Homeostatic controllers compensating for growth and perturbations

Cells and organisms have developed homeostatic mechanisms which protect them against a changing environment. How growth and homeostasis interact is still not well understood, but of increasing interest to the molecular and synthetic biology community to recognize and design control circuits which can oppose the diluting effects of cell growth. In this paper we describe the performance of selected negative feedback controllers in response to different applied growth laws and time dependent outflow perturbations of a controlled variable. The approach taken here is based on deterministic mass action kinetics assuming that cell content is instantaneously mixed. All controllers behave ideal in the sense that they for step-wise perturbations in volume and a controlled compound A are able to drive A precisely back to the controllers’ theoretical set-points. The applied growth kinetics reflect experimentally observed growth laws, which range from surface to volume ratio growth to linear and exponential growth. Our results show that the kinetic implementation of integral control and the structure of the negative feedback loop are two properties which affect controller performance. Best performance is observed for controllers based on derepression kinetics and controllers with an autocatalytic implementation of integral control. Both are able to defend exponential growth and perturbations, although the autocatalytic controller shows an offset from its theoretical set-point. Controllers with activating signaling using zero-order or bimolecular (antithetic) kinetics for integral control behave very similar but less well. Their performance can be improved by implementing negative feedback structures having repression/derepression steps or by increasing controller aggressiveness. Our results provide a guide what type of feedback structures and integral control kinetics are suitable to oppose the dilution effects by different growth laws and time dependent perturbations on a deterministic level.

Steady states and theoretical set-point for motif 2 zero-order controller Transporter-based compensatory flux with constant values ofV andk 3 We refer to the rate equations for A and E (Eqs. 43-44), which are written in the following form:Ȧ by setting in Eq. 44 M/(k 11 +M )=E/(k 10 +E)=1.

CalculatingÄ gives
Inserting Eq. S2 into Eq. S3 leads tö Setting theV /V terms in Eq. S5 to zero we geẗ Setting Eq. S1 to zero and neglecting theV /V terms gives the relationship between decreasing E and increasing V and k 3 to keep A at a constant steady state A ss , i.e.
Inserting (k 2 k 4 ) 2 from Eq. S7 into Eq. S6 and settingÄ=0 gives For constantk 3 and increasing values of V and k 3 the offset termk 3 k 2 k 4 /k 2 3 A ss V goes to zero and A ss is kept by the controller at its theoretical set-point A theor set =k 9 /k 8 as clearly seen in Fig. 13. Since a constant A ss level by this controller type is maintained by decreasing E values the negative feedback loop will break when E becomes low and the controller reaches its capacity limits (Eq. 48).

Cell-internal compensatory flux with constant values ofV and anḋ k 3
In this case the rate equations (Eqs. 62-63) are written aṡ InsertingĖ from Eq. S10 into Eq. S11 Setting Eq. S9 to zero and neglecting theV /V term, we have the condition how E has to decrease for increasing k 3 to keep A constant at A ss , i.e., Substituting (k 4 +E) 2 in Eq. S11 by (k 4 k 6 ) 2 /k 2 3 A 2 ss , setting the resulting equation to zero, and neglecting theV /V terms, leads to where A theor set =k 9 /k 8 and the offset term is zero fork 3 =0, and goes to zero whenk 3 is constant and k 3 increases.
Cell-internal compensatory flux with exponential increase ofV anḋ k 3 i) Exponential increase in V and constant k 3 (phase 2). We start again with the rate equationsȦ In the casek 3 =0, but V increases exponentially, sayV =κV , A ss and E ss show constant value, where A ss shows an offset above A theor set (overcompensation). The steady state in A can be calculated by setting Eq. S2 to zero, i.e., ii) Exponential increase in V and k 3 (phase 3). Assuming thatV =κV anḋ k 3 =ζk 3 with κ and ζ constants, we can calculateÄ assuming thatȦ=0. Inserting Eq. S2 (note thatV /V =κ) into Eq. S16 and setting Eq. S16 to zero gives the expression for the steady state of A, A ss , − k 4 k 6 (k 4 +E) 2 [k 8 ·A ss − k 9 − κ·E] −k 3 A ss = 0 (S17) leading to Note, that while A is in a steady state, E is decreasing (derepressing) in order to increase the compensatory flux. Eq. S18 can be rewritten as where A apparent set is an "apparent set-point". Thus, Eq. S19 can be written as  Fig. 25d). (c) Behaviors of A ss and A app set as a function of time (Eq. S20). (d) By the end of phase 3 E 2 decreases more rapidly than the exponential increase ofk 3 , which is indicated by the productk 3 E 2 going to zero.
Fig. S1c shows how γ 0 and A app set changes with respect to the controller's behavior when exposed to exponential increase in V and k 3 with the response shown in Fig. 25d. For convenience the perturbation profile and the controller's response are repeated in Figs. S1a and b. The derepression by decreasing E leads to an increase in γ 0 (Fig. S1c, curve outlined in blue). The increase in γ 0 is the result of E 2 decreasing more rapidly than the exponential increase ofk 3 . This is indicated in Fig. S1 where the productk 3 E 2 during phase 3 decreases and A ss → A theor set .