## Figures

## Abstract

When in a reaction kinetic integral controller a step perturbation is applied besides a constant background, the concentration of a controlled variable (described as *A*) will generally respond with decreased response amplitudes Δ*A* as backgrounds increase. The controller variable *E* will at the same time provide the necessary compensatory flux to move *A* back to its set-point. A typical example of decreased response amplitudes at increased backgrounds is found in retinal light adaptation. Due to remarks in the literature that retinal light adaptation would also involve a compensation of backgrounds we became interested in conditions how background compensation could occur. In this paper we describe novel findings how background influences can be robustly eliminated. When such a background compensation is active, oscillatory controllers will respond to a defined perturbation with always the same (damped or undamped) frequency profile, or in the non-oscillatory case, with the same response amplitude Δ*A*, irrespective of the background level. To achieve background compensation we found that two conditions need to apply: (i) an additional set of integral controllers (here described as *I*_{1} and *I*_{2}) have to be employed to keep the manipulated variable *E* at a defined set-point, and (ii), *I*_{1} and *I*_{2} need to feed back to the *A*-*E* signaling axis directly through the controlled variable *A*. In analogy to a similar feedback applied in quantum control theory, we term these feedback conditions as ‘coherent feedback’. When analyzing retinal light adaptations in more detail, we find no evidence of the presence of background compensation mechanisms. Although robust background compensation, as described theoretically here, appears to be an interesting regulatory property, relevant biological or biochemical examples still need to be identified.

**Citation: **Nygård M, Ruoff P (2023) Coherent feedback leads to robust background compensation in oscillatory and non-oscillatory homeostats. PLoS ONE 18(8):
e0287083.
https://doi.org/10.1371/journal.pone.0287083

**Editor: **Karthik Raman,
Indian Institute of Technology Madras, INDIA

**Received: **May 29, 2023; **Accepted: **August 14, 2023; **Published: ** August 28, 2023

**Copyright: ** © 2023 Nygård, Ruoff. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Data Availability: **All relevant data are within the paper and its Supporting information files.

**Funding: **The author(s) received no specific funding for this work.

**Competing interests: ** The authors have declared that no competing interests exist.

## Introduction

Homeostatic mechanisms play important roles in physiology and in the adaptation of organisms to their environments [1]. For example, vertebrate retinal photoreceptor cells contain negative feedback loops which participate in light adaptation [2–5]. A hallmark of vertebrate photoadaptation is that resetting kinetics accelerate and response amplitudes decrease as backgrounds increase [5, 6]. This behavior is seen in Fig 1 for a macaque monkey’s rod cell response towards a single light flash applied at different background light levels.

10 ms light flashes were applied to different light background intensities. Background intensities (in photons *μ*m^{−2}s^{−1}) were: **0**, 0; **1**, 3.1; **2**, 12; **3**, 41; **4**, 84; **5**, 162. The influence of the background on the response amplitude and the speed of resetting is clearly seen. *V*_{0} is the response amplitude for background **0**. Redrawn and modified after Fig 2A from Ref [7]. For a theoretical description of this behavior see Ref [5] and references therein.

Another retinal light adaptation example is shown in Fig 2. Here, the mean maximum firing rates of a cat ganglion cell was measured with respect to different step light perturbations (test spot luminance) which are applied at six different backgrounds [8].

Averaged maximum ganglion cell frequencies are shown as a function of six different background illuminations in response to applied light step perturbations (test spot luminance). The three colored curves show the averaged maximum frequencies at background illuminations 10^{−3}, 10^{−2} and 10^{−1} cd/m^{2}. A test spot luminance (perturbation) of 9 × 10^{−2} cd/m^{2} is indicated as the vertical dashed black line. The colored intersection points and vertical dashed lines indicate that for this perturbation strength the maximum mean response frequency decreases with increasing background illumination.

Kandel et al. [2] commented (see page 540, section *Light Adaptation Is Apparent in Retinal Processing and Visual Perception*) that Fig 2 would indicate a compensation of the background illumination and thereby causing the same response due to a lateral shifting along the perturbation (test spot luminance) axis. Based on this comment we became interested in mechanisms which would allow to compensate for background levels and thereby give the same response for a given perturbation irrespective of the applied background. In this paper we present results on how such a robust background compensation can be achieved in both oscillatory and non-oscillatory homeostats.

The paper is structured in the following way: We first show that a feedback type similar to what quantum physicists have termed ‘coherent feedback’ [9, 10] is required to obtain background compensation in both oscillatory and non-oscillatory homeostats. For oscillatory homeostats we show that coherent feedback control leads to background compensation and to frequency control. Although we previously observed robust frequency control [11], we did not recognize at that time the background-compensating property of coherent feedback. For non-oscillatory controllers or homeostats with damped oscillations, coherent feedback leads to conserved response profiles in the controlled variables, independent of an applied background. We then look at the situation of a ‘incoherent feedback’, where background compensation is lost, but oscillatory homeostats may still show robust frequency control. Finally, we analyze photoreceptor responses in terms of a model based on the Hill equation, as described by Dowling [3]. We show that parallel lines as in Fig 2, or as log-log plots, do not require the postulation of background compensation or additional adaptation mechanisms.

