Robust Concentration and Frequency Control in Oscillatory Homeostats

Homeostatic and adaptive control mechanisms are essential for keeping organisms structurally and functionally stable. Integral feedback is a control theoretic concept which has long been known to keep a controlled variable robustly (i.e. perturbation-independent) at a given set-point by feeding the integrated error back into the process that generates . The classical concept of homeostasis as robust regulation within narrow limits is often considered as unsatisfactory and even incompatible with many biological systems which show sustained oscillations, such as circadian rhythms and oscillatory calcium signaling. Nevertheless, there are many similarities between the biological processes which participate in oscillatory mechanisms and classical homeostatic (non-oscillatory) mechanisms. We have investigated whether biological oscillators can show robust homeostatic and adaptive behaviors, and this paper is an attempt to extend the homeostatic concept to include oscillatory conditions. Based on our previously published kinetic conditions on how to generate biochemical models with robust homeostasis we found two properties, which appear to be of general interest concerning oscillatory and homeostatic controlled biological systems. The first one is the ability of these oscillators (“oscillatory homeostats”) to keep the average level of a controlled variable at a defined set-point by involving compensatory changes in frequency and/or amplitude. The second property is the ability to keep the period/frequency of the oscillator tuned within a certain well-defined range. In this paper we highlight mechanisms that lead to these two properties. The biological applications of these findings are discussed using three examples, the homeostatic aspects during oscillatory calcium and p53 signaling, and the involvement of circadian rhythms in homeostatic regulation.


A-Activating Controller Motifs
The A-activating motifs are 1, 2, 5, and 6 ( Fig. 1, main paper). As an example we use the harmonic oscillator described in Fig. 2d. The rate equation for E is given as:Ė Integral control is introduced by zero-order kinetics when K Eset M becomes negligible in comparison to E (1,2). Under oscillatory conditions the setpoint in the average concentration of A, <A> c , is obtained by using the following condition where integration occurs along one orbit/cycle of stable oscillations with period P . By inserting the expression ofĖ (Eq. S1) into Eq. S2 we get Using ideal zero-order condition, K Eset M → 0, we have <E/(K Eset M +E)> c → 1, i.e., Inserting this result into Eq. S3, we get Note that <A> c is identical with the set-point of A when the system is nonoscillatory. This is shown in Fig. 3b and Fig. 5d for limit-cycle oscillators based on motif 2 and 5, respectively.

A-Inhibiting Controller Motifs
The A-inhibiting controller motifs are 3, 4, 7 and 8. As an example we use controller motif 3. Figure S1. Motif 3.
The rate equation for E is given as: Considering zero-order conditions in the Michaelis-Menten removal of E, the condition <Ė> c =0 gives: where P is the period of the oscillator and V Eset max /k 4 ·K A I is the homeostatic conserved property. In case the system becomes non-oscillatory the homeostatic set-point of A, A set , is given as (2):

Conservative Oscillator Types and Construction of their H-functions
We illustrate here the construction of the H-functions of the four different conservative oscillator types that can be constructed by using motif 2.
Oscillator with both A and E Removals Being Zero-Order Fig. 2a in the main paper shows the reaction scheme and rate equations for this case. For the sake of simplicity we assume that k 1 =0. The H-function obeys the following equations, which are analogous to the Hamilton-Jacobi The H-function is constructed by integratingȦ andĖ, i.e., Applying the zero-order conditions with respect to the removal kinetics of A and E, we get which leads to the final expression of H (see also Fig. S2)

Oscillator with Autocatalysis in A and E and First-Order Removals
In the case the degradation of E is first order, integral control in A can be implemented by first-order autocatalysis in E (3). To keep the system conservative with first-order degradation in A, the formation in A needs to be first-order autocatalytic as indicated in Fig. S3 and shown by the rate equations. Figure S3. Motif 2 with autocatalysis and first-order degradations in A and E.
The rate equations are:Ȧ Introducing the variables ξ = ln A and η = ln E, the rate equations can be transformed to:ξ By expressing E and A in Eqs. S16 and S17 in term of ξ and η, the function describes the kinetics of this conservative system and, after integration, is given by By using ξ = ln A and η = ln E, H can be expressed in terms of A and E, i.e,

Oscillator with Autocatalysis in A and Zero-order Removal of E
The scheme of this conservative oscillator is given in Fig. S5. The rate equations are:Ȧ . Motif 2 with autocatalysis and first-order degradation in A and and zero-order removal of E.
The H-function is given by the following integral: Inserting the expression ξ= ln(A) into Eq. S24 gives the final form of H(A, E): shows the numerically calculated oscillations and the constructed H-function describing these oscillations in phase-space.

Oscillator with Zero-order Removal of A and Autocatalysis in E
The reaction scheme of this conservative oscillator is given in Fig. S7. The rate equations are:Ȧ The H-function is given by the integral: Inserting η= ln(E) into Eq.S33 leads to Fig. S8 shows the numerically calculated oscillations and the constructed H-function.

