# Feedback Regulation and Its Efficiency in Biochemical Networks

## Abstract

Intracellular biochemical networks fluctuate dynamically due to various internal and external sources of fluctuation. Dissecting the fluctuation into biologically relevant components is important for understanding how a cell controls and harnesses noise and how information is transferred over apparently noisy intracellular networks. While substantial theoretical and experimental advancement on the decomposition of fluctuation was achieved for feedforward networks without any loop, we still lack a theoretical basis that can consistently extend such advancement to feedback networks. The main obstacle that hampers is the circulative propagation of fluctuation by feedback loops. In order to define the relevant quantity for the impact of feedback loops for fluctuation, disentanglement of the causally interlocked influences between the components is required. In addition, we also lack an approach that enables us to infer non-perturbatively the influence of the feedback to fluctuation in the same way as the dual reporter system does in the feedforward networks. In this work, we address these problems by extending the work on the fluctuation decomposition and the dual reporter system. For a single-loop feedback network with two components, we define feedback loop gain as the feedback efficiency that is consistent with the fluctuation decomposition for feedforward networks. Then, we clarify the relation of the feedback efficiency with the fluctuation propagation in an open-looped FF network. Finally, by extending the dual reporter system, we propose a conjugate feedback and feedforward system for estimating the feedback efficiency non-perturbatively only from the statistics of the system.

### Keywords

Fluctuation Linear noise approximation Noise decomposition Information flow Dual reporter system Variance decomposition## 1 Introduction

### 1.1 Dissecting Fluctuation in Biochemical Networks

Biochemical molecules in a cell fluctuate dynamically because of the stochastic nature of intracellular reactions, fluctuation of the environment, and the spontaneous dynamics of intracellular networks [18, 33, 41]. Some part of the fluctuation is noise that impairs or disturbs robust operation of the intracellular networks. The other part, however, conveys information on complex dynamics of various factors inside and outside of the cell [20, 32]. Dissecting fluctuation into distinct components with different biological roles and meanings is crucial for understanding the mechanisms how a cell controls and harnesses the noise and how information is transferred over apparently noisy intracellular networks [5, 9].

The intracellular fluctuation is generated from the intracellular reactions or comes from the environment, and then propagates within the intracellular networks. The fluctuation of individual molecular species within the networks is therefore a consequence of the propagated fluctuation from the different sources. Decomposition of the fluctuation into the contributions from the sources is an indispensable step for understanding their biological roles and relevance. When fluctuation propagates from one component to another unidirectionally without circulation, the fluctuation of the downstream can be decomposed into two contributions. One is the intrinsic part that originates within the pathway between the components. The other is the extrinsic part that propagates from the upstream component. Such decomposition can easily be extended for the network with cascading or branching structures in which no feedback exists. This fact drove the intensive anatomical analysis of the intracellular fluctuation.

### 1.2 Decomposition of Sources of Fluctuation

In order to dissect fluctuation into different components, two major strategies have been developed. One is to use the dependency of each component on different kinetic parameters of the network. By employing theoretical predictions on such dependency, we can estimate the relative contributions of different components from single-cell experiments with perturbations. Possible decompositions of the fluctuation were investigated theoretically for various networks such as single gene expression [3, 27, 28], signal transduction pathways [42, 48], and cascading reactions [46]. Some of them were experimentally tested [3, 27, 29].

The other strategy is the dual reporter system in which we simultaneously measure a target molecule with its replica obtained by synthetically duplicating the target. From the statistics of the target and the replica, i.e, mean, variance, and covariance, we can discriminate the intrinsic and extrinsic contributions to the fluctuation because the former is independent between the target and the replica whereas the latter is common to them. The idea of this strategy was proposed and developed in [28, 44], and verified experimentally for different organisms [11, 24, 34]. Its applicability and generality were further extended [4, 6, 16, 17, 36]. Now these strategies play the fundamental role for designing single-cell experiments and for deriving information on the anatomy of fluctuation from experimental observations [8, 15, 30, 35, 38, 45].