## Materials and methods

Computations were performed with the Fortran subroutine LSODE [12], which can be downloaded from https://computing.llnl.gov/projects/odepack. Graphical output was generated with gnuplot (www.gnuplot.info) and annotated with Adobe Illustrator (https://www.adobe.com/).

To make notations simpler, concentrations of compounds are denoted by compound names without square brackets. Time derivatives are generally indicated by the ‘dot’ notation. Rate parameters are in arbitrary units (au) and are presented as *k*_{i}’s (*i* = 1, 2, 3, …) irrespective of their kinetic nature, i.e. whether they represent turnover numbers, Michaelis constants, or inhibition/activation constants. To allow readers to redo calculations, the supporting information S1 Data contains python scripts for a set of selected results.

### Integral control, step perturbations and the background concept

In the calculations robust homeostasis of concentrations and frequencies is achieved by implementing integral control into the negative feedbacks, a concept which has its origin from control engineering [13–16], and has been indicated to occur also in biological systems [17–21]. Briefly, in integral control the difference (also termed error) between the actual concentration of a controlled variable *A* and its set-point is integrated in time. The integrated error can then be used to compensate precisely for stepwise perturbations [15, 16]. Fig 3a shows the control scheme of integral control. A reaction kinetic example is given in panel b using ‘motif 2’, which is one of eight basic negative feedback structures [22]. Note that the removal reactions of *A* have been divided into two parts: a perturbative removal by a stepwise change of *k*_{1} (indicated in dark-red), and a constitutive removal of *A*, termed ‘background’, with a constant *k*_{2} (indicated in dark-blue).

Panel a: Principle of integral control. The controlled variable *A* (outlined in blue) is compared with its set-point and the difference/error (*A*_{set}−*A*) is integrated. This leads to the integrated error *E*, which is able to compensate precisely for stepwise perturbations [15]. Panel b: Basic negative feedback loop (motif 2, [22]). Solid lines are chemical reactions, while dashed lines represent activations (plus sign) and inhibitions (negative sign). The removal of *A* is divided into two parts, a perturbative part where *k*_{1} can change stepwise (outlined in dark-red) and a background part where *k*_{2} is kept constant (outlined in dark-blue). Panel c: Rate equation of controller *E*. The zero-order removal of *E* introduces integral control. The set-point for *A* (*A*_{set}) is given as *k*_{5}/*k*_{4}, and the concentration of *E* becomes proportional to the integrated error [22]. For other ways to implement integral control, see text.

For example, when studying photoreceptor cells often a constant background light is used in addition to the application of a flash or step of light [2–4]. In fact, motif 2 has been found [5] to describe light adaptations and the influence of different background light intensities, as shown in Fig 1, relatively well. In this paper we describe novel findings how a background can be compensated for, such that the response kinetics become independent of different but constant backgrounds.

There are several kinetic requirements which can lead to integral control. Fig 3c shows how a zero-order removal of controller species *E* results in integral control with a defined set-point *A*_{set} of the controlled variable *A*.

In the following calculations we have used zero-order kinetics to implement integral control [17, 22–24]. However, it should be mentioned that there are other kinetics conditions to achieve integral control, such as antithetic control [21, 25, 26], which often will show identical resetting behaviors as in zero-order control [5, 26]. Also, autocatalytic reactions can be used to obtain integral control [27–29], which generally will show much faster resetting kinetics in comparison when integral control is introduced by zero-order kinetics [30, 31].

Recently, mathematical methods has been put forward based on either linear [32, 33] or nonlinear [34] generic models to describe and analyze physiological control systems. These more mathematically inclined methods differ from the approach taken here, which is based on mass action kinetics.

## Results and discussion

### Background compensation in negative feedback oscillators by coherent feedback

In this section we describe feedback conditions which can achieve background compensation in oscillatory homeostats. The here considered oscillators show frequency homeostasis based on a two-layered negative feedback structure. A center integral feedback layer ensures that the time average of a controlled variable *A*, defined by Eq 1, is kept robustly at set-point *A*_{set} by a controller species *E*.
(1)

A second ‘outer’ negative feedback layer keeps on its side the time average value of *E*, i.e. <*E*> (Eq 2), under robust homeostatic control by two additional controller variables *I*_{1} and *I*_{2}.
(2)

We previously showed [11] that such two-layered negative feedback structures enable robust frequency homeostasis. Here we report the additional and novel finding that when the *I*_{1} and *I*_{2} controllers feed back directly via *A* to control *E*, the oscillator has the capability to neutralize backgrounds. In analogy to a closely related feedback definition employed in quantum control theory and optics, we call this type of feedback for ‘coherent feedback’ (see [9, 10] and references therein).

#### Background compensation in a motif 2 based oscillatory homeostat.

Fig 4a shows an example of a frequency-compensated oscillator which can compensate for different but constant backgrounds.

Panel a: Reaction kinetic scheme based on derepression motif 2 (m2) [22] in the inner *A*-*e*-*E*-*A* negative feedback. Figs S12-S14 in the supporting information of Ref [11] describe some properties of this oscillator, but without having recognized at that time the ability to robustly compensate for backgrounds. Solid arrows indicate chemical reactions, while dashed lines show activations (plus signs) and one inhibition (minus sign). The addition of compound *e* was found [11] to promote limit-cycle oscillations. Panel b: Flow scheme indicating the additional control of *E* via *A* by controllers *I*_{1} and *I*_{2}.

