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Mathematics and 21st Century Biology
spondence with experiments. Models of the endocytic cycle, signal transduction cascades, and the cell division cycle have been productively used to organize data, extract “laws” for cellular processes, and even engineer cellular systems. Nevertheless, the models formulated over the past two decades build primarily on population-level measurements with poor spatial and temporal resolution. Without exception, the dynamics predicted by early models were richer than those predicted by the corresponding experimental data. For example, current models of cell cycle dynamics have more variables than can be measured experimentally. This experimental limitation is changing as a result of the rapid development of imaging techniques and high-throughput assays of cellular processes. The result should be an increased ability to evaluate models, which is the limiting step in improving them. It is now possible to collect multivariable and spatiotemporally resolved data on cellular processes ranging from molecular trafficking and signal transduction to integrated responses such as the cell division cycle and cell migration.
In spite of these improvements in experimental capabilities, the quantitative models that emerge will not be as resolved and detailed as the models used in, for example, the aerospace and semiconductor industries. This difference reflects the intrinsic variability of biology, the immense spatiotemporal complexity of cells, and our incomplete knowledge of cellular processes. In many areas of the physical sciences, a coarse model can be very simple and yet fairly accurate, and one only needs to develop heterogeneous, multivariable, spatially resolved models when dealing with second- or higher-order effects. In contrast, even the simplest cellular models must be extracted from heterogeneous, multivariable, spatially resolved experimental data. Learning how to manage these data and mine them to extract computationally manageable models of cellular functions is the key challenge to quantitative understanding of cells.
Exemplification of These Issues
In the early 1980s, Steven Wiley and his colleagues formulated kinetic models of receptor-mediated ligand internalization (Wiley and Cunningham, 1981). The models were initially developed for the epidermal growth factor (EGF) receptor, a key regulator of cell and tissue functions across species (Wiley, 2003). The models described the kinetics of ligand-receptor binding, internalization, recycling, and degradation. The mathematical models were in the form of small systems of ordinary differential equations that were integrated in time and coupled with standard optimization and parameter estimation routines to extract the model parameters (Figure 4.1).
Based on experiments with radioactively labeled EGF ligands, the