Extending the method of mathematically controlled comparison to include numerical comparisons

Motivation: The method of mathematically controlled comparison has been use d for some time to determine which of two alternative regulatory designs is better according to specific quantitative criteria for functional effectiv eness. In some cases, the results obtained using this technique are general and independent of parameter values and the answers are clear-cut. in othe rs, the result might be general, but the demonstration is difficult and num erical results with specific parameter values can help to clarify the situa tion. 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 alt ernatives can have the larger value for a given systemic property, dependin g on the specific values of the parameters. In any case, introduction of sp ecific values for the parameters reduces the generality of the results. The refore, we have been motivated to develop and apply statistical methods tha t would permit the use of numerical values for the parameters and yet retai n some of the generality that makes mathematically controlled comparison so attractive. Results: We illustrate this new numerical method in a step-by-step applicat ion using a very simple didactic example. We also validate the results by c omparison with the corresponding results obtained using the previously deve loped analytical method. The analytical approach is briefly present for ref erence purposes, since some of the same key concepts are needed to understa nd the numerical method and the results are needed for comparison. The nume rical method confirms the qualitative differences between the systemic beha vior of alternative designs obtained from the analytical method. In additio n, 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 unbra nched pathway makes the system more robust whereas it decreases the stabili ty margin of the steady state. The numerical method shows that the magnitud es of these differences are not comparable. The differences in stability ma rgins (1-2% on average) are small when compared to the differences in robus tness (50-100% on average). Furthermore, the numerical method shows that th e system with overall feedback responds more quickly to change than the oth erwise equivalent system without overall feedback. These results suggest re asons why overall feedback inhibition is such a prevalent regulatory patter n in unbranched biosynthetic pathways.