Feedback Regulation and its Efficiency in Biochemical Networks
NNoname manuscript No. (will be inserted by the editor)
Tetsuya J. Kobayashi · Ryo Yokota · Kazuyuki Aihara
Feedback Regulation and its Efficiencyin Biochemical Networks
Received: date / Accepted: date
Abstract
Intracellular biochemical networks fluctuate dynamically due tovarious internal and external sources of fluctuation. Dissecting the fluctuationinto biologically relevant components is important for understanding how acell controls and harnesses noise and how information is transferred overapparently noisy intracellular networks. While substantial theoretical andexperimental advancement on the decomposition of fluctuation was achievedfor feedforward networks without any loop, we still lack a theoretical basisthat can consistently extend such advancement to feedback networks. Themain obstacle that hampers is the circulative propagation of fluctuation byfeedback loops. In order to define the relevant quantity for the impact offeedback loops for fluctuation, disentanglement of the causally interlockedinfluence between the components is required. In addition, we also lack anapproach that enables us to infer non-perturbatively the influence of thefeedback to fluctuation as the dual reporter system does in the feedforward
Tetsuya J. Kobayashi, Ryo Yokota & Kazuyuki AiharaInstitute of Industrial Science, the University of Tokyo. 4-6-1 Komaba, Meguro-ku,Tokyo, 153-8505, JapanTel.: +81-3-5452-6798Fax: +81-3-5452-6798E-mail: [email protected] a r X i v : . [ q - b i o . M N ] M a y network. In this work, we resolve these problems by extending the work onthe fluctuation decomposition and the dual reporter system. For a single-loopfeedback network with two components, we define feedback loop gain as thefeedback efficiency that is consistent with the fluctuation decomposition forfeedforward networks. Then, we clarify the relation of the feedback efficiencywith the fluctuation propagation in an open-looped FF network. Finally,by extending the dual reporter system, we propose a conjugate feedbackand feedforward system for estimating the feedback efficiency only from thestatistics of the system non-perturbatively. Keywords
Fluctuation · Linear Noise Approximation · Noise Decomposi-tion · Information Flow · Dual Reporter System sources. Decomposition of the fluctuation into the contributions from thesources is the indispensable step for understanding their biological roles andrelevance. When fluctuation propagates from one component to another uni-directionally without circulation, the fluctuation of the downstream can bedecomposed into two contributions. One is the intrinsic part that originateswithin the pathway between the components. The other is the extrinsic partthat propagates from the upstream component. Such decomposition can eas-ily be extended for the network with cascading or branching structures inwhich no feedback exists. This fact drove the intensive anatomical analysisof the intracellular fluctuation in the last decade.1.2 Decomposition of sources of fluctuationIn order to dissect fluctuation into different components, two major strategieshave been developed. One is to use the dependency of each component ondifferent kinetic parameters in the network. By employing theoretical predic-tions on such dependency, we can estimate the relative contributions of dif-ferent components from single-cell experiments with perturbations. Possibledecompositions of the fluctuation were investigated theoretically for variousnetworks such as single gene expression [3, 26, 27], signal transduction path-ways [41,47], and cascading reactions [45]. Some of them were experimentallytested [3, 26, 28].The other is the dual reporter system in which we simultaneously measurea target molecule with its replica obtained by synthetically duplicating thetarget. From the statistics of the target and the replica, i.e, mean, variance,and covariance, we can discriminate the intrinsic and extrinsic contributionsto the fluctuation because the former is independent between the target andthe replica whereas the latter is common to them. The idea of this strat-egy was proposed and developed in [27, 43], and verified experimentally for different species [11, 23, 33]. Its applicability and generality were further ex-tended [4, 6, 16, 17, 35].Now these strategies play the fundamental role to design single-cell ex-periments and to derive information on the anatomy of fluctuation from theexperimental observations [8, 15, 29, 34, 37, 44].1.3 Feedback regulation and its efficiencyEven with the theoretical and the experimental