### 1.3 Feedback Regulation and Its Efficiency

Even with the theoretical and the experimental advancement in decomposing fluctuation, most of works focused on the feedforward (FF) networks in which no feedback and circulation exists. As commonly known in the control theory [7, 40], feedback (FB) loops substantially affect fluctuation of a network by either suppressing or amplifying it. Actually, the suppression of fluctuation in a single gene expression with a FB loop was experimentally tested in [2] earlier than the decomposition of fluctuation. While the qualitative and the quantitative impacts of the FB loops were investigated both theoretically and experimentally since then [1, 22, 25, 26, 36, 43, 47], we still lack a theoretical basis that can consistently integrate such knowledge with that on the fluctuation decomposition developed for FF networks.

The main problem that hampers the integration is the circulation of fluctuation in FB networks. Because fluctuation generated at a molecular component propagates the network back to itself, we need to disentangle the causally interlocked influence between the components in order to define the relevant quantity for the impact of the FB loops. From the experimental point of view, in addition, quantification of the impact of FB loops by perturbative experiments is not perfectly reliable because artificial blocking of the FB loops inevitably accompanies the change not only in fluctuation but also in the average level of the molecular components involved in the loops. It is quite demanding and almost impossible for most cases to inhibit the loops by keeping the average level unchanged. We still lack an approach that enables us to infer the influence of the FB non-perturbatively as the dual reporter system does for FF networks.

### 1.4 Outline of this Work

In this work, we address these problems by extending the work on the fluctuation decomposition [28, 42] and the dual reporter system [11, 44]. By using a single-loop FB network with two components and its linear noise approximation (LNA) [10, 19, 47], we firstly provide a definition of the FB loop gain as FB efficiency that is consistent with the fluctuation decomposition in [28, 42]. This extension relies on an interpretation of the fluctuation decomposition as source-by-source decomposition that is different from the variance decomposition proposed in the previous work [4, 6, 16, 17]. Then, we clarify the relation of the FB efficiency with the fluctuation propagation in a corresponding open-looped FF network. Finally, by extending the dual reporter system, we propose a conjugate FB and FF system for estimating the feedback efficiency only from the statistics of the system non-perturbatively. We also give a fluctuation relation among the statistics that may be used to check the validity of the LNA for a given network.

The rest of this paper is organized as follows. In Sect. 2, we review the decomposition of fluctuation for a simple FF system derived in [28, 42] by using the LNA. In Sect. 3, we extend the result shown in Sect. 2 to a FB network by deriving a source-by source decomposition of the fluctuation with feedback. Using this source decomposition, we define the FB loop gains that quantify the impacts of the FB to the fluctuation. In Sect. 4, we give a quantitative relation of the loop gains in the FB network with the fluctuation propagation in a corresponding open-looped FF network. In Sect. 5, we propose a conjugate FF and FB network as a natural extension of the dual reporter system. We clarify that the loop gains can be estimated only from the statistics, i.e., mean, variances, and covariances, of the conjugate network. We also show that a fluctuation relation holds among the statistics, which generalizes the relation used in the dual reporter system. In Sect. 6, we discuss a link of the conjugate network with the casual conditioning and the directed information, and finally give future directions of our work.

## 2 Fluctuation Decomposition and Propagation in a Small Biochemical Network

In this section, we summarize the result for the decomposition of fluctuation obtained in [28, 42] by using the LNA, and also its relation with the dual reporter system employed in [11, 24, 34] to quantify the intrinsic and extrinsic contributions from the experimental measurements.

### 2.1 Stochastic Chemical Reaction and Its Linear Noise Approximation

*N*different molecular species and

*M*different reactions. We assume that the stochastic dynamics of the network is modeled by the following chemical master equation:

*t*, and \(a_{k}(\varvec{n}) \in \mathbb {R}_{\ge 0}^{M}\) and \(\varvec{s}_{k}\in \mathbb {N}^{N}\) are the propensity function and the stoichiometric vector of the

*k*th reaction, respectively [12, 14, 19]. The propensity function, \(a_{k}\), characterizes the probability of occurrence of the

*k*th reaction when the number of the molecular species is \(\varvec{n}\), and the stoichiometric vector defines the change in the number of the molecular species when the

*k*th reaction occurs.

