An amplified derepression controller with multisite inhibition and positive feedback

How organisms are able to maintain robust homeostasis has in recent years received increased attention by the use of combined control engineering and kinetic concepts, which led to the discovery of robust controller motifs. While these motifs employ kinetic conditions showing integral feedback and homeostasis for step-wise perturbations, the motifs’ performance differ significantly when exposing them to time dependent perturbations. One type of controller motifs which are able to handle exponentially and even hyperbolically growing perturbations are based on derepression. In these controllers the compensatory reaction, which neutralizes the perturbation, is derepressed, i.e. its reaction rate is increased by the decrease of an inhibitor acting on the compensatory flux. While controllers in this category can deal well with different time-dependent perturbations they have the disadvantage that they break down once the concentration of the regulatory inhibitor becomes too low and the compensatory flux has gained its maximum value. We wondered whether it would be possible to bypass this restriction, while still keeping the advantages of derepression kinetics. In this paper we show how the inclusion of multisite inhibition and the presence of positive feedback loops lead to an amplified controller which is still based on derepression kinetics but without showing the breakdown due to low inhibitor concentrations. By searching for the amplified feedback motif in natural systems, we found it as a part of the plant circadian clock where it is highly interlocked with other feedback loops.

2) Reviewer #1: "Equation (1) and (2). Here is a suggestion: The authors should consider changing the use of variable E to U to (i) avoid potential associating E with Error and (ii) U is commonly known in control community to represent control action/signal. As a note, it took me quite a while to realise that E is not error but control action, which left me initially quite confused." Response: In our previous work we have kept the term E for the manipulated variable, because the concentration of E is proportional to the integrated error, i.e., E = t 0 (τ )dτ . This is the major reason why we wish to keep the E terminology for the manipulated variable in our A-E controller motifs, where A and E denote both the chemical/biochemical species and their concentrations. Another reason is that it appears awkward to refer to our earlier work and to have to explain our change in nomenclature. However, since U is consistently used in control-engineering as a symbol for the manipulated variable, we have included the following changes in the manuscript: i) In the legend of Fig 1, we mention that E is the manipulated variable and proportional to the integrated error =A set −A: "We will show below that the concentration of species E, in control engineering terms called the manipulated variable (and generally assigned the name U), is proportional to the integrated error =A set −A".
ii) On page 3 in the revised manuscript, Eq 2 is extended by Eq 3, showing thatĖ is proportional to (A set −A), i.e.,Ė and that the concentration of E is proportional to the integrated error .
3) Reviewer #1: "Line 47. The assumption made here is that K M E. It seems to me that the overall numerical simulation is carried out with K M E and it gives the impression the mechanism of Motif 2 will work based on this assumption. What if K M is not significantly smaller than E? How would this affect the overall analysis and conclusion? Has this point being considered as there is no certainty that in practice, K M E often hold."    Figure 3? In practice, how common is k 1 be subjected to this exponential type of perturbation and is it common that k 1 having such large value, in the range up to 10 7 ." Response: The parameters used to describe the exponential and hyperbolic increase of k 1 in Fig 3 and by Eq 14, respectively, are those as used before in References 32 and 33. Although exponential and hyperbolic increases have been found in certain natural systems (for example growth of cell volume and certain viral infections) there is to our understanding not much work which has focussed on time dependent perturbations with respect to robust homeostatic control. In this respect, it is difficult to give here a definite answer to the reviewer. Roughly, our (E/K I ) n term in Eq 6 will then be replaced by a polynomial of the form: where the K I i 's are dissociation constants. However, if the inhibitor has cooperative binding, then all terms with less than n factors are generally neglected, which leads to Eq 6 with an average  (7).: This is another suggestion: It is quite odd to label equation this manner where (2) is appearing between (6) and (7). I understand it is the same equation but it is rather odd. Why not just label them as (6)-(8) in a sequential manner?" Response: We have now labeled the equations in a sequential manner. 10) Reviewer #1: "Page 20, Line 340: Can the author clarifies this statement that RVE8 interacts with LHY/CCA1? I don't think that statement is correct as RVE8 does not interact with LHY/CCA1 as shown in the two articles below. If the authors look at the interaction of plant circadian genes shown in Figure 1 of both respective articles, there is no depiction of RVE8 interacting with LHY/CCA1." Response: Thank you for this! Actually, we wanted to change that sentence before submission, but apparently this was forgotten. We have now changed it and cite the two papers the reviewer refers to, in addition to a review paper by McClung; see sentence starting line 412: "RVE8 (as a homolog of LHY/CCA1) interacts with the promoters of PRR9, PRR5, TOC1, GI and the EC, which in their turn also have an influence on the plant circadian rhythm [64,68,69]." Changes made in the revised manuscript based on the comments by Reviewer #2 Changes made in the manuscript with respect to the comments by Reviewer #2 are indicated in blue in the marked-up copy of the revised manuscript.