The center oscillator in Fig 4a is given by the *A*-*e*-*E*-*A* feedback loop based on derepression motif 2 [22], where *E* keeps <*A*> under homeostatic control (see rate equations and definitions of set-points below). Oscillations are promoted, because the removals of *A* and *E* are zero-order with respect to *A* and *E*, and thereby construct a quasi-conservative oscillator [11]. The intermediate *e* has been included to obtain limit-cycle oscillations [11]. *I*_{1} and *I*_{2} are controller species, which keep <*E*> under homeostatic control. It is the control of <*E*> by *I*_{1} and *I*_{2}, which allows for the frequency homeostasis of the oscillator [11]. Their *A*-coherent feedback directly to *A* allows for robust background compensation. Since the central *A*-*e*-*E*-*A* negative feedback is an inflow controller it principally can only compensate for outflow perturbations [22]. The considered outflow perturbations of *A* split into two components: a constant (zero-order) background with rate constant *k*_{10} (Fig 4a outlined in blue) and a (zero-order) perturbation part where rate constant *k*_{2} undergoes a stepwise change (in Fig 4a outlined in red). Zero-order kinetics with respect to *A* are achieved by small *k*_{8} and *k*_{17} values, i.e. *A*/(*k*_{8} + *A*) ≈ 1 and *A*/(*k*_{17} + *A*) ≈ 1. Fig 4b shows the flow scheme and the control of *E* by *I*_{1} and *I*_{2} via the *A*-coherent part of the controller.

The rate equations are: (3) (4) (5) (6) (7)

The set-point of <*A*> (*A*_{set}) by controller *E* can be calculated from the steady state condition of the time averages:
(8)

In Eq 8 the zero-order condition with respect to *E* ensures a robust perfect adaptation of <*A*_{ss}> to *A*_{set} when the system oscillates, or a perfect adaptation of *A* to *A*_{set} in case the feedback loop is non-oscillatory [11]. Since the control of *A* by *E* is an inflow controller based on the derepression flux *k*_{3}⋅*k*_{5}/(*k*_{5} + *E*), the controller is active whenever <*A*> is below *A*_{set}.

Variable *E* is controlled by *I*_{1} and *I*_{2}. They act as respectively outflow or inflow controllers [22] with respect to <*E*> (if oscillatory) or *E* (if non-oscillatory). Also here zero-order removals of both *I*_{1} and *I*_{2} ensure robust set-points. For controller *I*_{1} the steady state condition gives:
(9)

The *I*_{1} outflow controller becomes active whenever <*E*> is higher than .

The set-point for the *I*_{2} inflow controller is determined by the steady state condition:
(10)

The *I*_{2} controller becomes active whenever <*E*> is lower than . It should be noted that the values of the inflow/outflow set-points and need to follow certain rules to guarantee that inflow and outflow controllers cooperate. In this case should be lower than , otherwise *I*_{1} and *I*_{2} will work against each other and windup will occur. For a discussion about windup in combined controllers, see Ref [22].

In the following we describe how the above oscillator behaves in presence of a stepwise perturbation at different but constant backgrounds. Fig 5 shows the oscillator’s behavior for a stepwise perturbation in *k*_{2} from 1.0 (phase 1) to 10.0 (phase 2) at a background *k*_{10} = 0.0. The time of change in *k*_{2} is indicated in each panel by a vertical arrow. Panel a shows the oscillations in *A* together with its average <*A*> (Eq 1), while panel b shows *E* and <*E*> (Eq 2). Panel c shows the changes in *I*_{1} and *I*_{2}, and panel d shows the frequency (i.e. the inverse of the period length). The resetting of the frequency to its pre-perturbation value is clearly seen. If *I*_{1} and *I*_{2} would not be present, <*A*> would be kept at *A*_{set} = 2.0 by a reduced (derepressed) *E* as seen in panel b at 100 time units. However, since <*E*> is also controlled by *I*_{1} and *I*_{2}, i.e. between 5.0 () and 4.99 (), *I*_{1} and *I*_{2} take over the control of both <*A*> and <*E*>.

Vertical arrows indicate where the change in *k*_{2} occurs. Other rate constants: *k*_{3} = 100.0, *k*_{4} = 1.0, *k*_{5} = 0.1, *k*_{6} = 2.0, *k*_{7} = *k*_{8} = *k*_{13} = *k*_{16} = *k*_{17} = 1 × 10^{−6}, *k*_{9} = 20.0, *k*_{11} = 1.0, *k*_{12} = 5.0, *k*_{14} = 4.99, *k*_{15} = 1.0, and *k*_{g} = *k*_{g3} = 1 × 10^{−2}. Initial concentrations: *A*_{0} = 0.3780, *E*_{0} = 2.4784, *e*_{0} = 1.5993 × 10^{−2}, *I*_{1,0} = 4.5727 × 10^{2}, *I*_{2,0} = 2.9817 × 10^{2} (see S1 Data for python script).