advancement in decomposingfluctuation, most of works focused on the feedforward (FF) networks in whichno feedback and circulation exist. As commonly known in the control theory[7, 39], feedback (FB) loops substantially affect fluctuation of a network byeither suppressing or amplifying it. Actually, the suppression of fluctuationin a single gene expression with a FB loop was experimentally tested in [2]earlier than the decomposition of fluctuation. While the qualitative and thequantitative impacts of the FB loops were investigated both theoreticallyand experimentally [1, 21, 24, 25, 35, 42, 46], we still lack a theoretical basisthat can consistently integrate such knowledge with that on the fluctuationdecomposition developed for the FF networks.The main problem that hampers the integration is the circulation of fluc-tuation in the FB network. Because fluctuation generated at a molecularcomponent propagates the network back to itself, we need to disentanglethe causally interlocked influence between the components to define the rel-evant quantity for the impact of the FB loops. From the experimental pointof view, in addition, quantification of the impact of FB loops by perturba-tive experiments is not perfectly reliable because artificial blocking of theFB loops inevitably accompanies the change not only in fluctuation but alsoin the average level of the molecular components involved in the loops. It isquite demanding and almost impossible for most cases to inhibit the loops bykeeping the average level unchanged. We still lack an approach that enables us to infer the influence of the FB non-perturbatively as the dual reportersystem does.1.4 Outline of this workIn this work, we resolve these problems by extending the work on the fluctu-ation decomposition [27, 41] and the dual reporter system [11, 43]. By usinga single-loop FB network with two components and its linear noise approxi-mation (LNA) [10,19,46], we first provide a definition of the FB loop gain asFB efficiency that is consistent with the fluctuation decomposition in [27,41].Then, we clarify the relation of the FB efficiency with the fluctuation prop-agation in a corresponding open-looped FF network. Finally, by extendingthe dual reporter system, we propose a conjugate FB and FF system forestimating the feedback efficiency only from the statistics of the system non-perturbatively. We also give a fluctuation relation among the statistics thatmay 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 thedecomposition of fluctuation for a simple FF system derived in [27, 41] byusing the LNA. In Sect. 3, we extend the result shown in Sect. 2 to a FBnetwork by deriving a decomposition of the fluctuation with feedback. Usingthis decomposition, we define the FB loop gain that is relevant for quantify-ing the impact of the FB to the fluctuation. In Sect. 4, we give a quantitativerelation of the loop gain in the FB network with the fluctuation propagationin a corresponding open-looped FF network. In Sect. 5, we propose a conju-gate FF and FB network as a natural extension of the dual reporter systemsused mainly for the FF networks. We clarify that the loop gain can be es-timated only from the statistics, i.e., mean, variances, and covariances, ofthe conjugate network. We also show that a fluctuation relation holds amongthe statistics, which generalizes the relation used in the dual reporter sys- tem. In Sect. 6, we discuss a link of the conjugate network with the directedinformation, and give future directions of our work.
In this section, we summarize the result for the decomposition of fluctuationobtained in [27, 41] by using the LNA, and also its relation with the dualreporter system that was employed in [11, 23, 33] to quantify the intrinsicand extrinsic contributions from the experimental measurements.2.1 Stochastic chemical reaction and its linear noise approximationLet us consider a chemical reaction network consisting of N different molecu-lar species and M different reactions. We assume that the stochastic dynamicsof the network is modeled by the following chemical master equation:d P ( t, n )d t = M (cid:88) k =1 [ a k ( n − s k ) P ( t, n − s k ) − a k ( n ) P ( t, n )] , (1)where n = ( n , . . . , n N ) T ∈ N N ≥ is the numbers of the molecular species, P ( t, n ) is the probability that the number of molecular species is n at t ,and a k ( n ) ∈ R M ≥ and s k are the propensity function and the stoichiometricvector of the k th reaction, respectively [12, 14, 19]. The propensity functioncharacterizes the probability of occurrence of the k th reaction when the num-ber of the molecular