*n*, i.e., the average of

*n*, as

^{1}Even though the propensity function \(\varvec{a}(\varvec{n})\) is not affine, Eq. (3) (or Eq. (6)) can produce a good approximation of the fluctuation, provided that the fixed point, \(\bar{\varvec{n}}\), is a good approximation of the average, \(\left<\varvec{n} \right>\), and that the local dynamics around the fixed point is approximated enough by its linearization. In addition, compared with other approximations, the LNA enables us to obtain an analytic representation of the fluctuation because Eq. (3) is a linear algebraic equation with respect to \(\varvec{\Sigma }\). Owing to this property, the LNA and its variations played the crucial role to reveal the analytic representation of the fluctuation decomposition in biochemical reaction networks [21, 28, 47]. As in these previous works, we employ the LNA to obtain an analytic representation for the feedback efficiency.

### 2.2 Decomposition of Fluctuation

*x*,

*x*regulates

*y*unidirectionally. Then, for a fixed point \((\bar{x},\bar{y})\) of Eq. (2), that satisfies

*K*and

*D*in Eq. (3) becomes

*x*and

*y*, respectively. \(d_{x}\) and \(d_{y}\) are the minus of the diagonal terms of

*K*and represent the effective degradation rates of

*x*and

*y*. \(k_{yx}\) is the off-diagonal term of

*K*that represents the interaction from

*x*to

*y*. \(H_{ij}\) is the susceptibility of the

*i*th component to the perturbation of the

*j*th one. Except this section, we mainly use

*d*s and

*k*s as the representation of the parameters rather than

*H*s and \(\tau \)s introduced in [28].

^{2}By solving Eq. (3) analytically, the following fluctuation-dissipation relation was derived in [28]:

^{3}and describes how the fluctuation generates and propagates within the network. (

*I*) is the intrinsic fluctuation of

*x*that originates from the stochastic birth and death of

*x*. \(1/\bar{x}\) reflects the Poissonian nature of the stochastic birth and death, and \(1/H_{xx}\) is the effect of auto-regulatory FB. Similarly, (

*II*) is the intrinsic fluctuation of

*y*that originates from the stochastic birth and death of

*y*. (

*III*), on the other hand, accounts for the extrinsic contributions to the fluctuation of

*y*due to the fluctuation of

*x*. The term (

*III*) is further decomposed into (

*i*), (

*ii*), and (

*iii*). (

*i*) is the fluctuation of

*x*, and therefore, identical to (

*I*). (

*ii*) and (

*iii*) determine the efficiency of the propagation of the fluctuation from

*x*to

*y*, which are the time-averaging and sensitivity of the pathway from

*x*to

*y*, respectively. This representation captures the important difference of the intrinsic and the extrinsic fluctation such that the intrinsic one, the term (

*II*), can be always reduced by increasing the average of \(\bar{y}\) whereas the extrinsic one, the term (

*III*), cannot.

^{4}

*x*to the fluctuation of

*y*is described by the CV of

*x*as the term (

*i*). Because the fluctuation of

*x*and

*y*depend mutually if we have a FB between

*x*and

*y*, we need to characterize the fluctuation of

*y*without directly using the fluctuation of

*x*. To this end, we adopt the variances and covariances as the measure of the fluctuation and use the following decomposition of the fluctuation for the FF network:

*I*), (

*II*), and (

*III*) in Eq. (11) correspond to those in Eq. (10). The interpretation of the terms within (

*I*), (

*II*), and (

*III*) is, however, different. In Eq. (11), the terms (

*i*) are interpreted as the fluctuation purely generated by the birth and death reactions of

*x*and

*y*by neglecting any contribution of the auto-FBs. Because the simple birth and death of a molecular species without any regulation follow the Poissonian statistics, the intensity of the fluctuation is equal to the means of the species. Thus, \(\bar{x}\approx \left<x\right>\) and \(\bar{y}\approx \left<y\right>\) in the terms (

*i*) represent the generation of the fluctuation by the birth and death of

*x*and

*y*, respectively. The fluctuation generated is then amplified or suppressed by the auto-regulatory FBs. \(G_{xx}\) and \(G_{yy}\) in the terms (

*ii*) account for this influence, and are denoted as auto-FB gains in this work. Finally, \(G_{yx}\) in the term (

*iii*) quantifies the efficiency of the propagation of the fluctuation from

*x*to

*y*. We denote \(G_{yx}\) as path gain from

*x*to

*y*. If we use the notation in Eq. (10), \(G_{yx}\) is described as

*y*into (

*II*) and (

*III*) is consistent between Eqs. (10) and (11) while the further decompositions within (

*II*) and (

*III*) are different.