Fig 6 shows the same perturbation in *k*_{2} as in Fig 5, but with a background of *k*_{10} = 2048.0. The increased removal of *A* by the background is compensated by an increase of *I*_{2} and a decrease of *I*_{1}, which keep <*A*> and <*E*> at their respective set-points. The maximum frequency, which occurs directly after the *k*_{2} step is not affected by the changed background.

Vertical arrows indicate where the change in *k*_{2} occurs. Other rate constants as in Fig 5. Initial concentrations: *A*_{0} = 2.1377, *E*_{0} = 7.6720, *e*_{0} = 1.0996 × 10^{−1}, *I*_{1,0} = 3.4304, *I*_{2,0} = 2.0465 × 10^{5} (see S1 Data for python script).

Fig 7 shows how the maximum frequency depends on *k*_{2} steps at different but constant backgrounds *k*_{10}. The parallel lines which occur due to the unchanged maximum frequencies for a given background are indicative of “background compensation”.

The maximum frequency (see Figs 5d or 6d) is plotted as a function of , where is the *k*_{2} value during phase 2. The maximum frequency is determined for different stepwise *k*_{2} changes, i.e. for 1.0→2.0, 1.0→3.0, …, 1.0→9.0, up to 1.0→10.0. The steps occur at time t = 100 at different but constant *k*_{10} backgrounds (bgs). Calculations have been performed analogous to Figs 5 and 6. The *k*_{10} background values are 0, 1, 2, 4, up to 2048 (indicated in the figure). Other rate constants as in Fig 5. Initial concentrations: bgs 0–128, as in Fig 5; bg 256, *A*_{0} = 0.9866, *E*_{0} = 7.3508, *e*_{0} = 5.2447 × 10^{−2}, *I*_{1,0} = 5.8243, *I*_{2,0} = 2.5447 × 10^{4}; bg 512, *A*_{0} = 8.3872 × 10^{−4}, *E*_{0} = 4.8793, *e*_{0} = 3.9572 × 10^{−5}, *I*_{1,0} = 7.6544, *I*_{2,0} = 5.1046 × 10^{4}; bg 1024, *A*_{0} = 1.7657, *E*_{0} = 7.6866, *e*_{0} = 9.1430 × 10^{−2}, *I*_{1,0} = 4.2379, *I*_{2,0} = 1.0225 × 10^{5}; bg 2048, as in Fig 6.

#### Background compensation in a motif 8 (m8) based oscillatory homeostat.

To provide an additional example of a frequency-compensated negative feedback oscillator with background compensation we use a m8 outflow control motif [22] for the center feedback loop. The scheme of this oscillator is shown in Fig 8. In this motif, the controlled variable *A* inhibits the generation of the controller *E*. Controller *E* on its side inhibits the removal of *A*. The outer controllers, *I*_{1} and *I*_{2}, feed directly back to *A*. As for the m2 controller, oscillations in the central m8 oscillator are facilitated by removing *A* and *E* by zero-order processes. The rate equations are (‘pert’ stands for perturbation and ‘bg’ for background):
(11) (12) (13) (14) (15)

Solid arrows indicate chemical reactions, while dashed lines show activations (plus signs) and inhibitions (minus sign).

The inflow to *A* is divided into a stepwise perturbative component *k*_{1} (indicated in Eq 11 by ‘pert’ and outlined in red in Fig 8) and a background *k*_{3} (indicated in Eq 11 by ‘bg’ and outlined in blue in Fig 8). All other in- and outflows to and from *A* are compensatory fluxes.

As for the m2 oscillator, we can calculate the set-points for <*E*> by *I*_{1} and *I*_{2} by setting Eqs 14 and 15 to zero and assume that *I*_{1} and *I*_{2} are removed by zero-order reactions, which leads to:
(16)

Unfortunately, for this scheme the oscillatory *A*_{set} cannot be calculated analytically. The closest analytical expression we can obtain is by setting Eqs 12 and 13 to zero, eliminating the *k*_{11}⋅*e* term, and then calculating the time average of 1/(*k*_{10} + *A*):
(17)

While calculations easily verify the right-hand side of Eq 17, <*A*> needs to be calculated numerically.

Fig 9 shows that the oscillator described in Fig 8 shows frequency homeostasis at different but constant *k*_{3} backgrounds. In panel a the background is *k*_{3} = 0.0, while in panel b we have *k*_{3} = 1024.0. In both cases the maximum frequency for a *k*_{1} step of 1.0→100.0 is unchanged, indicating that the maximum frequency is background compensated.

