Calcium plays a crucial role in a huge range of cellular processes including muscle contraction, secretion, neuronal ring and many other functions. Of particular interest are the oscillations seen in free intracellular calcium concentration, which are known to act as intracellular messages, relaying information within cells to regulate cell activity.
A key feature of intracellular calcium dynamics is that some physiological processes occur much faster than others. This leads to models with variables evolving on very dierent time scales. Using geometric singular perturbation techniques (GSPT) it is possible to exploit this separation in time scales to analyse the models. These techniques can be used to explain the observed dynamics, including oscillatory patterns known as mixed-mode oscillations and complicated bifurcation structures.
I will discuss my approach to doing mathematical biology, which is by no means the best and hopefully not the worst, based on a simple rule: we have made a contribution when our collaborators say we have. * Thus far, I have developed four inspirational (for me) collaborations in math biology: a huge effort called the Virtual Lung Project; a study of single cell mechanochemical oscillations; a study of the yeast mitotic spindle in metaphase; and a study of viral-antibody interactions. I will discuss what I find cool about each of these projects, biologically and mathematically, and in particular why they are attractive for young mathematicians. For young researchers, it is important to know how to start, even more so how to sustain, a meaningful relationship and collaboration in math biology.
* A theme I borrowed from Fred Brooks, who started the Computer Science Department at UNC.
Jae Kyoung Kim
The development of luciferase markers and other experiment techniques has allowed
measurement of the timecourses of the expression of genes and proteins with remarkable
accuracy. Since this data has been used to construct many mathematical models, it is important
to ask if this problem of model building is well-posed. Here, we focus on a common form of
ordinary differential equation (ODE) models for biological clocks, which consist of production
and degradation terms, and assume we have an accurate measurement of their solution. Given
these solutions, do ODE models exist? If they exist, are they unique? We show that timecourse
data can sometimes, but not always determine the unique quantitative relationships (i.e.
biochemical rates) of network species. In other cases, our techniques can rule out functional
relationships between network components and show how timecourses can reveal the underlying
network structure. We also show that another class of models is guaranteed to have existence and
uniqueness, although its biological application is less clear. Our work shows how the
mathematical analysis of the process of model building is an important part of the study of
mathematical models of biological clocks
I will present a number of problems in macromolecular cellular, and tissue level biology that can be modeled using approaches from statistical physics, stochastic processes, and membrane mechanics. First, I will consider a simple model of nucleosome positioning that predicts the coverage of DNA by histones. At the cellular level, the evolution of cell populations can be described by nonequilibrium statistical mechanical models such as the zero-range process, with birth and death. We have applied this type of model to describe cancer progression. Finally, at the tissue level, basic membrane mechanics will be used to a mathematical framework for retinal detachments. These examples are meant to highlight the versatility of using basic paradigms from condensed matter and statistical physics to distill complex problems in cell biology and physiology.