*II*) and (

*III*) as in Eq. (11) by solving the Lyapunov equation, Eq. (3), with \(\bar{a}_{x}\) and \(\bar{a}_{y}\) being replaced with 0, respectively. Because \(\bar{a}_{x}\) and \(\bar{a}_{y}\) represent the noise sources associated with each variable, we call our decomposition as source-by-source decomposition or more simply as source decomposition [47].

^{5}At the same time, it was also revealed in [4, 6, 16, 17] that the decomposition into (

*II*) and (

*III*) is identical to the variance decomposition formula in statistics as

*x*(

*t*), and \(\mathbb {E}\) and \(\mathbb {V}\) are the expectation and the variance, respectively.

^{6}While the source decomposition and the variance decomposition are different in general, they coincide in this special case of the simple FF network. Because the variance decomposition formula holds generally even for a non-stationary situation without using the linearization by the Lyapunov equation, the variance decomposition, Eq. (14), has wider applicability than the source decomposition, Eqs. (10) and (11), as a decomposition formula.

^{7}Nonetheless, the source decomposition has advantages as a decomposition formula when we extend the decompositions to FB networks as discussed in the next section.

### 2.3 Dual Reporter System

The decomposition, Eqs. (10) or (11), guides us how to evaluate the intrinsic and the extrinsic fluctuation in *y* experimentally. When we can externally control the mean of *y* without affecting the term (*III*), we can estimate the relative contributions of (*II*) and (*III*) as the intrinsic and the extrinsic fluctuation by plotting \(\sigma _{y}^{2}\) as a function of the mean of *y*. When *x* and *y* correspond to mRNA and protein in the single gene expression, the translation rate works as such a control parameter [23]. This approach was intensively employed to estimate the efficiency of the fluctuation propagation in various intracellular networks [3, 27, 29].

^{8}In the dual reporter system, a replica of

*y*is attached to the downstream of

*x*as in Fig. 1b where \(y'\) denotes the molecular species of the replica. The replica \(y'\) must have the same kinetics as

*y*, and must be measured simultaneously with

*y*. If

*y*is a protein whose expression is regulated by another protein

*x*as in Fig. 1c, then \(y'\) can be synthetically constructed by duplicating the gene of

*y*and attaching fluorescent probes with different colors to

*y*and \(y'\) as in Fig. 1d [11]. Under the LNA, the covariance between

*y*and \(y'\) can be described as

*y*can be estimated as

*III*) is \(\mathbb {V}[\mathbb {E}[y(t)|\mathcal {X}(t)]]\). As shown in [4], this correspondence between \(\mathbb {V}[\mathbb {E}[y(t)|\mathcal {X}(t)]]\) and \(\sigma _{y,y'}\) for the dual reporter system is more generally derived as

*y*(

*t*) and \(y'(t)\) are conditionally independent given the history, \(\mathcal {X}(t)\). As discussed in [4], the term (

*II*) is also directly estimated by calculating \(\mathbb {E}[d_{\bot }^{2}]:\,=\frac{1}{2}\mathbb {E}[(y(t)-y'(t))^{2}]\) because

## 3 Feedback Loop Gain in a Small Biochemical Network

*K*and

*D*in the Lyapunov equation (Eq. (3)) for the FB network can be described as

*y*back to

*x*as

*x*and

*y*as

*y*to

*x*does not exist, i.e., \(k_{xy}=0\), then \(L_{x}=L_{y}=G_{xy}=0\) and Eq. (21) is reduced to Eq. (11). Thus, \(L_{x}\) and \(L_{y}\) account for the effect of the FB. \(L_{x}\) and \(L_{y}\) are denoted as FB loop gains in this work. Eq. (21) clearly demonstrates that the representation in Eq. (10) does not work with the FB because \(\sigma _{y}^{2}\) cannot be described with \(\sigma _{x}^{2}\) any longer. In contrast, we can interpret the terms in Eq. (21) consistently with those in Eq. (11) because all the terms, (

*I*), (

*II*) and (

*III*), are unchanged in Eq. (21). The new term, (

*IV*), in the expression for \(\sigma _{x}^{2}\) appears to account for the propagation of the fluctuation generated by the birth and death events of

*y*back to

*x*.

