The principal investigator continues research on cellular control systems and delay differential equations, focussing on three main topics. The first area is an interdisciplinary study done in collaboration with Dr. J. W. Zyskind on the modeling of initiation of DNA replication in Escherichia coli. Using mathematical models, they attempt to explain which are the most important steps in the beginning of the DNA replication cycle. Several other regulatory processes for growing cells are examined, providing additional insight into cellular control systems and introducing new dynamical systems to study. The second area of inquiry analyzes the stability region for a linear differential equation with two delays. Both analytical and numerical techniques are used to determine how the stability region varies with the parameters in this equation. The stability region varies dramatically for even small changes in some parameters, so studying these anomalies is very important from a modeling point of view. The final area of investigation examines mathematical models for erythropoiesis and thrombopoiesis. Previous models have included two delays linking this study to the previous one. More recently, models have been developed using either state-dependent delays or age-structured modeling. The principal investigator has developed a new age-structured model that more accurately reflects the biology in hematopoietic systems. He will undertake mathematical studies to determine the qualitative behavior of this model and to examine how it compares to models with either multiple delays or state-dependent delays. Parameter identification methods may provide quantitative information for how these systems can become unstable in certain disease states. The principal objective of this project is to develop mathematical models that correspond to known biological behavior. The models will provide insight into the mechanisms that govern the specific biological problem. DNA replication marks the beginning of the cell cycle in bacteria. An understanding of the controls underlying the cell cycle is important for other areas of research, including genetic engineering and cancer research. The mathematical studies provide support for certain biological experiments, suggest further avenues of experimentation, and show how some biological theories are impossible. The computer simulations of cellular processes require less time to perform than experiments and are significantly less costly. However, it is the interaction between the experimenter and the mathematical modeler in this study that will be most important to the advancement in understanding the cell cycle. Mathematical models for erythropoiesis or red blood cell production can elucidate the effects of certain changes in this complicated process that begins with simple undifferentiated cells in the bone marrow and proceeds to mature red blood cells, which carry oxygen throughout the body. The models may explain the mechanisms underlying certain blood diseases and suggest possible therapeutic procedures. The models could aid in developing new procedures to efficiently collect blood, which is significant for patients who choose to provide their own blood supply for elective surgery and avoid risks of HIV infection. The work includes several novel mathematical techniques that will advance fundamental research tools available for future mathematical studies. These tools will be applicable to a wide range of problems in applied mathematics.
