Bruce Kessler

At least four distinct circadian regulatory mechanisms are required for all phases of rhythms in mRNA amount.

Since the advent of techniques to investigate gene expression on a large scale, numerous circadian rhythms in mRNA amount have been reported. These rhythms generally differ in amplitude and phase. The authors investigated how far a parameter not regulated by the circadian clock can influence the phase of a rhythm in RNA amount arising from a circadian rhythm of transcription. Using a discrete-time approach, they modeled a sinusoidal rhythm in transcription with various constant exponential RNA decay rates. They found that the slower the RNA degradation, the later the phase of the RNA amount rhythm compared with the phase of the transcriptional rhythm. However, they also found that the phase of the RNA amount rhythm is limited to a timeframe spanning the first quarter of the period following the phase of the transcriptional rhythm. This finding is independent of the amplitude and vertical shift of the transcriptional rhythm or even of the way constant RNA degradation is modeled. The authors confirmed their results with a continuous-time model, which allowed them to derive a simple formula relating the phase of the RNA amount rhythm solely to the phase and period of its sinusoidal transcriptional rhythm and its constant RNA half-life. This simple formula even holds true for the best sinusoidal approximations of a nonsinusoidal rhythm of transcription and RNA amount. When expanding the model to include additional events with constant exponential kinetics, such as RNA processing, they found that each event expands the phase limit by another quarter of the period when it occurs in sequence but not when it occurs as a competing process. However, the limit expansion comes at the price of minuscule amplitudes. When using a discrete-time approach to model constant rates of transcription with a sinusoidal RNA half-life, the authors found that the phase of the RNA amount rhythm is unaffected by changes in the constant rate of transcription. In summary, their data show that at least 4 distinct circadian regulatory mechanisms are required to allow for all phases in rhythms of RNA amount, one for each quarter of the period.

Improved automated monitoring and new analysis algorithm for circadian phototaxis rhythms in Chlamydomonas.

Automated monitoring of circadian rhythms is an efficient way of gaining insight into oscillation parameters like period and phase for the underlying pacemaker of the circadian clock. Measurement of the circadian rhythm of phototaxis (swimming towards light) exhibited by the green alga Chlamydomonas reinhardtii has been automated by directing a narrow and dim light beam through a culture at regular intervals and determining the decrease in light transmittance due to the accumulation of cells in the beam. In this study, the monitoring process was optimized by constructing a new computer-controlled measuring machine that limits the test beam to wavelengths reported to be specific for phototaxis and by choosing an algal strain, which does not need background illumination between test light cycles for proper expression of the rhythm. As a result, period and phase of the rhythm are now unaffected by the time a culture is placed into the machine. Analysis of the rhythm data was also optimized through a new algorithm, whose robustness was demonstrated using virtual rhythms with various noises. The algorithm differs in particular from other reported algorithms by maximizing the fit of the data to a sinusoidal curve that dampens exponentially. The algorithm was also used to confirm the reproducibility of rhythm monitoring by the machine. Machine and algorithm can now be used for a multitude of circadian clock studies that require unambiguous period and phase determinations such as light pulse experiments to identify the photoreceptor(s) that reset the circadian clock in C. reinhardtii.

A construction of compactly supported biorthogonal scaling vectors and multiwavelets on R 2

In Kessler (Appl. Comput. Harmonic Anal. 9 (2000), 146-165), a construction was given for a class of orthogonal compactly supported scaling vectors on R2, called short scaling vectors, and their associated multiwavelets. The span of the translates of the scaling functions along a triangular lattice includes continous piecewise linear functions on the lattice, although the scaling functions are fractal interpolation functions and possibly nondifferentiable. In this paper, a similar construction will be used to create biorthogonal scaling vectors and their associated multiwavelets. The additional freedom will allow for one of the dual spaces to consist entirely of the continous piecewise linear functions on a uniform subdivision of the original triangular lattice.

Multiwavelets for Quantitative Pattern Matching

The purpose of this paper is to provide an in- troduction to the concepts of wavelets and multi- wavelets, and explain how these tools can be used by the analyst community tond patterns in quan- titative data. Three multiwavelet bases are intro- duced, the GHM basis from (3), a piecewise poly- nomial basis with approximation order 4 from (2), and a smoother approximation-order-4 basis devel- oped by the author in previous work (6). The tech- nique of using multiwavelets tond patterns is il- lustrated in a trac-analysis example.