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.

Design of potent and selective human cathepsin K inhibitors that span the active site

Potent and selective active-site-spanning inhibitors have been designed for cathepsin K, a cysteine protease unique to osteoclasts. They act by mechanisms that involve tight binding intermediates, potentially on a hydrolytic pathway. X-ray crystallographic, MS, NMR spectroscopic, and kinetic studies of the mechanisms of inhibition indicate that different intermediates or transition states are being represented that are dependent on the conditions of measurement and the specific groups flanking the carbonyl in the inhibitor. The species observed crystallographically are most consistent with tetrahedral intermediates that may be close approximations of those that occur during substrate hydrolysis. Initial kinetic studies suggest the possibility of irreversible and reversible active-site modification. Representative inhibitors have demonstrated antiresorptive activity both in vitro and in vivo and therefore are promising leads for therapeutic agents for the treatment of osteoporosis. Expansion of these inhibitor concepts can be envisioned for the many other cysteine proteases implicated for therapeutic intervention.

Microsatellite variability and genetic distances.

We analyze the within- and between-population dynamics of the distribution of the number of repeats at multiple microsatellite DNA loci subject to stepwise mutation. Analytical expressions for moments up to the fourth order within a locus and the variance of between-locus variance at mutation-drift equilibrium have been obtained. These statistics may be used to test the appropriateness of the one-step mutation model and to detect between-locus variation in the mutation rate. Published data are compatible with the one-step mutation model, although they do not reject the two-step model. Using both multinomial sampling and diffusion approximations for the analysis of the genetic distance introduced by Goldstein et al. [Goldstein, D. B., Linares, A. R., Cavalli-Sforza, L. L. & Feldman, M. W. (1995) Proc. Natl. Acad. Sci. USA 92, 6723-6727], we show that this distance follows a chi 2 distribution with degrees of freedom equal to the number of loci when there is no variation in mutation rates among the loci. In the presence of such variation, the variance of the distance is obtained. We conclude that the number of microsatellite loci required for the construction of phylogenetic trees with reliable branch lengths may be several hundred. Also, mutations that change repeat scores by several units, even though extremely rare, may dramatically influence estimates of population parameters.