1 (a) Reviewer #2: "Line 58: Why the breakdown takes place when E = KI ??" Response: As E decreases with increasing k 1 the compensatory flux k 2 /(1+ E K I ) will increase and reach finally its maximum value k 2 . As long as E/K I 1 the controller is operative. However, the situation becomes critical when E/K I ≤1 and k 1 continues to increase. Controller breakdown occurs when the maximum compensatory flux k 2 is reached and k 1 grows further. To estimate the k 1 value when controller performance becomes critical, k crit 1 , we use the condition E/K I =1. Inserting E/K I =1 into Eq 1 withȦ=0, gives Eq 4.  Response: The treatment is analogous to that described in 1(a) above. We have tried to make this more explicit in the manuscript by the following addition, lines 136-141, outlined in blue: "...the controller starts to break down upon increasing k 1 values. As for the controller without C (Fig 2, Eq 5) we can estimate a critical k 1 value when breakdown starts by setting E/K I =1 and inserting it into Eq 7. Solving for k 1 and noting that A is still at its set-point, gives Fig 6 shows that the controller's lifetime is now dependent on C and the values of the rate constants k 5 and k 6 ." 1 (d) Reviewer #2: "Line 140: How can we guarantee that A will stay at the set-point when k 1 grows further? As Fig 3, A can encounter a breakdown if k 1 grows further. Without mathematical proof or any intuitive descriptions, there is no way to guarantee this. I think this can be shown simply by solvingĖ = 0 andĊ = 0. Then by solvingȦ = 0, we could prove that A is approximately equal to the set point in the long run." Response: Please note that the conditionĊ = 0 will only work for step-wise perturbations in k 1 . When k 1 increases with time, we have thatĊ> 0.
An intuitive description of C's role when k 1 increases with time (linearly, exponentially, or hyperbolically) is based on using appropriate kinetic rate laws in generating C, which allows C to follow and compensate the increase in k 1 . There is a kind of hierarchical kinetic order, by which time-dependent increases/perturbations of k 1 can be ruled by a controller. We have: • hyperbolic controller kinetics (second-order autocatalysis) can compensate first-order autocatalytic growth • first-order autocatalytic controller kinetics can compensate linear growth If perturbations have linear, autocatalytic (first-order), or hyperbolic kinetics, then respectively, linear, autocatalytic, or hyperbolic controller kinetics can also oppose the perturbations, but for some controller types an offset, i.e., a deviation from the set-point is observed. This type of kinetic hierarchy has been described and applied in Ref 32.
To address the reviewer's point for an intuitive description we have included the following paragraph (lines 161-176): "To get an intuitive understanding about the role of C and the required kinetics to oppose different rate laws of k 1 , we note that linear and first-order autocatalytic growth rates in k 1 can be compensated by second-order autocatalytic (hyperbolic) generation of C. In other words, compensatory growth based on second-order autocatalysis will dominate over perturbative first-order autocatalytic or linear growth and thereby control it. The reason for this, is because second-order autocatalysis is eventually more rapid than first-order autocatalysis and linear growth. Also, a linear increase of k 1 can be opposed when the compensatory flux is based on first-order autocatalysis, because first-order autocatalysis will in the end become more rapid than linear growth. Ref 32 gives a more detailed description of how such hierarchies between rate laws has been applied with respect to to different time-dependent perturbations and controller motifs. Thus, the role of C, which is generated by an appropriate rate law, is to be "ahead" of the time-dependent perturbation and oppose it by its influence to the compensatory flux. As a result, E will go into a steady state and A can be kept, due to Eq 8 and due to the zero-order removal of E, at its homeostatic set-point. Indeed,Ȧ=0 in Eq 7 and constant E implies that C=k 1 ×constant." 1 (e) Reviewer #2: "Line 154: With Fig 8 left panel it seems C is almost k1+constant. But indeed,Ȧ = 0 in Eq (7) implies that C = k1 × constant. I think the authors should mention this." Response: Thank you for pointing out the implication for C fromȦ=0. We have now mentioned this in the last sentence starting line 175: "Indeed,Ȧ=0 in Eq 7 and constant E implies that C=k 1 ×constant.".

