for processing. Rather, the total efficiency is a function of all elements and, importantly, an inefficient word in one element can be compensated for by strong words in other elements. These complex patterns indicate that effective tools to identify 3′-end-processing signals will require more than consensus sequence identification.

A streptavidin mutant with altered ligand-binding specificity

The biotin-binding site of streptavidin was modified to alter its ligand-binding specificity. In natural streptavidin, the side chains of N23 and S27 make two of the three hydrogen bonds with the ureido oxygen of biotin. These two residues were mutated to severely weaken biotin binding while attempting to maintain the affinity for two biotin analogs, 2-iminobiotin and diaminobiotin. Redesigning of the biotin-binding site used the difference in local electrostatic charge distribution between biotin and these biotin analogs. Free energy calculations predicted that the introduction of a negative charge at the position of S27 plus the mutation N23A should disrupt two of the three hydrogen bonds between natural streptavidin and the ureido oxygen of biotin. In contrast, the imino hydrogen of 2-iminobiotin should form a hydrogen bond with the side chain of an acidic amino acid at position 27. This should reduce the biotin-binding affinity by approximately eight orders of magnitude, while leaving the affinities for these biotin analogs virtually unaffected. In good agreement with these predictions, a streptavidin mutant with the N23A and S27D substitutions binds 2-iminobiotin with an affinity (Ka) of 1 × 106 M−1, two orders of magnitude higher than that for biotin (1 × 104 M−1). In contrast, the binding affinity of this streptavidin mutant for diaminobiotin (2.7 × 104 M−1) was lower than predicted (2.9 × 105 M−1), suggesting the position of the diaminobiotin in the biotin-binding site was not accurately determined by modeling.

Growth of new brainstem connections in adult monkeys with massive sensory loss

Somatotopic maps in the cortex and the thalamus of adult monkeys and humans reorganize in response to altered inputs. After loss of the sensory afferents from the forelimb in monkeys because of transection of the dorsal columns of the spinal cord, therapeutic amputation of an arm or transection of the dorsal roots of the peripheral nerves, the deprived portions of the hand and arm representations in primary somatosensory cortex (area 3b), become responsive to inputs from the face and any remaining afferents from the arm. Cortical and subcortical mechanisms that underlie this reorganization are uncertain and appear to be manifold. Here we show that the face afferents from the trigeminal nucleus of the brainstem sprout and grow into the cuneate nucleus in adult monkeys after lesions of the dorsal columns of the spinal cord or therapeutic amputation of an arm. This growth may underlie the large-scale expansion of the face representation into the hand region of somatosensory cortex that follows such deafferentations.

Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

Isotropic isotopy and symplectic null sets

Capacity is an important numerical invariant of symplectic manifolds. This paper studies when a subset of a symplectic manifold is null, i.e., can be removed without affecting the ambient capacity. After examples of open null sets and codimension-2 non-null sets, geometric techniques are developed to perturb any isotopy of a loop to a hamiltonian flow; it follows that sets of dimension 0 and 1 are null. For isotropic sets of higher dimensions, obstructions to the perturbation are found in homotopy groups of the orthogonal groups.

DNA sequencing and genotyping by transcriptional synthesis of chain-terminated RNA ladders and MALDI-TOF mass spectrometry

Sets of RNA ladders can be synthesized by transcription of a bacteriophage-encoded RNA polymerase using 3′-deoxynucleotides as chain terminators. These ladders can be used for sequencing of DNA. Using a nicked form of phage SP6 RNA polymerase in this study substantially enhanced yields of transcriptional sequencing ladders. Matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) of chain-terminated RNA ladders allowed DNA sequence determination of up to 56 nt. It is also demonstrated that A→G and C→T variations in heterozygous and homozygous samples can be unambiguously identified by the mass spectrometric analysis. As a step towards single-tube sequencing reactions, α-thiotriphosphate nucleotide analogs were used to overcome problems caused by chain terminator-independent, premature termination and by the small mass difference between natural pyrimidine nucleotides.

A DNA sequencing strategy that requires only five bases of known terminal sequence for priming.

We have previously reported an enhanced version of sequencing by hybridization (SBH), termed positional SBH (PSBH). PSBH uses partially duplex probes containing single-stranded 3' overhangs, instead of simple single-stranded probes. Stacking interactions between the duplex probe and a single-stranded target allow us to reduce the probe sizes required to 5-base single-stranded overhangs. Here we demonstrate the use of PSBH to capture relatively long single-stranded DNA targets and perform standard solid-state Sanger sequencing on these primer-template complexes without ligation. Our results indicate that only 5 bases of known terminal sequence are required for priming. In addition, the partially duplex probes have the ability to capture their specific target from a mixture of five single-stranded targets with different 3'-terminal sequences. This indicates the potential utility of the PSBH approach to sequence mixtures of DNA targets without prior purification.

Calculating the probability of multitaxon evolutionary trees: bootstrappers Gambit.

The reconstruction of multitaxon trees from molecular sequences is confounded by the variety of algorithms and criteria used to evaluate trees, making it difficult to compare the results of different analyses. A global method of multitaxon phylogenetic reconstruction described here, Bootstrappers Gambit, can be used with any four-taxon algorithm, including distance, maximum likelihood, and parsimony methods. It incorporates a Bayesian-Jeffreys'-bootstrap analysis to provide a uniform probability-based criterion for comparing the results from diverse algorithms. To examine the usefulness of the method, the origin of the eukaryotes has been investigated by the analysis of ribosomal small subunit RNA sequences. Three common algorithms (paralinear distances, Jukes-Cantor distances, and Kimura distances) support the eocyte topology, whereas one (maximum parsimony) supports the archaebacterial topology, suggesting that the eocyte prokaryotes are the closest prokaryotic relatives of the eukaryotes.

Intersubunit contacts made by tryptophan 120 with biotin are essential for both strong biotin binding and biotin-induced tighter subunit association of streptavidin.

In natural streptavidin, tryptophan 120 of each subunit makes contacts with the biotin bound by an adjacent subunit through the dimer-dimer interface. To understand quantitatively the role of tryptophan 120 and its intersubunit communication in the properties of streptavidin, a streptavidin mutant in which tryptophan 120 is converted to phenylalanine was produced and characterized. The streptavidin mutant forms a tetrameric molecule and binds one biotin per subunit, as does natural streptavidin, indicating that the mutation of tryptophan 120 to phenylalanine has no significant effect on the basic properties of streptavidin. However, its biotin-binding affinity was reduced substantially, to approximately 10(8) M-1, indicating that the contact made by tryptophan 120 to biotin has a considerable contribution to the extremely tight biotin binding by streptavidin. The mutant retained bound biotin over a wide pH range or with the addition of urea up to 6 M at neutral pH. However, bound biotin was efficiently released by the addition of excess free biotin due, presumably, to exchange reactions. Electrophoretic analysis revealed that the intersubunit contact made by tryptophan 120 to biotin through the dimer-dimer interface is the major interaction responsible for the biotin-induced, tighter subunit association of streptavidin. In addition, the mutant has weaker subunit association than natural streptavidin even in the absence of biotin, indicating that tryptophan 120 also contributes to the subunit association of tetramers in the absence of biotin.