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.

Responses of the phototransduction cascade to dim light.

The biochemistry of visual excitation is kinetically explored by measuring the activity of the cGMP phosphodiesterase (PDE) at light levels that activate only a few tens of rhodopsin molecules per rod. At 23 degrees C and in the presence of ATP, the pulse of PDE activity lasts 4 s (full width at half maximum). Complementing the rod outer segments (ROS) with rhodopsin kinase (RK) and arrestin or its splice variant p44 does not significantly shorten the pulse. But when the ROS are washed, the duration of the signal doubles. Adding either arrestin or p44 back to washed ROS approximately restores the pulse width to its initial value, with p44 being 10 times more efficient than arrestin. This supports the idea that, in vivo, capping of phosphorylated R* is mostly done by p44. When myristoylated (14:0) recoverin is added to unwashed ROS, the pulse duration and amplitude increase by about 50% if the free calcium is 500 nM. This effect increases further if the calcium is raised to 1 microM. Whenever R* deactivation is changed--when RK is exogenously enriched or when ATP is omitted from the buffer--there is no impact on the rising slope of the PDE pulse but only on its amplitude and duration. We explain this effect as due to the unequal competition between transducin and RK for R*. The kinetic model issued from this idea fits the data well, and its prediction that enrichment with transducin should lengthen the PDE pulse is successfully validated.

Preassembly of interleukin 2 (IL-2) receptor subunits on resting Kit 225 K6 T cells and their modulation by IL-2, IL-7, and IL-15: A fluorescence resonance energy transfer study

Assembly and mutual proximities of α, β, and γc subunits of the interleukin 2 receptors (IL-2R) in plasma membranes of Kit 225 K6 T lymphoma cells were investigated by fluorescence resonance energy transfer (FRET) using fluorescein isothiocyanate- and Cy3-conjugated monoclonal antibodies (mAbs) that were directed against the IL-2Rα, IL-2Rβ, and γc subunits of IL-2R. The cell-surface distribution of subunits was analyzed at the nanometer scale (2–10 nm) by FRET on a cell-by-cell basis. The cells were probed in resting phase and after coculture with saturating concentrations of IL-2, IL-7, and IL-15. FRET data from donor- and acceptor-labeled IL-2Rβ-α, γ-α, and γ-β pairs demonstrated close proximity of all subunits to each other in the plasma membrane of resting T cells. These mutual proximities do not appear to represent mAb-induced microaggregation, because FRET measurements with Fab fragments of the mAbs gave similar results. The relative proximities were meaningfully modulated by binding of IL-2, IL-7, and IL-15. Based on FRET analysis the topology of the three subunits at the surface of resting cells can be best described by a “triangular model” in the absence of added interleukins. IL-2 strengthens the bridges between the subunits, making the triangle more compact. IL-7 and IL-15 act in the opposite direction by opening the triangle possibly because they associate their private specific α receptors with the β and/or γc subunits of the IL-2R complex. These data suggest that IL-2R subunits are already colocalized in resting T cells and do not require cytokine-induced redistribution. This colocalization is significantly modulated by binding of relevant interleukins in a cytokine-specific manner.