On the Existence and Uniqueness of Biological Clock Models Matching Experimental Data

Jae Kyoung Kim (August 29, 2012)

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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