species is n , and the stoichiometric vector defines thechange in the number of the molecular species when the k th reaction occurs.In general, it is almost impossible to directly solve Eq. (1) both analyti-cally and numerically because it is a high-dimensional or infinite-dimensionaldifferential equation. To obtain insights for the dynamics of the reaction net-work, several approximations have been introduced [12, 19]. Among others, the first-order approximation is the deterministic reaction equation that isdescribed for the given propensity functions and the stoichiometric vector asd Ω u ( t )d t = S a ( Ω u ( t ) ) , (2)where Ω is the system size, u ∈ R n ≥ is the concentration of n as u = n /Ω , a ( n ) := ( a ( n ) ) , . . . , a M ( n )) T , and S ∗ ,k := s k is the stoichiometric matrix.Equation (2) was successfully applied for chemical reaction networks withlarge system size where the fluctuation of the concentration of the molecu-lar species can be neglected. When the system size is not sufficiently large,however, Eq. (2) is not relevant for analyzing the fluctuation of the network.The LNA is a kind of the second-order approximation of the Eq. (1) thatcharacterizes the fluctuation around a fixed point, ¯ n , of Eq. (2) that satis-fies S a ( ¯ n ) = 0. The stationary fluctuation of the network around ¯ n is thenobtained by solving the following Lyapunov equation [10, 19, 46]( K ( ¯ n ) Σ ) + ( K ( ¯ n ) Σ ) T + D ( ¯ n ) = 0 , (3)where K ∗ ,j ( n ) := ∂S a ( n ) ∂n j , D i,j ( n ) := (cid:88) k s i,k s j,k a k ( n ) , (4)and Σ is the covariance matrix of n . When the propensity function a ( n )is affine with respect to n , the dynamics of Ω u ( t ) determined by Eq. (2) isidentical to that of the first cumulant of n , i.e., the average of n , asd (cid:104) n (cid:105) d t = S a ( (cid:104) n (cid:105) ) , (5)where (cid:104) n (cid:105) := (cid:80) n n P ( t, n ). In addition, the second cumulant, i.e., the covari-ance matrix, follows the Lyapunov equation asd Σ d t = ( K ( (cid:104) n (cid:105) ) Σ ) + ( K ( (cid:104) n (cid:105) ) Σ ) T + D ( (cid:104) n (cid:105) ) . (6) Therefore, if the propensity function a ( n ) is affine, the stationary fluctuationof n is exactly described by Eq. (3). For a non-affine a ( n ), Eq. (5) and Eq. (6)can also be regarded as an approximation of the full cumulant equationsby the cumulant closure [12] under which we ignore the influence of thesecond and the higher order cumulants to Eq. (5), and that of the thirdand the higher order ones to Eq. (6) . Even though the propensity function a ( n ) is not affine, Eq. (3) (or Eq. (6)) can produce a good approximationof the fluctuation, provided that the fixed point, ¯ n , is a good approximationof the average, (cid:104) n (cid:105) , and that the local dynamics around the fixed point isapproximated enough by its linearization. In addition, compared with otherapproximations, the LNA enables us to obtain an analytic representation ofthe fluctuation because Eq. (3) is a linear algebraic equation with respectto Σ . Owing to this property, the LNA and its variations played the crucialrole to reveal the analytic representation of the fluctuation decomposition inbiochemical reaction networks [27,46]. As in these previous works, we employthe LNA to obtain an analytic representation for the feedback efficiency.2.2 Decomposition of fluctuationStarting from the LNA, Paulsson derived an analytic result on how noise isdetermined in a FF network with two components (Fig. 1 (A)) [27]. Here,we briefly summarize the result derived in [27]. Let n = x and n = y fornotational simplicity, and consider the FF reaction network (Fig. 1 (A)) withthe following propensity function and stoichiometric matrix, a ( x, y ) = ( a + x ( x ) , a − x ( x ) , a + y ( x, y ) , a − y ( x, y )) , S = +1 − − . (7) We prefer this interpretation of the Lyapunov equation because the LNA hasbeen applied for various intracellular networks whose system size is not sufficientlylarge. X (A)(B) mRNAProtein X Y Protein XProtein Y
Signle Gene Expression Two Gene Regula;on
Receptormessanger
Two Gene Regula;on
X Y
Gene of X Gene of YBinding regionof X (C)
Y Y’ (D)
X Y
Gene of YBinding regionof X Y’ Gene of Y’ Gene of X
Fig. 1 (A) The structure of the two-component FF network. Interpretations of thisnetwork as single-gene expression [11,22,26,27], two-gene regulation [46], and signaltransduction [47] are shown. (B) The structure of the dual reporter system [4, 11,17,27,43]. (C) A schematic diagram of the FF network for two-gene regulation.(D)A schematic diagram of the dual reporter network for the two-gene regulation.
