Modeling, Simulating, and Parameter Fitting of Biochemical Kinetic Experiments

In many chemical and biological applications, systems of differential equations containing unknown parameters are used to explain empirical observations and experimental data. The DEs are typically nonlinear and difficult to analyze, requiring numerical methods to approximate the solutions. Compounding this difficulty are the unknown parameters in the DE system, which must be given specific numerical values in order for simulations to be run. Estrogen receptor protein dimerization is used as an example to demonstrate model construction , reduction, simulation, and parameter estimation. Mathematical, computational, and statistical methods are applied to empirical data to deduce kinetic parameter estimates and guide decisions regarding future experiments and modeling. The process demonstrated serves as a pedagogical example of quantitative methods being used to extract parameter values from biochemical data models. 1. Introduction. The empirical study of many current biological problems generates large and complex data sets. How best to use this data to generate and improve scientific hypotheses is a subject of great interest to biologists, mathematicians, statis-ticians, and computational scientists. Combining these various scientific and quantitative disciplines requires careful communication. Figure 1.1 illustrates the flow of information for a typical problem in quantitative biology. The modeling process described in the present work began with the formation of a scientific hypothesis based on laboratory experiments and intuition. This theory was used by biochemists to construct an experimental protocol and generate data. The same theory was used by mathematicians to develop a mathematical model, which was studied analytically and simulated computationally. Both models were intended to confirm or improve hypotheses. The theory and experiments are described in §2 while the mathematical and computational models are described in §3 and §4, respectively. For the biological model to feed back on theory, its output data needed to be analyzed. For the quantitative model to feed back on theory, its unknown parameter values needed to be estimated, so that meaningful simulations could be performed. Data modeling allowed empirical evidence to be combined with computational simulation as a means of generating parameter estimates and furthering biological theories. This process is described in §5. As Figure 1.1 suggests, the deduction of parameter estimates and confirmation of scientific hypotheses is not the end of the modeling process. Indeed, the data model answered some questions while raising others, prompting a new cycle of modeling which is now underway. Conclusions and future modeling directions are discussed in §6.