With the advent of fast and easily accessible computers, mathematical and computer modeling of biological phenomena are becoming powerful tools in biomedical research. K. A. Dill recently stated, "The future of biomedicine depends on strength of its underpinning sciences, including chemistry, physics and mathematics" (Strengthening biomedicine's roots, Nature , 400 (1999)309-310).
My education and previous career in theoretical physics (associate professor level) influenced my present interests. These interests broadly cover mathematical modeling of processes and structures underlying the biological phenomena and data analysis in biochemistry, biophysics, and biomedical sciences in general. The main features of mathematical models are that they describe large amounts of data in an interpretable and quantitative way, and that they often make possible predictions for experiments that are too difficult, costly, or ethically impossible. For example, it is not ethical to measure rate of natural growth of the breast cancer in a patient to whom the disease has already been diagnosed. Rather, a mathematical model of tumor growth should be developed and used for prognostic purposes, as well as for the optimization of therapy.
My specific research interests revolve around the following topics.
- Mathematical modeling of tumor growth. I proposed several models for quantitation of feedback control processes that characterize tumor growth when stimulated by secreted growth factors. In this context, different approaches to modeling of multi-cell tumor spheroids were compared. Theoretical foundation for Gompertz growth model is elucidated. Most recently I developed a mathematical model for radio-viral therapy of multiple myeloma proposed by Mayo researchers. This model describes quantitatively the effects of the therapy and can be used for prediction of the outcome. Based on this model, I am currently working on optimization of therapy scheduling.
- Mathematical models based on polyexponential functions . These functions are widely used in biochemistry and biophysics (e.g., fluorescence intensity and anisotropy decay, NMR signal decay, transient birefringence decay, chemical relaxation signals, single-ion channel dwell time histograms, etc.). In spite of frequent use, the polyexponential models are still puzzling due to peculiar numerical ill-conditioning. The latter leads to fundamental non-uniqueness that is critical for interpretation. I am analyzing these models from different aspects and, in particular, their application to fluorescence decay. Earlier work was focused on Pade-Laplace and maximum likelihood methods for parameter estimation. Currently, I am developing methodology for exploration of parameter space with the goal of finding all sets of possible parameter values compatible with the data, and thus improve our understanding of limits in data interpretation. These investigations are applied to studies of structure and function of proteins by fluorescence and NMR measurements. We have also developed a specific polyexponential model for quenching of fluorescence in proteins, as it occurs by various modes of energy transfer. Researchers in the field have accepted this model as possible basis for interpretation of complicated phenomena related to fluorescence decay in macromolecules.
- Mathematical aspects of kinetic studies of plasma membrane calcium pump and enzyme kinetics in general. Various methods for estimation of rate constants are investigated. A method based on formaly integrated kinetic equations is under development. Plasma membrane calcium pump is involved in regulation of various cell functions, often critical for several pathophysiological conditions.
- Mathematical modeling of binding kinetics of macromolecules monitored by biosensors based on surface plasmon resonance technology. A new method to analyze binding data obtained by BIACORE instrument is proposed. It involves fractal kinetics and flow effects of circulating solution. Binding of breast-cancer associated protein (BRCA1) to phosphorylated DNA repair hlicase (BACH1) is analyzed using proposed method.
- Mathematical model for microcirculatory functional status of heart as evaluated by fast X-ray imaging . It is expected that the diagnostic methods based on these studies will make possible earlier detection of arterial heart disease.
- Development of new algorithms for global optimization. The algorithm involves combination of inductive and simplex search. Global optimization is required for most of the above studies, as well as in the whole spectrum of disciplines from physics and chemistry to biology and medicine and even in social sciences.
- Characterization of proteomic maps via biodecriptors based on graph theory. Mathematical graphs are associated with proteins characterized by mass, charge, and abundance in a protemic map. Graphs are further characterized by invariants of associated matrices that serve as biodescriptors. It is hoped that this method of data reduction will be effective in diagnostic capabilities, based on proteomics maps for individual patients. This is my most recent research interest.
The most exciting revelation in the above research is that by using mathematical and computational methods one can obtain results that are often very surprising, and otherwise impossible to obtain.