^{9}

*x*and

*y*. First, the fluctuation is suppressed when \(L_{x}\) and \(L_{y}\) are negative whereas it is amplified when they are positive. When \(d_{x}\) and \(d_{y}\) are positive,

^{10}the sign of \(L_{x}\) and \(L_{y}\) are determined only by the sign of \(k_{xy}k_{yx}\). Thus, the FB loop is negative when

*x*regulates

*y*positively and

*y*does

*x*negatively or vise versa. This is consistent with the normal definition of the sign of a FB loop. Second, the efficiency of the FB depends on the source of the fluctuation. For example, when \(L_{x}\ll L_{y}\), e.g., the time-scale of

*x*is much faster than

*y*as \(d_{x} \gg d_{y}\), then Eq. (21) can be approximated as

*I*) that is the part of fluctuation of

*x*whose origin is the birth and death events of

*x*itself. This result reflects the fact that the slow FB from

*x*to itself via

*y*cannot affect the fast component of the fluctuation of

*x*.

^{11}Finally, when \(L_{x}\) and \(L_{y}\) satisfy \(1-L_{x}-L_{y}=1-k_{xy}k_{yx}/d_{x}d_{y}=0\), the fluctuation of both

*x*and

*y*diverges due to the FB. Since this condition means that the determinant of

*K*becomes 0 and

*K*is the Jacobian matrix of Eq. (2), the fluctuation of

*x*and

*y*diverges due to the destabilization of the fixed point, \(\bar{\varvec{n}}\), by the FB. The Lyapunov equation is no longer valid around an unstable fixed point, and thereby, the source decomposition cannot be applied to such a situation.

^{12}

*y*to

*x*as \(k_{xy}=0\).

^{13}While the variance decomposition is defined quite generally, it may not reflect the way how the fluctuation of the target is determined in the system. This fact is demonstrated clearly by noting that \(\mathbb {E}[\mathbb {V}[y(t)|\mathcal {X}(t)]]\) is independent of \(k_{yx}\). Even without feedback from

*x*to

*y*as \(k_{yx}=0\), the variance of

*y*(

*t*) can be decomposed into two terms via the history of \(\mathcal {X}(t)\) because we can infer the behavior of

*y*(

*t*) from the history of \(\mathcal {X}(t)\) by reversing the causal relation from

*x*(

*t*) to

*y*(

*t*) by Bayes’ theorem. Because the causal relation is reversed by Bayes’ theorem, the interpretation of each decomposed component by the variance decomposition does not reflect the regulatory relation between

*x*(

*t*) and

*y*(

*t*). This property of the variance decomposition by history should be noted for its application to FB networks. In contrast, even for the FB network, the source decomposition can provide a consistent decomposition that reflects the way how the fluctuation is determined by regulatory relations within the system. We should emphasize, however, that the source decomposition requires the stationarity and the linearization of the system by the Lyapunov equation that are not necessary for the variance decomposition.

^{14}

## 4 Relation Between Fluctuation Propagation and Feedback Gain

*x*to

*y*and from

*y*back to

*x*. However, the meaning of the individual gains, \(L_{x}\) and \(L_{y}\), is still ambiguous.

*x*is regulated not by

*y*but by its replica, \(y'\). We assume that \(y'\) is not driven by

*x*and that

*y*does not drive

*x*. Thereby, \(y' \rightarrow x \rightarrow y\) forms a FF network.

*K*and

*D*in Eq. (3) for this network become

*y*, we assume that all the kinetic parameters of \(y'\) are equal to those of

*y*as \(k_{xy'}=k_{xy}\), \(d_{y'}=d_{y}\), and \(\bar{a}_{y'}=\bar{a}_{y}\).

