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Sensitivity Analysis
Sensitivity analysis attempts to determine how the output changes with respect to small changes in the model parameters. In this case, we can represent this as $d[vitamin]/d[parameter_i]$. We normalize these sensitivities in two ways. First, we non-dimensionalize them by dividing by the vitamin concentration and multiplying by the starting value of the parameter. Then we obtain relative sensitivities by scaling them so the maximum sensitivity is 1.
We noticed that sensitivity to the Michaelis-Menten binding constant km is nearly zero in all pathways. We can interpret this as meaning the substrate concentration is high enough to saturate the enzyme at each step of the pathway. Vmax, which is the product of kcat and the enzyme concentration, controls the reaction rate. Thus in our analyses we show the sensitivities with respect to kcat.
Parameter fitting
While a model takes known qualitative information and organizes it systematically, parameter-fitting allows us to compare models with measured data, giving our models predictive power. Parameter fitting can be construed as an optimization problem. Here, we attempt to minimize an error function between our model and our observations. A common error function is the squared error:
\[||simulation-observation||{_{2}}^{2}=\sum_{i,j}^{N}(simulation_{i})^{2}-(observation_{j})^{2}\]
While mathematically elegant, it does not explicitly bound how badly a particular point might be fit. To ensure that even our worst points are reasonably well-fit by the model, we use the minimax error function:
\[max_{i,j}[(simulation_{i})^{2}-(observation_{j})^{2}]\]
In addition using the SimBiology Toolbox in Matlab, we have access to a convenient graphical interface for parameter fitting. Matlab uses the Levenberg–Marquardt Algorithm (LMA), which minimizes the least-squares error function mentioned above. While fairly robust, LMA has two drawbacks: it finds only a local minimum of the error function, and it assumes no measurement error in the measurements parameters. Nonetheless, it is a popular method that works well on most models.
We are currently awaiting the result of experiments to quantify the production of vitamins and other products using spectroscopy and HPLC. Once the data is available, our modeling group will be able to format the data and fit our model using LMA and minimax optimization.