In both panels a stepwise perturbation in *k*_{1} from 1.0 (phase 1) to 100.0 (phase 2) is applied (the step is indicated by the vertical arrows on top of the plots). In panel a, a constant background of *k*_{3} = 0.0 is applied (at both phases 1 and 2), while in panel b the background is 1024.0. Other rate constant values are: *k*_{4} = 1 × 10^{4}, *k*_{5} = *k*_{8} = *k*_{14} = *k*_{17} = *k*_{18} = 1 × 10^{−6}, *k*_{6} = 1 × 10^{3}, *k*_{7} = 50.0, *k*_{9} = 0.1, *k*_{11} = 1.0, *k*_{12} = 5.0, *k*_{13} = 50.00, *k*_{15} = 50.0, *k*_{16} = 1.0 and *k*_{g1} = *k*_{g2} = 1 × 10^{−2}. Initial concentrations (*k*_{3} = 0.0): *A*_{0} = 3.3568 × 10^{2}, *E*_{0} = 2.6209 × 10^{1}, *e*_{0} = 7.3942, *I*_{1,0} = 2.4840 × 10^{4}, *I*_{2,0} = 1.2768 × 10^{4}. Initial concentrations (*k*_{3} = 1024.0): *A*_{0} = 3.6188, *E*_{0} = 1.8696 × 10^{1}, *e*_{0} = 1.7115 × 10^{2}, *I*_{1,0} = 4.6869, *I*_{2,0} = 9.0420 × 10^{4}. Two python scripts, which in addition show the variations of *A*, *E*, *I*_{1}, and *I*_{2}, are included in S1 Data.

Fig 10 shows the maximum frequencies for different *k*_{1} steps and *k*_{3} backgrounds. There, different but constant *k*_{3} backgrounds are applied with values 0, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024. Variable *k*_{1} step perturbations are applied by starting with a 1.0 (phase 1) to 10.0 (phase 2) step and ending with a 1.0 (phase 1) to 500.0 (phase 2) step by successively increasing the *k*_{1} values in phase 2 by 10.0. The figure shows that the maximum frequencies, although dependent on the *k*_{1} step, remain unchanged with respect to an applied background *k*_{3}. This behavior results in parallel lines when maximum frequencies are plotted against .

The maximum frequency (see Fig 9) is plotted as a function of the sum of and background *k*_{3}, where is the *k*_{1} value in phase 2. In analogy with the calculations in Figs 5 and 6, the maximum frequency is determined for different stepwise *k*_{1} changes, i.e. for 1.0→10.0, 1.0→20.0, …, 1.0→30.0, up to 1.0→500.0, which occur at time t = 100 at different but constant *k*_{3} backgrounds (bgs). The *k*_{3} background values are 0, 2, 4, up to 1024 (indicated in the figure). Other rate constants as described in Fig 9. Initial concentrations: bg 0: as in Fig 9a; bg 2: *A*_{0} = 3.4008 × 10^{2}, *E*_{0} = 2.3906 × 10^{1}, *e*_{0} = 7.0224, *I*_{1,0} = 2.4739 × 10^{4}, *I*_{2,0} = 1.2869 × 10^{4}; bg 4: *A*_{0} = 3.4461 × 10^{2}, *E*_{0} = 2.1393 × 10^{1}, *e*_{0} = 6.6417, *I*_{1,0} = 2.4637 × 10^{4}, *I*_{2,0} = 1.2971 × 10^{4}; bg 8: *A*_{0} = 4.8073 × 10^{2}, *E*_{0} = 6.0165 × 10^{1}, *e*_{0} = 1.0914 × 10^{2}, *I*_{1,0} = 2.4401 × 10^{4}, *I*_{2,0} = 1.3207 × 10^{4}; bg 16: *A*_{0} = 4.3570, *E*_{0} = 1.9953 × 10^{1}, *e*_{0} = 1.6964 × 10^{2}, *I*_{1,0} = 2.4005 × 10^{4}, *I*_{2,0} = 1.3603 × 10^{4}; bg 32: *A*_{0} = 3.9151 × 10^{1}, *E*_{0} = 5.4663 × 10^{1}, *e*_{0} = 1.1875 × 10^{2}, *I*_{1,0} = 2.3201 × 10^{4}, *I*_{2,0} = 1.4407 × 10^{4}; bg 64: *A*_{0} = 3.0270 × 10^{2}, *E*_{0} = 4.1000 × 10^{1}, *e*_{0} = 1.0456 × 10^{1}, *I*_{1,0} = 2.1646 × 10^{4}, *I*_{2,0} = 1.5962 × 10^{4}; bg 128: *A*_{0} = 3.2021 × 10^{2}, *E*_{0} = 3.3534 × 10^{1}, *e*_{0} = 8.7470, *I*_{1,0} = 1.8443 × 10^{4}, *I*_{2,0} = 1.9165 × 10^{4}; bg 256: *A*_{0} = 6.4511 × 10^{1}, *E*_{0} = 6.8158 × 10^{1}, *e*_{0} = 9.3623 × 10^{1}, *I*_{1,0} = 1.2002 × 10^{4}, *I*_{2,0} = 2.5606 × 10^{4}; bg 512: *A*_{0} = 2.6297 × 10^{2}, *E*_{0} = 5.5584 × 10^{1}, *e*_{0} = 1.5294 × 10^{1}, *I*_{1,0} = 3.2525 × 10^{3}, *I*_{2,0} = 4.2375 × 10^{4}; bg 1024: as in Fig 9b. Panel a shows an overview of the maximum frequencies up to background 64, while panel b shows a blown-up part indicated in panel a. Panel c shows the maximum frequencies for backgrounds in the range between 64 to 1024.