*y*as

*IV*) for

*x*is similar to the propagation of the fluctuation from

*y*to

*x*in the FB network, but it represents the propagation of the fluctuation from the replica \(y'\) in this opened FF network. The new term (

*V*) accounts for the propagation of the fluctuation from \(y'\) down to

*y*. The gain of this propagation has two terms, \(G_{yxy'}\) and \(G_{yy'}\). The first term, \(G_{yxy'}\), is the total gain of the propagation from \(y'\) to

*x*and from

*x*to \(y'\) because \(G_{yxy'}=G_{yx}G_{xy'}\) holds. In order to see the meaning of the term \(G_{yy'}\), we need to rearrange Eq. (25) as

*y*that cannot be reflected to the fluctuation of the intermediate component,

*x*. In addition, by solving Eq. (3), we can see that the gain \(G_{yy'}\) is directly related to the covariance between

*y*and \(y'\) as

*x*, and then the absorbed fluctuation propagates to the downstream, i.e.,

*y*. This component does not convey the information of the upstream because that does not affect the covariance between the upstream and the downstream. The other type described by \(G_{yy'}\) is that the fluctuation of the upstream propagates to the downstream without affecting the intermediate component. This fluctuation conveys the information on the upstream to the downstream because it is directly linked to their covariance. The fact that \(L_{y}\) is related to the latter indicates that the FB efficiency is directly linked to the information transfer in the opened loop from \(y'\) to

*y*. By considering another opened loop network where the replica of

*x*is introduced as in Fig. 3c, we can obtain the following result:

*x*by keeping all the other properties and kinetic parameters the same as those of

*y*. For example, if

*x*and

*y*are regulatory proteins and if they regulate each other as transcription factors as in Fig. 2b, the replica, \(y'\), must not be regulated by

*x*but its expression rate must be equal to the average expression rate of

*y*under the regulation of

*x*as in Fig. 3b. This requires fine-tuning of the expression rate of \(y'\) by modifying the DNA sequences relevant for the rate. In order to conduct this tuning, we have to measure several kinetic parameters of the original FB networks that undermines the advantage of the opened network such that measurements of the kinetic parameters are unnecessary to estimate \(L_{x}\) and \(L_{y}\) via Eq. (32).

## 5 Estimation of Feedback Loop Gain by a Conjugate FB and FF Network

*x*and

*y*are the same as the original FB network. The replica, \(y'\), is regulated by

*x*as

*y*is but does not regulate

*x*back. Thus,

*x*and the replica \(y'\) form an FF network. If

*x*and

*y*are regulatory proteins as in Fig. 2b, the replica \(y'\) can be engineered by duplicating the gene

*y*with its promoter site, and by modifying the coding region of the replica so that \(y'\) looses the affinity for binding to the regulatory region of

*x*as in Fig. 4b. This modification is much easier than that required for designing the opened network in Fig. 3.

*K*and

*D*in Eq. (3) become

*x*nor

*y*, the fluctuation of

*x*and that of

*y*are the same as those of the FB network in Eq. (21). The variance of the replica \(y'\) can be decomposed as

*y*and \(y'\) as

*x*as in Fig. 4c, d.

*x*and

*y*and normalized covariances. This result indicates that we have multiple ways, (a), (b), and (c), to estimate \(L_{x}\) and \(L_{y}\) from the statistics of the conjugate network. In addition, we also have a fluctuation relation that holds between the statistics as

### 5.1 Verification of the Relations by Numerical Simulation

*x*and

*y*are involved to measure \(L_{x}\) and \(L_{y}\), simultaneously as in Fig. 5a. For the variable \(\varvec{n}=(x,y,x',y')^{\mathbb {T}}\), the stoichiometric matrix and the propensity function are

