Homeostatic Controllers Compensating for Growth and Perturbations

Cells and organisms have developed homeostatic mechanisms to maintain internal stabilities which protect them against a changing environment. How cellular growth and homeostasis interact is still not well understood, but of increasing interest to the synthetic and molecular biology community where molecular control circuits are sought and tried to maintain homeostasis that opposes the diluting effects of cell growth. In this paper we describe the performance of four negative feedback (inflow) controllers, which, for different observed growth laws (time-dependent increase in the cellular volume V ) are able to compensate for various time-dependent removals of the controlled variable A . The four implementations of integral control are based on zero-order, first-order autocatalytic, second-order (antithetic), and derepressing inhibition kinetics. All controllers behave ideal in the sense that they for step-wise perturbations in V and A are able to drive the controlled variable precisely back to the controller’s theoretical set-point The applied increase in cellular volume includes linear, exponential and saturating growth and reflect experimentally observed growth laws of single cell organisms and other cell types. During the increase in V , additional linear or exponential time-dependent perturbations which remove A are applied, and controllers are tested with respect to their ability to compensate for both the increase in volume V and the applied perturbations removing A . Our results show that the way how integral control is kinetically implemented and the structure of the negative feedback loop are essential determinants of a controller’s performance. The results provide a ranking between the four tested controller types. Considering exponential volume increases together with an exponentially increasing removal rate of A controllers based on derepression kinetics perform best, but break down when the control-inhibitor’s concentration gets too low. The first-order autocatalytic controller is able to defend time-dependent exponential growth and removals in A , but generally with a certain offset below its theoretical set-point The controllers based on zero-order and second-order (antithetic) integral feedback can only manage linear increases in V and removals in A , in dependence of the controllers’ aggressiveness. Our results provide a theoretical basis what controller kinetics are needed in order to compensate for different growth laws.