#### Background compensation in non-oscillatory homeostats.

In this section we look at background compensation in non-oscillatory homeostats where *E* is controlled by *I*_{1} and *I*_{2} via coherent feedback. We show two examples: in the first one the controller’s response after a step perturbation is significantly damped, while in the other example the response shows a larger train of (damped) oscillations. In both cases the response profiles of the controlled variables *A* and *E* are preserved and independent of the background.

For the first example we use the oscillator scheme from Fig 4a. To go over to a non-oscillatory mode, we change the kinetics for all *A*-removing reactions from zero-order to first-order kinetics with respect to *A*. The rate equation of *A* becomes (compare with Eq 3):
(18)
while the rate equations for the other components (Eqs 4–7) remain the same.

Fig 11 gives an overview of the results. In panel a the maximum excursions Δ*A* after the step (see inset) are plotted for different backgrounds *k*_{10} as a function of the sum of the phase 2 *k*_{2} value and *k*_{10}. For each background, the nine *k*_{2} steps 1→2, 1→3, ….,1→9, and 1→10 are applied and Δ*A* is determined. For a certain defined *k*_{2}-step Δ*A* is independent of the background (*k*_{10}), which leads to the parallel lines in panel a. Panel b shows the situation for a *k*_{2} 1→ 10 step when background *k*_{10} = 0. In panel c the same step is applied, but the background has been increased to *k*_{10} = 10. Comparing Fig 11b and 11c shows that profiles in both *A* and *E* are the same with *A*_{set} = 2.0 and *E*_{set} = 100.

Panel a: Each colored curve shows the values of Δ*A* for the nine *k*_{2} steps: 1→2, 1→3, …,1→10 at *k*_{10} background levels: 0, 2, 4, …,8, 10. Inset shows how Δ*A* is defined. is the value of *k*_{2} during phase 2. Panels in b: Time profiles of *A*, *E* (left panel) and *I*_{1}, *I*_{2} (right panel) for a 1→10 *k*_{2} step at background *k*_{10} = 0.0. The change in *k*_{2} is applied at time t = 500, which is indicated by the vertical arrows. Panels in c are similar to the panels in b with the difference that background *k*_{10} is 10.0. Other rate constants: *k*_{3} = 5 × 10^{3}, *k*_{4} = 1.0, *k*_{5} = 0.5, *k*_{6} = 2.0, *k*_{7} = 1 × 10^{−5}, *k*_{9} = 2.0, *k*_{11} = 0.1, *k*_{12} = 10.0, *k*_{13} = *k*_{16} = 1 × 10^{−4}, *k*_{14} = 1.0, *k*_{15} = 0.01, *k*_{g} = 0.01, *k*_{g3} = 1 × 10^{−3}. Initial concentrations for panel b: *A*_{0} = 2.0, *E*_{0} = 100.0, *e*_{0} = 1.0, *I*_{1,0} = 2.5684 × 10^{3}, *I*_{2,0} = 2.8492 × 10^{4}. Initial concentrations for panel c: *A*_{0} = 2.0, *E*_{0} = 100.0, *e*_{0} = 1.0, *I*_{1,0} = 1.5734 × 10^{3}, *I*_{2,0} = 2.8592 × 10^{4}. For python scripts showing the results of panels b and c, please see supporting information S1 Data.

Fig 13 shows another example of coherent feedback. Here we have two inflow controllers *E*_{1} and *E*_{2}, but only *E*_{2} is connected to *A* via *I*_{1} and *I*_{2} through a coherent feedback. The reason why we looked at two *E*-controllers was to see whether *E*_{2} alone, i.e. without the help of *I*_{1} and *I*_{2}, was able to compensate backgrounds. This, however, turned out not to be the case and *I*_{1} and *I*_{2} were included to control *E*_{2}.

Compound *a* is a precursor to *A*, which has been found to promote limit-cycle oscillations, once the system is fully oscillatory [11]. Stepwise perturbations in *k*_{2} (outlined in red) are applied at constant backgrounds levels *k*_{10} (outlined in blue).

The rate equations are:
(19) (20) (21) (22) (23) (24) *E*_{1} and *E*_{2} provide two set-points for *A*: one, , by setting Eq 21 to zero and solving for the steady state level of *A* under zero-order conditions, and the other, , by doing the same for Eq 22. This gives:
(25)
and
(26)

In the calculations we have set and . Since the *E*_{2} inflow controller has a lower set-point in comparison with *E*_{1}, *E*_{2} will take over the control of *A* [22], while *E*_{1} will be inactive and allow a constant inflow to *A* via *a*.

Fig 14 shows that in this system a 1→10 perturbation in *k*_{2} induces a train of damped oscillations with background (*k*_{10}) independent concentration profiles. In panels a and b Δ*A* (for definition see inset in Fig 11) is shown as a function of increasing *k*_{2} steps at different but constant *k*_{10} backgrounds. Panel c shows the time profile in *A* for a 1→10 *k*_{2} step with a *k*_{10} background of 0. In panel d the same step is applied but now with a background of *k*_{10} = 1024. One clearly sees the conserved background-independent transition profiles in *A*.