*p*(

*x*,

*y*), \(p(x,y')\), and \(p(x',y)\) are plotted for different parameter values of \(K_{x}\). The FB is strong for small \(K_{x}\) whereas it is weak for large \(K_{x}\). \((L_{x}, L_{y})\) estimated by Eq. (38) for the parameter values are plotted in Fig. 5d. The three estimators, (a), (b), and (c) in Eq. (38), are used for comparison. For this simulation, all the estimators work well, but they have slightly larger variability in \(L_{x}\) compared with that in \(L_{y}\) for large values of \(|L_{x}|\) and \(|L_{y}|\). In addition, when \(|L_{x}|\) and \(|L_{y}|\) are very small, i.e. much less than 1, the estimators show relatively larger variability and bias, suggesting that the estimation of very weak FB efficiency requires more sampling. For the same parameter values, we also test Eq. (39) in Fig. 5e. Both \(F_{x}+F_{x'}-2 F_{x,x'}\) and \(F_{y}+F_{y'}-2 F_{y,y'}\) localize near 2 irrespective of the parameter values. As Figs. 5d, e demonstrate, the estimators obtained from the simulations agree with the analytical values of \((L_{x}, L_{y})\), and the fluctuation relation also holds very robustly.

In order to test how nonlinearity affects the estimation, we also investigated a non-linear negative feedback regulation defined by \(f(y):\,=f_{0} \frac{1}{1+(y/K_{y})^{n_{y}}}\), and \(g(x):\,=g_{0} \frac{(x/K_{x})^{n_{x}}}{1+(x/K_{x})^{n_{x}}}\). We change the Hill coefficients, \(n_{x}\) and \(n_{y}\), by keeping the fixed point unchanged as in Fig. 6a. Compared with the linear case, the variability of the estimators is almost similar even though the feedback regulation is nonlinear (Fig. 6b). In addition, the estimators show a good agreement with the analytical value except for very large value of \(|L_{x}|\) and \(|L_{y}|\). This suggests that Eq. (38) works as good estimators when the trajectories of the system are localized sufficiently near the fixed point as in Fig. 6a. For the very large value of \(|L_{x}|\) and \(|L_{y}|\) where \(|n|=2^{7/2}\), all the estimators are slightly biased towards smaller values, and the estimator (b) in Eq. (38) has larger variance than the others. In addition, similarly to the linear case, the estimators require larger sampling when \(|L_{x}|\) and \(|L_{y}|\) are much less than 1. Even with the nonlinear regulation, the fluctuation relation holds robustly as shown in Fig. 6c, which is consistent with the good agreement of the estimators of \(L_{x}\) and \(L_{y}\) with their analytical values as in Fig. 6b.

All these results indicate that the estimators obtained in Eq. (38) can be used to estimate the FB efficiency as long as the efficiency is moderate and the trajectories of the system are localized near the fixed point.

## 6 Discussion

In this work, we extended the fluctuation decomposition obtained for FF networks to FB networks. In this extension, the FB loop gains are naturally derived as the measure to quantify the efficiency of the FB. By considering the opened FF network obtained by opening the loop of the FB network, the relation between the loop gains and the fluctuation propagation in the FF network was clarified. In addition, we proposed the conjugate FB and FF network as a methodology to quantify the loop gains by showing that the loop gains are estimated only from the statistics of the conjugate network. By using numerical simulation, we demonstrated that the loop gains can actually be estimated by the conjugate network while we need more investigation on the bias and variance of the estimators. Furthermore, the fluctuation relation that holds in the conjugate network was also verified. We think that our work gives a theoretical basis for the conjugate network as a scheme for experimental estimation of the FB loop gains.

*x*and

*y*,

- 1.
duplicate them to obtain their replicas, \(x'\) and \(y'\), by genetic engineering as in Fig. 5a;

- 2.
modify the replicas, \(x'\) and \(y'\), so that they cannot regulate back

*y*and*x*, respectively; - 3.
measure the expression levels of

*x*and \(x'\) or*y*and \(y'\) simultaneously by single-cell measurements;^{15} - 4.
calculate the variances and covariances of

*x*, \(x'\),*y*, and \(y'\) from the data; - 5.
estimate the loop gains by using their estimators, (a), (b), and (c) in Eq. (38);

- 6.
calculate the fluctuation relation, Eq. (39), in order to check the validity of the source decomposition.

