## IMA Postdoc Seminar (September 25, 2007)

**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.

**Slides:** PDF