Extending the method of mathematically controlled comparison to include numerical comparisons

The method of mathematically controlled comparison has been used for some time to determine which of two alternative regulatory designs is better according to specific quantitative criteria for functional effectiveness. In some cases, the results obtained using this technique are general and independent of parameter values and the answers are clear-cut. In others, the result might be general, but the demonstration is difficult and numerical results with specific parameter values can help to clarify the situation. In either case, numerical results with specific parameter values can also provide an answer to the question of how much larger the values might be. In contrast, a more ambiguous result is obtained when either of the alternatives can have the larger value for a given systemic property, depending on the specific values of the parameters. In any case, introduction of specific values for the parameters reduces the generality of the results. Therefore, we have been motivated to develop and apply statistical methods that would permit the use of numerical values for the parameters and yet retain some of the generality that makes mathematically controlled comparison so attractive.

We illustrate this new numerical method in a step-by-step application using a very simple didactic example. We also validate the results by comparison with the corresponding results obtained using the previously developed analytical method. The analytical approach is briefly present for reference purposes, since some of the same key concepts are needed to understand the numerical method and the results are needed for comparison. The numerical method confirms the qualitative differences between the systemic behavior of alternative designs obtained from the analytical method. In addition, the numerical method allows for quantification of the differences and it provides results that are general in a statistical sense. For example, the older analytical method showed that overall feedback inhibition in an unbranched pathway makes the system more robust whereas it decreases the stability margin of the steady state. The numerical method shows that the magnitudes of these differences are not comparable. The differences in stability margins (1-2% on average) are small when compared to the differences in robustness (50-100% on average). Furthermore, the numerical method shows that the system with overall feedback responds more quickly to change than the otherwise equivalent system without overall feedback. These results suggest reasons why overall feedback inhibition is such a prevalent regulatory pattern in unbranched biosynthetic pathways.