Coherent feedback leads to robust background compensation in oscillatory and non-oscillatory homeostats

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 ΔA 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 ΔA, 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 I1 and I2) have to be employed to keep the manipulated variable E at a defined set-point, and (ii), I1 and I2 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.


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][3][4][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.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].
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-  [7].For a theoretical description of this behavior see Ref [5] and references therein.https://doi.org/10.1371/journal.pone.0287083.g001oscillatory 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.
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][14][15][16], and has been indicated to occur also in biological systems [17][18][19][20][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).
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][3][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][23][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][28][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.

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 (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.controller species E.
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 .
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.
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  [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]  The rate equations are: The set-point of <A> (A set ) by controller E can be calculated from the steady state condition of the time averages: |ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl } � 1 ðzeroÀ orderÞ 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: The I 1 outflow controller becomes active whenever <E> is higher than E I 1 set .
The set-point for the I 2 inflow controller is determined by the steady state condition: The I 2 controller becomes active whenever <E> is lower than E I 2 set .It should be noted that the values of the inflow/outflow set-points E I 1 set and E I 2 set need to follow certain rules to guarantee that inflow and outflow controllers cooperate.In this case E I 2 set should be lower than E I 1 set , 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.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): 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: 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): While calculations easily verify the right-hand side of Eq 17, <A> needs to be calculated numerically.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 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 firstorder kinetics with respect to A. The rate equation of A becomes (compare with Eq 3): while the rate equations for the other components (Eqs 4-7) remain the same.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 .
The rate equations are:  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).https://doi.org/10.1371/journal.pone.0287083.g013 E 1 and E 2 provide two set-points for A: one, A E 1 set , by setting Eq 21 to zero and solving for the steady state level of A under zero-order conditions, and the other, A E 2 set , by doing the same for Eq 22.This gives: and In the calculations we have set A E 1 set ¼ 2:1 and A E 2 set ¼ 2:0.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 ΔA (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.
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.
The rate equations are ('pert' stands for perturbation and 'bg' for background):

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 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] V 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 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 If a light perturbation I 1 is applied in addition to background I 0 the total response amplitude is Subtracting Eq 34 from Eq 35 gives 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.

Fig 1 .
Fig 1.Light adaptation in a macaque monkey's rod cell.10 ms light flashes were applied to different light background intensities.Background intensities (in photons μm −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.

Fig 2 .
Fig 2. Light adaptation of an on-center ganglion cell in the cat retina (redrawn from Fig 8, Ref [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.https://doi.org/10.1371/journal.pone.0287083.g002

Fig 3 .
Fig 3. Integral control and the definition of background.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. https://doi.org/10.1371/journal.pone.0287083.g003

Fig 4 .
Fig 4. Frequency-compensated oscillator with background compensation by coherent feedback.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 .
Fig 4. Frequency-compensated oscillator with background compensation by coherent feedback.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 .https://doi.org/10.1371/journal.pone.0287083.g004 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

Fig 8 .
Fig 8. Frequency-compensated oscillator with background compensation by coherent feedback based on derepression motif m8 in the inner A-e-E-A negative feedback.Solid arrows indicate chemical reactions, while dashed lines show activations (plus signs) and inhibitions (minus sign).https://doi.org/10.1371/journal.pone.0287083.g008

Fig 12 .
Fig 12. Same scheme as Fig 4a, but to facilitate a non-oscillatory homeostat all A-removing reactions are changed to first-order kinetics with respect to A. https://doi.org/10.1371/journal.pone.0287083.g012

Fig 16
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

Fig 18 shows
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.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 Fig18shows, 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].

Fig 18 .
Fig 18. Photoadaptation behaviors in rods and cones described by the Michaelis-Menten equation (Eq 33).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.https://doi.org/10.1371/journal.pone.0287083.g018