On the matching and proportional laws of cybernetic models

The Matching and Proportional Laws are heuristic control policies that have found widespread use in cybernetic models of biological systems. Within this context, the laws serve as optimization surrogates for predicting the response of metabolic control circuits that modulate enzyme levels and activities. The key result of the current contribution is to demonstrate clearly the optimality properties of these laws and the assumptions that underlie their development. In doing so, we arrive at generalized versions of the Matching and Proportional Laws that are shown to collapse to the forms originally derived by Kompala et al. (Biotechnol. Bioeng. 1986, 28, 1044-1055) when certain simplifications are applied. As a further line of investigation, we show how Kompala et al.'s cybernetic laws compare with alternative control policies in their ability to describe diauxic growth behavior of microbial cultures. We find that Kompala et al.'s model describes the experimental observations more accurately than other limiting-case models that are either too aggressive or too passive in capturing the mixed-substrate growth rates and intermediate lag periods. Monte Carlo analysis of computational growth experiments in which strains obeying different regulatory policies directly compete for available nutrients reveals that the Matching and Proportional Law policy does not maximize the average growth rate of the culture. However, it allocates metabolic resources more frugally than other policies that outperform it and may be more realistic in reflecting the cell's true fitness-to-cost tradeoff as judged by its agreement with experimental growth data.