Purpose This project helps investigators to cope with complex equations that model biological systems. PROJECT 1:David Ruskin and Judy Walters, NINDS. In order to monitor variations in neuron firing rates in a live animal, this group wanted a MATLAB version of the Lomb periodogram, as published by Press et al. "Numerical Recipes in C", Cambridge Univ. Press, Cambridge 1988. The authors describe this program as "turgid code", and indeed, it took quite an effort to get it right. I wrote two MATLAB versions, one based on the above book, and a simpler but less efficient version based on a paper by Jeffrey Scargle: Astrophys. J. 263 (1982), 835-853. The object was to allow complete control over the test cases to insure bug-free code. In the process, I found that the method is flawed, and can give misleading information about the presence and power of a given frequency. I wrote a paper about how to correct the problem, and submitted it to the journal "Astrophysics and Space Science", who published the original Lomb paper, and has now accepted mine, subject to minor revisions. This year, the final galleys were recieved and approved. PROJECT 2: Richard Hendler, NHLBI. Bacteriorhodopsin (bR)is an important biological energy transducer. It converts light into electrochemical potential by pumping protons across the cell membrane. bR is one of those molecules, like hemoglobin and cytochrome oxidase, in which a lot is known about the structure, but important aspects of the mechanism are still unclear. We are using mathematical modeling to decide the merits of several models that have appeared in the literature in recent years, including our own. The experiment consists of exposing bR to a laser flash, and measuring absorbance spectra as a function of time. We have developed new techniques that will enable investigators to extract more information from their data than was previously thought possible. For example, the fraction of bR reacting to the flash and the fraction in eachof the fast and slow subcycles are now resolvable, along with absolute spectra rather than difference spectra. We have spent much of the year refining our methods and writing two papers, the first about the methods themselves, and the second about results specific to bR. These two papers have now been published. We are in the process of trying these methods on mutated cells and on cells exposed to various disrupting agents. PROJECT 3: Peter Basser NICHD In the past, we have shown that at least seven MRI images are required to get complete information about the orientation of diffusion, e.g. in nerve fibers. However, when the situation is complicated by more than one kind of sample in a voxel, and especially by more than one orientation (e.g. when nerve fibeers cross), seven images no longer suffice. We are designing imaging strategies to resolve these more complex cases. We are currently testing the software by classifying voxels into three classes: those well-fitted by (A) a one-term isotropic model, (B) a one-term anisotropic model, and (C) neither. PROJECT 4:Microarrays: for DNA microarray analysis is written mostly in MATLAB. This year, I have been an occasional consultant in upgrading the performance of microarray analysis functions. PROJECT 5: Charles Spivak NIDA. Confocal microscopy is used to monitor uptake of drug in cell cultures with more than one layer of cells on a slide. We aim to model differences in drug uptake kinetics which are layer-dependent. PROJECT 6: MATLAB. MATLAB is a major numerical software package in use by hundreds of investigators at the NIH. The 2-day MATLAB course had been given in spring and fall for the past ten years, undergoing revision as the language expanded in scope. The class notes, originally a small pamphlet, have grown to 147 pages requiring continual revision as the language evolves. Recently, the course was expanded from 2 to 3 days, and the response has been most favorable. PROJECT 7: Consulting. Each year, investigators call me about a variety of mathematical or software problems. Most questions are easily answered, but some have taken a day or more of effort. The most extended consultations this year have been with Peter Basser, Robert Berger, Daniel Flatow, Richard Hendler, Paul Kienzle, Mark Lewis, Allen Minton, Philip Ross, Harry Saroff, Charles Spivak, and Norman Watts.