Speaker: Peter Hinow (IMA)
Title: Inverse problems in nanobiology and analysis of an age-structured population model
Abstract: In this talk, two topics from my dissertation research will be presented - inverse problems in nanobiology and the analysis of an age-structured population model.
The density profile of an elastic fiber like DNA will change in space and time as ligands associate with it. This observation affords a new direction in single molecule studies provided that density profiles can be measured in space and time. In fact, this is precisely the objective of seismology, where the mathematics of inverse problems have been employed with success. We argue that inverse problems in elastic media can be directly applied to biophysical problems of fiber--ligand association, and demonstrate that robust algorithms exist to perform density reconstruction in the condensed phase.
In the second part of the talk we will turn to a nonlinear partial differential equation model for an age--structured population. A characteristic of many growth processes is that as the number of individuals inreases, the population growth slows. In the case of a tumor cell mass, cells can belong to two distinct subpopulations, namely those of proliferating versus nonproliferating cells. It was shown previously that a linear model of the same structure has the property of asynchronous exponential growth. That is, the age distribution approaches a limit shape and the total number of cells grows to infinity or decays to zero. The situation is different in the nonlinear model under consideration, where transition rates between proliferating and nonproliferating classes depend on the total population. Under a certain natural condition, namely that both $0$ and $\infty$ are repelling in the total population space, we show that the nonlinear population dynamic model based on chronological age must have a nontrivial equilibrium solution.