Panels a and b: Δ*A* as a function of for different but constant backgrounds. is the *k*_{2} value in phase 2. The background values (*k*_{10}) are indicated above the colored curves. Panels c and d: Concentration profiles in *A* when a *k*_{2} 1→10 step is applied at respective backgrounds of *k*_{10} = 0.0 and 1024.0. Other rate constants: *k*_{3} = 2.5 × 10^{3}, *k*_{4} = 1.0, *k*_{5} = 0.1, *k*_{6} = 2.1, *k*_{7} = 1 × 10^{−5}, *k*_{9} = 0.5, *k*_{11} = 0.5, *k*_{12} = 200.0, *k*_{13} = 100, *k*_{14} = *k*_{17} = *k*_{20} = 1 × 10^{−5}, *k*_{15} = 1 × 10^{3}, *k*_{16} = 10.0, *k*_{18} = 1.0, *k*_{19} = 99.99, *k*_{g} = *k*_{g3} = 0.1. Initial concentrations for c: *A*_{0} = 2.0, *E*_{1,0} = 2.0 × 10^{−4}, *E*_{2,0} = 100.0, *a*_{0} = 4.99 × 10^{3}, *I*_{1,0} = 2.5684 × 10^{3}, *I*_{2,0} = 2.8492 × 10^{4}. Initial concentrations for d: *A*_{0} = 2.0, *E*_{1,0} = 2.0 × 10^{−4}, *E*_{2,0} = 100.0, *a*_{0} = 4.99 × 10^{3}, *I*_{1,0} = 3.5195 × 10^{3}, *I*_{2,0} = 2.0888 × 10^{3}. Supporting information S1 Data includes the python scripts showing the results of panels c and d.

#### Frequency homeostasis without background compensation.

Fig 15a shows an oscillator scheme which we previously described in relation to robust frequency homeostasis [11]. We wondered whether frequency homeostasis without coherent feedback would also lead to background compensation, but realized that this is not the case. In Fig 15a *I*_{1}, *I*_{2}, and *E* do not feed back coherently to *A*, but (incoherently) to *a*, which is a precursor of *A*.

Panel a: reaction scheme. Panel b: Flow scheme. For rate equations, see main text.

The rate equations are (’pert’ stands for perturbation and ‘bg’ for background): (27) (28) (29) (30) (31)

Fig 16 shows an example of oscillations for the scheme in Fig 15 with background *k*_{10} = 0 and a step perturbation in *k*_{2} from 1 (phase 1) to 10 (phase 2). It may be noted that in this case oscillations occur although the removal reactions of *a* and *A* are first-order with respect to *a* and *A*, indicating that first-order processes are only a ‘weak’ condition to abolish oscillatory behavior, as has been indicated in the above section ‘Background compensation in non-oscillatory homeostats’.

Background *k*_{10} = 0.0. A *k*_{2} step 1→10 occurs at time t = 500 indicated by the vertical arrows. Panel a: Concentration of *A* and average <*A*> as a function of time. Panel b: Concentration of *E* and average <*E*> as a function of time. Panel c: Concentrations of *I*_{1} and *I*_{2} as a function of time. Panel d: Frequency as a function of time. Other rate constants: *k*_{3} = 1 × 10^{6}, *k*_{4} = 1.0, *k*_{5} = 1 × 10^{−6}, *k*_{6} = 2.0, *k*_{7} = *k*_{13} = *k*_{16} = 1 × 10^{−6}, *k*_{9} = 2.0, *k*_{11} = 5.0, *k*_{12} = 100.0, *k*_{14} = 99.99, *k*_{15} = 5.0, *k*_{g} = 1 × 10^{−3}, and *k*_{g3} = 100.0. Initial concentrations: *A*_{0} = 5.6920 × 10^{−3}, *E*_{0} = 6.1163, *a*_{0} = 3.6221 × 10^{−3}, *I*_{1,0} = 4.4051 × 10^{4}, *I*_{2,0} = 2.7566 × 10^{2}. See S1 Data for python scripts.

Fig 16 clearly shows the occurrence of frequency homeostasis. However, when the oscillator is tested for different but constant *k*_{10} backgrounds with changed *k*_{2} steps the maximum frequency decreases with increasing backgrounds. Fig 17a shows the decrease of the maximum frequency and loss of robust background compensation at four different *k*_{10} backgrounds when *k*_{2} steps are applied from 1→2 up to 1→10. When using a logarithmic ordinate (Fig 17b) lines appear more or less parallel, which may give the illusion that the system responds in a background compensated way.

Calculations were performed with rate constants as described in Fig 16. Panel a show results with linear scaling of axes, while panel b shows the same data set as double-logarithmic plots. Initial concentrations: bg (*k*_{10}) = 0.0: see legend of Fig 16; bg (*k*_{10}) = 1.0: *A*_{0} = 2.5946 × 10^{−3}, *E*_{0} = 25.4830, *a*_{0} = 2.6844 × 10^{−3}, *I*_{1,0} = 3.0980 × 10^{4}, *I*_{2,0} = 1.3296 × 10^{4}; bg (*k*_{10}) = 2.0: *A*_{0} = 5.0041 × 10^{−3}, *E*_{0} = 15.8930, *a*_{0} = 7.8102 × 10^{−3}, *I*_{1,0} = 2.2995 × 10^{4}, *I*_{2,0} = 2.1181 × 10^{4}; bg (*k*_{10}) = 4.0: *A*_{0} = 4.7328 × 10^{−3}, *E*_{0} = 21.6050, *a*_{0} = 1.2043 × 10^{−2}, *I*_{1,0} = 1.3516 × 10^{4}, *I*_{2,0} = 3.0610 × 10^{4}.