*x*(

*t*) and

*y*(

*t*), \(\mathbb {P}[\mathcal {X}(t),\mathcal {Y}(t)]\). From the definition of the conditional probability, we can decompose this joint probability as

*x*(

*t*) and

*y*(

*t*) are causally interacting, we can decompose the joint probability differently as

*y*back to

*x*, then this decomposition is reduced to

*y*to

*x*is defined as

*y*to

*x*, and measures the directional flow of information from

*y*back to

*x*. In the conjugate network, the replica \(y'\) is driven only by

*x*. Thus, the joint probability between \(\mathcal {X}(t)\) and \(\mathcal {Y}'(t)\) becomes

^{16}the directed information can be calculated by obtaining the joint distributions, \(\mathbb {P}[\mathcal {X}(t),\mathcal {Y}(t)]\) and \(\mathbb {P}[\mathcal {X}(t),\mathcal {Y}'(t)]\), of the conjugate network. This relation of the conjugate network with the directed information strongly suggests that the directed information and the causal decomposition are related to the loop gains. Resolving this problem will lead to more fundamental understanding of the FB in biochemical networks because the directed information is found fundamental in various problems such as the information transmission with FB [49], gambling with causal side information [31], population dynamics with environmental sensing [37], and the information thermodynamics with FB [39]. This problem will be addressed in our future work.

## Footnotes

- 1.
We prefer this interpretation of the Lyapunov equation because the LNA has been applied for various intracellular networks whose system size is not sufficiently large.

- 2.
Because we can obtain a notationally simpler result.

- 3.
CV is defined by the ratio of the standard deviation to the mean as \(\sigma _{x}/\left<x\right>\).

- 4.
For example, when the average of \(\bar{y}\) is increased by increasing the translation rate, the term does not decrease. Note that the term (III) can also decrease when, e.g., the average of \(\bar{y}\) is increased by reducing the degradation rate of

*y*. - 5.
See also Appendix 1 for more detailed discussion.

- 6.
The correspondence of Eqs. (14) with (10) or (11) is valid only when \(\sigma _{y}^{2}\) is decomposed under the conditioning with respect to the history of

*x*(*t*), \(\mathcal {X}(t)\), rather than the instantaneous state of*x*(*t*) at*t*. Note that the variance decomposition is general enough to hold not only for FF but also FB networks. - 7.
Because of its definition, the source decomposition is valid at least when the fluctuation of the system is well approximated by the stationary solution of the Lyapunov equation.

- 8.
- 9.
We can also extend this result for a FB network with more than two components. See Appendix 2 for the detail.

- 10.
For biologically relevant situations, \(d_{x}\) and \(d_{y}\) are positive because they can be regarded as the effective degradation rates.

- 11.
Note that the term (

*III*) is affected by the FB even though its origin is the fast birth and death events of*x*. This can be explained as follows. In general, the path gain from*x*to*y*, \(G_{yx}\), becomes very small compared to the others when*x*has much faster time scale than*y*as \(d_{x} \gg d_{y}\). Thus, the term (*III*) becomes quite small and little fluctuation propagates from*x*to*y*because of the averaging effect of the slow dynamics of*y*. (*III*) represents, therefore, the slow component in the fluctuation of the birth and death of*x*that has the comparative timescale as that of*y*. This is why the slow FB can affect the term (*III*). - 12.
Note that the numerators in Eq. (21), \(1-L_{x}\) and \(1-L_{y}\), cannot be 0 under the condition that \(1-L_{x}-L_{y}>0\) because \(L_{x}\) and \(L_{y}\) have the same sign.

- 13.
See also Appendix 4.

- 14.
Both variance and source decompositions are insufficient to address feedback structures perfectly.

- 15.
The measurement should not necessarily be time-lapse as long as it is conducted at the stationary state.

- 16.
In practice, measuring the joint probability of histories is almost impossible.

- 17.
This assumption can be weakened such that \(D(\bar{\mathbf {n}})\) is diagonalizable by linear transformation as discussed in [47].

- 18.
Note that the decomposition is also possible when

*D*is not diagonal. Please refer to [47] for more detailed discussion.

## Notes

### Acknowledgments

We thank Yoshihiro Morishita, Ryota Tomioka, Yuichi Wakamoto, and Yuki Sughiyama for discussion. This research is supported partially by Platform for Dynamic Approaches to Living System from MEXT, Japan, the Aihara Innovative Mathematical Modelling Project, JSPS through the FIRST Program, CSTP, Japan, the JST CREST program, the Specially Promoted Project of the Toyota Physical and Chemical Research Institute, and the JST PRESTO program.

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