Because a ± x depends only on x , x regulates y unidirectionally. Then, for afixed point (¯ x, ¯ y ) of Eq. (2), that satisfies a + x (¯ x ) = a − x (¯ x ) , a + y (¯ x, ¯ y ) = a − y (¯ x, ¯ y ) , (8) K and D in Eq. (3) becomes K = − d x k yx − d y = − H xx /τ x − ¯ y ¯ x H yx /τ y − H yy /τ y , D = a x
00 2¯ a y , (9)where ¯ a x := a + x (¯ x ) = a − x (¯ x ), ¯ a y = a + y (¯ x, ¯ y ) = a − y (¯ x, ¯ y ), and H i,j := ln a − i /a + i ∂ ln j (cid:12)(cid:12)(cid:12) (¯ x, ¯ y ) for i, j ∈ { x, y } . d x and d y are the minus of the diagonal terms of K andrepresent the effective degradation rates of x and y . k yx is the off-diagonalterm of K that represents the interaction from x to y . H ij is the susceptibility of the i th component to the perturbation ofthe j th one. τ x := ¯ x/ ¯ a x and τ y := ¯ y/ ¯ a y are the effective life-time of x and y , respectively. Except this section, we mainly use d s and k s as the representation of the parameters rather than H s and τ s introduced in [27] .By solving Eq. (3) analytically, the following fluctuation-dissipation relationwas derived in [27] as σ x ¯ x = 1¯ xH xx (cid:124) (cid:123)(cid:122) (cid:125) ( I ) , σ y ¯ y = 1¯ yH yy (cid:124) (cid:123)(cid:122) (cid:125) ( II ) + ( iii ) (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) H yx /τ y H yy /τ y (cid:19) ii ) (cid:122) (cid:125)(cid:124) (cid:123) H yy /τ y H yy /τ y + H xx /τ x ( i ) (cid:122)(cid:125)(cid:124)(cid:123) σ x ¯ x (cid:124) (cid:123)(cid:122) (cid:125) ( III ) . (10)This representation measures the intensity of the fluctuation by the coefficientof variation (CV) , and describes how the fluctuation generates and propa-gates within the network. ( I ) is the intrinsic fluctuation of x that originatesfrom the stochastic birth and death of x . 1 / ¯ x reflects the Poissonian natureof the stochastic birth and death, and 1 /H xx is the effect of auto-regulatoryFB. Similarly, ( II ) is the intrinsic fluctuation of y that originates from thestochastic birth and death of y . ( III ), on the other hand, corresponds tothe 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 thefluctuation of x , and therefore, identical to ( I ). ( ii ) and ( iii ) determine theefficiency of the propagation of the fluctuation from x to y , which correspondto the time-averaging and sensitivity of the pathway from x to y , respectively.This representation captures the important difference of the intrinsic and theextrinsic fluctation such that the intrinsic one, the term ( II ), can be reducedby increasing the average of ¯ y whereas the extrinsic one, the term ( III ),cannot.While Eq. (10) provides an useful interpretation on how the fluctuationpropagates in the FF network, it is not appropriate for the extension to theFB network because the contribution of x to the fluctuation of y is describedby the CV of x as the term ( i ). Because the fluctuation of x and y depend The former gives a notationally simpler result. The CV is defined by the ratio of the standard deviation to the mean as σ x / (cid:104) x (cid:105) .1 mutually if we have a FB between x and y , we need to characterize thefluctuation of y without directly using the fluctuation of x . To this end, weadopt the variances and covariances as the measure of the fluctuation anduse the following decomposition of the fluctuation for the FF network: σ x = ( ii ) (cid:122)(cid:125)(cid:124)(cid:123) G x,x ( i ) (cid:122)(cid:125)(cid:124)(cid:123) ¯ x (cid:124) (cid:123)(cid:122) (cid:125) ( I ) , σ y = ( ii ) (cid:122)(cid:125)(cid:124)(cid:123) G yy ( i ) (cid:122)(cid:125)(cid:124)(cid:123) ¯ y (cid:124) (cid:123)(cid:122) (cid:125) ( II ) + ( iii ) (cid:122)(cid:125)(cid:124)(cid:123) G yx ( ii ) (cid:122)(cid:125)(cid:124)(cid:123) G x,x ( i ) (cid:122)(cid:125)(cid:124)(cid:123) ¯ x (cid:124) (cid:123)(cid:122) (cid:125) ( III ) , (11)where we define G xx , G yy , and G yx as G xx = 1 H xx , G yy = 1 H yy , G yx = k yx ( d x + d y ) d y . (12)The terms ( 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, dif-ferent. In Eq. (11), the terms ( i ) are interpreted as the fluctuation purelygenerated by the birth and death reactions of x and y by neglecting any con-tribution of the auto-FBs. Because the simple birth and death of a molecularspecies without any regulation follow the Poissonian statistics, the intensityof the fluctuation is equal to the means of the species. Thus, ¯ x ≈ (cid:104) x (cid:105) and¯ y ≈ (cid:104) y (cid:105) in the terms ( i ) represent the generation of the fluctuation by thebirth and death of x and y , respectively. The fluctuation generated is thenamplified or suppressed by the auto-regulatory FBs. G xx and G yy in theterms ( ii ) account for this influence, and are denoted as auto-FB gains inthis work. Finally, G yx in the term ( iii ) quantifies the efficiency of the prop-agation of the fluctuation from x to y . We denote G yx as the path gain from x to y . If we use the notation in Eq. (10), G yx is described as G yx = (cid:18) ¯ yH yx /τ y ¯ xH yy /τ y (cid:19) H yy /τ y H yy /τ y + H xx /τ x . (13)The decomposition of the fluctuation of y into ( II ) and ( III ) is consis-tent between Eq. (10) and Eq. (11) while the further decompositions within ( II ) and ( III ) are different. It was recently revealed in [4, 6, 16, 17] that thedecomposition into ( II ) and ( III ) is linked to the variance decompositionformula in statistics as σ y = V [ E [ y ( t ) |X ( t )]] (cid:124) (cid:123)(cid:122) (cid:125) ( II ) + E [ V [ y ( t ) |X ( t )]] (cid:124) (cid:123)(cid:122) (cid:125) ( III ) , (14)where X ( t ) := { x ( τ ); τ ∈ [0 , t ] } is the history of x ( t ), and E and V are theexpectation and the variance, respectively . Because the variance decompo-sition formula holds generally, Eq. (14) is more fundamental than Eq. (10)and Eq. (11) as the decomposition of fluctuation. This decomposition wasfurther analyzed in [4, 6, 16].2.3 Dual reporter systemThe decomposition, Eq. (10) or Eq. (11), guides us how to evaluate the intrin-sic and the extrinsic fluctuation in y experimentally. When we can externallycontrol the mean of y without affecting the term ( III ), we can estimatethe relative contributions of ( II ) and ( III ) as the intrinsic and the extrin-sic fluctuation by plotting σ y as a function of the mean of y . When x and y correspond to a mRNA and a protein in the single gene expression, thetranslation rate works as such a control parameter [22]. This approach wasintensively employed to estimate the efficiency of the fluctuation propagationin various intracellular networks [3, 26, 28].Another way to quantify the intrinsic and the extrinsic fluctuation is thedual reporter system adopted in [11,23,33] whose network structure is shownin Fig. 1 (B) . In the dual reporter system, a replica of y is attached to thedownstream of x as in Fig. 1 (B) where y (cid:48) denotes the molecular species The correspondence of Eq. (14) with Eq. (10) or Eq. (11) is valid only when σ y is decomposed under the conditioning with respect to the history of x ( t ), X ( t ),rather than the instantaneous state of x ( t ) at t . The dual reporter system is also called a conjugate reporter system [4, 6].3 of the replica. The replica, y (cid:48) , must have the same kinetics as y , and mustbe measured simultaneously with y . If y is a protein whose expression isregulated by another protein, x , as in Fig. 1 (C), then y (cid:48) can be syntheticallyconstructed by duplicating the gene of y and attaching fluorescent probeswith different colors to y and y (cid:48) as in Fig. 1 (D) [11]. Under the LNA, thecovariance between y and y (cid:48) can be described as σ y,y (cid:48) = G yx G xx ¯ x. (15)Thus, by using only the statistics of the dual reporter system, the intrinsicand the extrinsic components in y can be estimated as σ y = G yy (cid:104) y (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ( II ) + σ y,y (cid:48) (cid:124)(cid:123)(cid:122)(cid:125) ( III ) , (16)where it is unnecessary to control any kinetic parameters externally. Thisapproach is generalized as the conjugate reporter system together with themore general decomposition, Eq. (14), in [4, 6, 16]. While we had substantial progress in the decomposition of the fluctuationand its experimental measurement for the FF networks in the last decade,its extension to FB networks is yet to be achieved. In this section, we extendthe decomposition of the fluctuation in the simple FF network (Eq. (11)) tothe corresponding FB one depicted in Fig. 2 (A) . As in Eq. (9), K and D in the Lyapunov equation (Eq. (3)) for the FB network can be described as K = − d x k xy k yx − d y = − H xx /τ x − ¯ x ¯ y H xy /τ x − ¯ y ¯ x H yx /τ y − H yy /τ y , D = a x
00 2¯ a y . (17) (A) XY (B) Y X
Fig. 2 (A) The structure of the two-component FB network. (B) A schematicdiagram of the FB network for two-gene regulation.