### Is retinal light adaptation background compensated?

Based on the comment in Ref [2] that the parallel lines in Fig 2 indicate the same response at different backgrounds and involve a form of compensation mechanism, we became interested to look into the conditions how background compensation could occur. This requires of course how the term ‘background compensation’ is defined. In our definition, background compensation implies the presence of a compensatory mechanism, which enables the same response of a controlled system, *independent* of different but constant backgrounds, which are applied in parallel to a perturbation. The result shown in Fig 2, however, does not meet this definition. As indicated by the vertical dashed bar in the figure, an increase of the background from the red to the blue curve, and from the blue to the green curve, a reduction in the average maximum frequency is observed when a test spot luminance of 9 × 10^{−2} cd/m^{2} is applied. In fact, the adaptation behavior shown in Fig 1 can show an analogous behavior as in Fig 2.

To see this we use a model described by Dowling [3], where the response amplitude *V* of retinal cells with respect to a light perturbation *I* is given by the Hill equation [35]
(32)

The cooperativity *α* is 1.0 for photoreceptor cells (changing the Hill equation into a Michaelis-Menten equation), but found to be 0.7 to 0.8 for horizontal cells, 1.2–1.4 for bipolar and sustained ganglion cells, and about 3.4 for transient ganglion cells (for an overview see [3]).

We consider here the response kinetics of rods and cones, i.e. *α* = 1 with
(33)

As pointed out by Naka and Rushton [36], in the presence of a background *I*_{0} the response *V*_{1} upon a perturbation *I*_{1} of a single pigment system will follow Eq 33, but with an increase of *σ* to *σ*_{1} = *σ* + *I*_{0} and a scaling of *V*_{max} by a factor of *σ*/(*σ* + *I*_{0}). This can be shown as follows:

In the presence of a constant background *I*_{0} Eq 33 gives
(34)

If a light perturbation *I*_{1} is applied in addition to background *I*_{0} the total response amplitude is
(35)

Subtracting Eq 34 from Eq 35 gives (36)

Fig 18 shows Eq 33 with six different *σ* values which mimick six different background levels. For the sake of simplicity we have set *V*_{max} = 1. In panel a both axes are linear, while in panel b the ordinate is logarithmic and the abscissa is linear. In panel c the ordinate is linear and the abscissa is logarithmic. Finally, in panel d both axes are logarithmic.

The colored lines in the panels show Eq 33 with *σ* values ranging over six orders of magnitudes from *σ* = 1 × 10^{−4} up to *σ* = 10. For simplicity, *V*_{max} = 1. Panel a: Both axes are linear. Panel b: Ordinate is logarithmic and abscissa is linear. Panel c: Ordinate is linear and abscissa is logarithmic. Panel d: Both axes are logarithmic. The dashed vertical lines indicate an perturbation intensity of *I* = 1. The colored intersection points with the vertical dashed lines show the responses of *V* for the different backgrounds with the same color.

Fig 18c is analogous to the results in Fig 2 when for a given perturbation (indicated by the vertical dashed lines) an increased background or an increased *σ* leads to a reduction in the averaged maximum frequency. No background compensation, as indicated by Kandel et al. in Ref [2] appears necessary.

When studying the photoadaptation of gecko photoreceptors, Kleinschmidt and Dowling [37] showed log-log relationships analogous to Fig 18d. Dowling interpreted the parallel lines as a result of a second adaptive mechanism in the receptor, which shifts the photoreceptor intensity-response curves along the intensity axis, thus extending the range over which the receptor responds (see page 222 in Ref [3], bottom section).

Clearly, as Fig 18 shows, the parallel lines in panels c or d neither require the need for a background compensation mechanism nor additional adaptive mechanisms. While adaptation mechanisms compensating for a background cannot be excluded, the observation of parallel lines in semi-logarithmic or double-logarithmic plots appear not sufficient to indicate additional background compensation mechanisms besides of the negative feedbacks which lead to the responses in Fig 1 [5].

## Conclusion and outlook

We have shown how robust background compensation in oscillatory and non-oscillatory homeostatic controllers can be realized. The needed feedback condition has been termed ‘coherent feedback’ in analogy to a corresponding concept applied in quantum control theory. Although the property of robust background compensation appears interesting, we are presently not aware of any biological or biochemical example that shows or applies this property. Background compensation may become of interest in synthetic biology to design cellular responses, which by some reason are needed to become background independent. Concerning the case of retinal light adaptation, parallel lines in semi-logarithmic or double-logarithmic plots do not necessarily imply the presence of background compensating mechanisms as defined in this paper.

## Supporting information

### S1 Data. Documentation.

A zip-file with python scripts describing the results for Figs 5, 6, 9a, 9b, 12b, 12c, 14c, 14d, and 16.

https://doi.org/10.1371/journal.pone.0287083.s001

(ZIP)

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