Harmonic Approximation of Frequency in Conservative Oscillatory Controllers
The harmonic approximation of the frequency in conservative controllers provides insights why oscillatory controllers based on even-numbered motifs ( Fig. 1) increase their frequency upon increased perturbation strengths.
As an example we show how the harmonic approximation of the frequency can be obtained for the conservative oscillatory controller based on motif 2 (Fig. 2a). We assume zero-order removal in A and E and k 1 = 0. The rate equations read then:Ȧ Taking the second time derivative of Eq. S36 gives: Eq. S38 can be rearranged into the following form: When E in Eq. S39 is replaced by E ss we get the equation of a harmonic oscillator, i.e.,Ä/ω 2 +A=constant, with frequency ω given as and which approximately describes the frequency of the conservative oscillator (Eq. S39). A corresponding second-order differential equation can be derived for E:Ë A ss and E ss denote the steady state concentrations whenȦ=0 andĖ=0.
When replacing E by E ss in Eq. S41 the same harmonic frequency approximation as described by Eq. S40 is obtained. Similar expressions are found for the other E-inhibiting oscillatory controllers. Because the level of E decreases with increasing perturbation strength, Eq. S40 indicates that the E-inhibiting controllers will increase their frequency when perturbations are increased as shown in Fig. 3 for the motif-2-based controller.
For the conservative oscillators based on motifs 4 and 8, i.e., when both A and E are inhibiting, the harmonic oscillator approximations are: where k i and k j denote rate constants of the reactions which are inhibited by A and E.
For conservative oscillators based on motifs 1 and 5, the frequency is not dependent on either A or E giving harmonic oscillators (see next section). We consider the rate equations:

Harmonic Oscillations (Inflow Controller Motif 1)
In case of zero-order conditions in the removal of A and E they reduce to: Taking the second time derivative of Eq. S46 and inserting the expression oḟ E into it, leads to:Ä Dividing Eq. S48 by k 3 ·V Eset max gives the equation of a harmonic oscillator around the set-point <A> c : where A(t) is given as: A amp denotes the A-amplitude of the oscillations, ω is the frequency, and φ is a phase angle.

First-Order Degradation Restrains Oscillations
The amount of uncontrolled first-order degradation of A has a major influence on the size of the parameter space for the extended motif 5 (Figs. 4a and 5d in main paper) in which sustained oscillations are observed.Our results show that first-order degradation has the ability to quench the oscillations and does so for a large range of parameters. Fig. S10 shows how an increasing firstorder rate constant in the removal of A reduces the oscillatory behavior in the k 1 (uncontrolled inflow of A)-k 5 (conversion of precursor e into E) parameter space. The parameter space in which sustained oscillations are observed shrinks markedly when when the first-order degradation rate constant k 3 is increased by one order of magnitude, i.e. from 0.01 to 0.1 (Fig. S10, panel c) Figure S10. Period of oscillations for varying k 1 and k 5 values using the limit-cycle verion of motif 5 described in the main paper (Fig. 4a). The period is set to −1 when there are no oscillations (black area). Panels (a), (b), and (c) show the results for three different values for the first-order Aremoving rate constant k 3 . The parameter values used are: Fig. S10 also shows the propagation towards quasi-harmonic kinetics. With increasing k 5 values the periods approaches the harmonic values of 2π/ √ k 2 k 4 . When the conversion from e to E is fast (high k 5 ), the motif gives quasi-harmonic oscillatory period homeostasis as discussed in the main paper. A very fast conversion from e to E does however lead to a non-oscillatory homeostasis in A, i.e., the range of k 1 values that give oscillations shrinks with increasing k 5 .

Zero-Order Degradation Facilitates Oscillations
How close the controlled degradation of A is to a perfect zero-order degradation is another factor that influences the size of the oscillatory regime. With a controlled degradation of A by the compensatory flux j comp K A M becomes an indicator of how close the degradation is to perfect zeroorder, i.e. when K A M → 0. Fig. S11 shows the size of the parameter space in which one observes oscillatory behavior in the extended motif 5 (Fig 4a in the main paper) for three different values of K A M . The uncontrolled first-order degradation rate constant k 3 is 1 × 10 −2 in all cases. Figure S11. Period of oscillations for varying k 1 and k 5 values using the limit-cycle verion of motif 5 described in the main paper (Fig. 4a). The period is set to −1 when there are no oscillations (black area). Panels Robust Frequency Control With Inflow Controller Motif 2 and Alternative I 1 /I 2 Feedback In the main manuscript the feedbacks from I 1 and I 2 were applied on intermediate a (Fig. 7) or were "mixed", i.e. were applied to both "a" and "A" (Fig. 8). In the following we show a model where the feedbacks from I 1 /I 2 are returned to A only. The scheme is given in Fig. S12. Figure S12. Alternative I 1 /I 2 feedback arrangement in a motif-2-based oscillator. The feedbacks from I 1 and I 2 act on A only.
The rate equations are: The controller molecules (manipulated variables) E, I 1 , and I 2 define the set-points <A> set , <E> I 1 set , and <E> I 2 set , respectively, which are given by Fig. S13 shows the oscillator's behavior, i.e., <A>, <E>, and the frequency as a function of the perturbation k 2 when <A> set =2.0, <E> I 1 set = 5.0, and <E> I 2 set = 2.0. Due to the two set-points <E> I 1 set and <E> I 2 set the frequency has a corresponding homeostatic regulation at two frequencies. Note that, although <E> changes between different set-points when k 2 is changed, <A> is kept at its homeostatic set-point <A> set =2.0. Fig. S14 shows the oscillations and the I 1 /I 2 regulation of the system when k 2 is changed from a relative low value (k 2 =1.0) to a relative high value (k 2 =7.0) at time t=50.0 (dashed line). At low k 2 values controller I 1 is dominant and removes A such that this controller's set-point in <E> is maintained. At high k 2 values I 2 is up-regulated and I 1 downregulated. I 2 now adds A to the system in order to keep the <E> level at the set-point determined by controller I 2 .