Past and Current Research Projects

of research interests. My main research interests are mathematical biology, stochastic processes, probability theory, dynamical systems, and scientific computing. A major concern of my research is to apply and develop mathematical theories to model biological problems occurring in molecular biology, developmental biology, and biochemistry. My professional aim is to construct mathematical models to understand problems in biology and to promote communication and collaboration between scientists and mathematicians in interdisciplinary areas. My research has consistently focused on stochastic modeling of biochemical networks involving both stochastic and deterministic analysis. The biochemical system consisting of species with large numbers can be modeled deterministically, since randomness averages out. However, when biochemical systems contain small numbers of some species, stochastic effects may play an important role in both time and spatial evolution of the system state, such as in gene transcription/translation networks and signal transduction. In biochemical systems, commonly large in size, constructing a simple stochastic model which captures important characteristics of the system state would be very helpful to understand the complicated mechanism underlying biological observation. In my thesis [1] and in the publication [2], I extended and developed a method to reduce and approximate complicated stochastic biochemical networks using differences between time scales of species and reactions under the assumption that the system is well mixed. In [3], I applied the reduction method to the real biological model called “A heat shock response model of E. coli.” and analyzed the system behavior in time. In [4], we approximated the magnitude of fluctuation of the system state in time, which may not be captured by the reduction method using multiscale nature. In case biochemical systems are not well mixed, the local behavior in the spatially-distributed system has a huge effect on forming patterns in concentration of signaling molecules, for example, commonly seen in the systems of developmental biology. When the diffusion in the reacting system is not too slow, a compartmentalized model can be used to describe the time and spatial evolution of species numbers with random fluctuations. In [5], we used a statistic which measures relative fluctuations of the overall system, and suggested criteria to decide the number of computational cells based on the magnitude of rates for diffusion and reaction. In the spatially-distributed biochemical systems with production from a source, abundance of signaling molecules in one region becomes larger than one in the other region as time passes. Based on the spatial distribution of signaling molecules, these regions may have different features expressed as output of the signal transduction. In [6], we constructed and analyzed a mathematical model which can explain how localization happens effectively, though the number of signaling molecules is usually low. In biochemical systems, fluctuation exists due to a small number of signaling molecules. However, the result or the output of the signal transduction is robust from the noise. In [7], we considered what factors make the signal transduction procedure robust in the specific biological problem of drosophila embryo. In the next section, I will discuss past and current research projects in detail, and in the last section, I will give detailed direction of future work.