Improved methods for the mathematically controlled comparison of biochemical systems

Abstract

The method of mathematically controlled comparison provides a structured approach for the comparison of alternative biochemical pathways with respect to selected functional effectiveness measures. Under this approach, alternative implementations of a biochemical pathway are modeled mathematically, forced to be equivalent through the application of selected constraints, and compared with respect to selected functional effectiveness measures. While the method has been applied successfully in a variety of studies, we offer recommendations for improvements to the method that (1) relax requirements for definition of constraints sufficient to remove all degrees of freedom in forming the equivalent alternative, (2) facilitate generalization of the results thus avoiding the need to condition those findings on the selected constraints, and (3) provide additional insights into the effect of selected constraints on the functional effectiveness measures. We present improvements to the method and related statistical models, apply the method to a previously conducted comparison of network regulation in the immune system, and compare our results to those previously reported.

Figure 1

Reference and alternative systems. Biochemical maps for the reference system (with suppression) and the alternative (without suppression) are given in A and B respectively. Adapted from Irvine and Savageau [21].

Figure 2

Example mapping: pathways to S-systems. The S-system framework provides for a straightforward mapping of biochemical pathway maps into systems of equations. The pathway and equations for cases A and B differ only in the feedback inhibition of the first step in the process. This inhibition is represented by a single parameter, g_1,3.

Figure 3

Biosynthetic pathway alternatives. Biosynthetic pathways similar to that illustrated were compared using the method of mathematically controlled comparison by Alves and Savageau [13]. These biosynthetic pathways differ only in the reversibility of the first step.

Figure 4

Step responses to source antigen increase. Step responses to a four-fold increase in source antigen are presented for both the nominal values (panel A) (from Irvine and Savageau) and for a case in which the values were drawn from a narrow distribution about those nominal values (panel B). Systemic antigen responses are shown on the left and effector on the right. Solid lines indicate the response for the reference system (with suppression) and dashed lines are used for the alternative. Step responses for the nominal values indicate a preference for the system with suppression. The step responses for the sampled case indicate a preference for the alternative when considering dynamic peaks for effector concentration.

Figure 5

Performance measure distributions. Smoothed histograms were computed using R (density function) and are presented for each of the 7 functional performance measures for each of the equivalence levels. The dashed vertical line indicates the median of the distribution for the reference system. The alternative is preferred when more of its probability mass is distributed to the left of the dashed line.

Figure 6

Pairwise comparison of selected performance measures. Scatter plots of 840 pairs were produced in R and show basal systemic antigen level (top), basal effector level (middle), and sensitivity (bottom) in the reference (x-axis) versus the same value in the associated alternative (y-axis) for all equivalence levels. The dashed diagonal line corresponds to the null condition where the reference and the alternative are equal. The reference is preferred over the alternative when more pairs fall above the diagonal. System sensitivity shows a clear preference for the reference with all cases preferring the reference under full equivalence.