Competition in the chemostat with an undesirable lethal competitor.

In this study, we compare the effects of competitors in a chemostat when one of the competitors is lethal to the other. The first competitor ("the mutant") is the desired organism because it provides a benefit, such as a substance that is harvested. However, when the mutant undergoes cell division the result may return to the original ("wild type") organism that produces a substance ("toxin") that is lethal to the mutant. We introduce an external inhibitor that negatively affects the growth of the wild type organism but that does not affect the mutant. The goal is for the mutant to dominate in the competition while co-existing with its wild type relative that is controlled. In this manner, we hope that understanding the dynamics of the system will help in designing methods to control the purity of the harvesting vessel without having to periodically restart the process more than necessary. We show that it is possible for co-existence in which the undesirable wild-type coexists with the mutant. However, it is also possible to destabilize the system and cause the extinction of the mutant.

A fractal analysis of human cranial sutures.

OBJECTIVES: Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure their exponents of similarity (fractal dimensions). DESIGN: One hundred three documented human skulls from the Terry Collection of the Smithsonian Institution were used. Their sagittal sutures were digitized and the data converted to bitmap images for analysis using box-counting method of fractal software. RESULTS: The log-log plots of the number of boxes containing the sutural pattern, N(r), and the size of the boxes, r, were all linear, indicating that human sagittal sutures possess scale-invariant features and thus are fractals. The linear portion of these log-log plots has limits because of the finite resolution used for data acquisition. The mean box dimension, D(b), was 1.29289 +/- 0.078457 with a 95% confidence interval of 1.27634 to 1.30944. CONCLUSIONS: Human sagittal sutures are self-similar and have a fractal dimension of 1.29 by the box-counting method. The significance of these findings includes: sutural morphogenesis can be described as a repeated iteration function, and mathematical models can be constructed to produce self-similar curves with such D(b). This elucidates the mechanism of actual pattern formation. Whatever the mechanisms at the cellular and molecular levels, human sagittal suture follows the equation log N(r) = 1.29 log 1/r, where N(r) is the number of square boxes with sides r that are needed to contain the sutural pattern and r equals the length of the sides of the boxes.