(f ) Reviewer #2
: "Line 181-183: The derivative of C is not zero for t large enough. Eq (15) is only valid when k1 is a step-wise change. To address this rigorously, the author should use the fact thatĊ C 2 goes to zero as t goes to infinity rather than Eq(10)=0." Response: Actually, we have been using this fact. We have now reworded the sentence in the revised manuscript to (starting line 213): "E's set-point can be calculated by dividing Eq 12 with C 2 and setting the resulting expression to zero, which leads to 1 (g) Reviewer #2: "Line 282-286: This part was really interesting to me. If the authors can provide any intuition behind this, that could make this paper better pretty much." Response: We have added a paragraph (starting line 312) outlining the idea that period homeostasis can be achieved by mechanisms which keep the flux through the oscillator constantly regulated/balanced : Response: Correct, but please note that <C>=0.001=constant and that <Ċ>=0. The derivation of K I K I +E = k 6 k 5 =0.01 (Eq 30 in revised ms) is based on <Ċ>=0 (Eq 25). Eq 25 is then divided by C leading to Eq 30. Eq 30 is in agreement with the numerical calculations shown in Fig 21. 1 (i) Reviewer #2: "Eq (28): Why we need to see this quantity? Any meaning?" Response: The quantity K I K I +E = k 6 k 5 =0.01 (now Eq 30) is conserved and we looked for an analytical solution/expression for <E>. However, as we mention in the ms we were not able to find one, and therefore used the numerical solution of < E >. Response: Yes, this is in agreement with the comment by Reviewer #1, no. 1 above, that our motivation for this work should be stated more clearly. Please, see our response to 1) Reviewer #1 above and the red-outlined paragraph in the revised manuscript starting at line 33. Response: Yes, the reviewer is correct, as long as a steady state exists,Ċ = 0 implies (11) (now Eq 13). We have reworded that sentence in this way (starting line 149), but referred to a calculation where A is under homeostatic control, but E is not: "Note that for any changes in k 1 , as long as a steady state exists, E becomes homeostatic controlled in addition to A, because the conditionsĊ/C=0 orĊ/C 2 =0 in respectively Eqs 11 or 12 imply that

Minor issues
independent of the perturbation k 1 . However, a situation where A is homeostatic controlled, but E is not, is given below, when a controller with first-order autocatalysis in C meets a hyperbolically increasing k 1 ." 6. Reviewer #2: "Line 170: Before I see Fig 12, I did not think the system maintains homeostasis for A because A eventually encounters breakdown. So it would better to show Fig 12, especially the plots without C before Figure 9 for a better flow of the paper." Response: In the Introduction we mention (line 43) that motif 2 (m2, Fig 2) can balance exponentially, even hyperbolically increasing perturbations, and refer to [32]. As an example, we show in Fig 3 the case when m2 opposes exponentially increasing k 1 . When C has been introduced, we asked the question how well does m2 without C behave in comparison with controllers that have first-or second-order autocatalysis in C? Fig 12 is   12. Reviewer #2: "Line 244: Why we consider this? Why the paper considers this only for the second-order autocatalysis?" Response: In order to get a homeostatic response and to avoid overcompensation, we found that when C is generated by second-order autocatalysis we sometimes had to adjust the k 2 and k 5 values that were used by first-order autocatalysis. We have now changed that part slightly to (see also changed parts outlined in blue in revised ms, starting line 275): "When C is generated by second-order autocatalysis (Eq 12) the resulting controller is, as for first-order autocatalysis, able to defend A set , but values for k 2 and k 5 had to be adjusted to get a homeostatic response and to avoid overcompensation. Fig 18 shows the case when k 1 increases exponentially. We found that an