By defining the path gain from y back to x as G xy := k xy d x ( d x + d y ) , (18)we can derive a decomposition of the fluctuation of x and y as σ x = 1 − L y − L x − L y ( I ) (cid:122) (cid:125)(cid:124) (cid:123) G xx ¯ x + 11 − L x − L y ( IV ) (cid:122) (cid:125)(cid:124) (cid:123) G xy G yy ¯ y,σ y = 1 − L x − L x − L y G yy ¯ y (cid:124) (cid:123)(cid:122) (cid:125) ( II ) + 11 − L x − L y G yx G xx ¯ x (cid:124) (cid:123)(cid:122) (cid:125) ( III ) , (19)where we define L x := k xy k yx d x ( d x + d y ) , L y := k xy k yx ( d x + d y ) d y . (20)If the FB from y to x does not exist, i.e., k xy = 0, then L x = L y = G xy = 0and Eq. (19) is reduced to Eq. (11). Thus, L x and L y account for the effectof the FB. L x and L y are denoted as the FB loop gains in this work. Eq. (19)clearly demonstrates that the representation in Eq. (10) does not work withthe FB because σ y cannot be described with σ x any longer. In contrast, wecan interpret the terms in Eq. (19) consistently with those in Eq. (11) becauseall the terms, ( I ), ( II ) and ( III ), are unchanged in Eq. (19). The new term,( IV ), in the expression for σ x appears to account for the propagation of thefluctuation generated by the birth and death events of y back to x . Equations (19) and (20) indicate how the FB affects the fluctuation of x and y . First, the fluctuation is suppressed when L x and L y are negativewhereas it is amplified when they are positive. When d x and d y are positive ,the sign of L x and L y are determined by the sign of k xy k yx . Thus, the FBloop is negative when x regulates y positively and y does x negatively or viseversa. 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 (cid:28) L y , e.g., the time-scale of x is much faster than y ,then Eq. (19) can be approximated as σ x = ( I ) (cid:122) (cid:125)(cid:124) (cid:123) G xx ¯ x + 11 − L y ( IV ) (cid:122) (cid:125)(cid:124) (cid:123) G xy G yy ¯ y, σ y = 11 − L y G yy ¯ y (cid:124) (cid:123)(cid:122) (cid:125) ( II ) + G yx G xx ¯ x (cid:124) (cid:123)(cid:122) (cid:125) ( III ) . Thus, the FB does not work for the term ( I ) that is the part of fluctuation of x whose origin is the birth and death events of x itself. This result reflects thefact that the slow FB from x to itself via y cannot affect the fast componentof the fluctuation of x . 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 theFB. 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 tothe destabilization of the fixed point, ¯ n , by the FB. For biologically relevant situations, d x and d y are positive because they can beregarded as the effective degradation rates. Note that the term (
III ) is affected by the FB even though its origin is thefast 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 others when x has much faster time scale than y . Thus, the term ( III ) becomes quite small andlittle fluctuation propagates from x to y because of the averaging effect of the slowdynamics of y . ( III ) represents, therefore, the slow component in the fluctuationof the birth and death of x that has the comparative timescale as that of y . Thisis why the slow FB can affect the term ( III ).6 (A) XY (B) XY’ YY’ (C) XY (D) YX’ X’
No interactionNo interactionNo interactionNo interaction X Fig. 3 (A) The structure of the the opened FF network obtained by replicating y inthe FB network. (B) A schematic diagram of the network in (A) for gene regulation.(C) The structure of the the opened FF network obtained by replicating x in theFB network. (D) A schematic diagram of the network in (C) for gene regulation. As shown in Sect. 3, L x and L y are quantitatively related to the efficiency ofthe FB. Because L x L y = G xy G yx holds, the loop gains are also linked to thepropagation of the fluctuation from x to y and from y back to x . However,the meaning of the individual gains, L x and L y , is still ambiguous.To clarify the relation between the loop gains and the propagation offluctuation, we consider a three-component FF network shown in Fig. 3 (A)and (B) that are obtained by opening the FB network in Fig. 2 (A). In thenetwork shown in Fig. 3 (A), x is regulated not by y but by its replica denotedas y (cid:48) . We assume that y (cid:48) is not driven by x , and thereby, y (cid:48) → x → y forms a FF network. K and D in Eq. (3) for this network become K = − d y (cid:48) k y (cid:48) x − d x k xy − d y , D = diag (cid:16) ¯ a y (cid:48) ¯ a x ¯ a y (cid:17) . (21)Because y (cid:48) is the replica of y , we assume that all the kinetic parameters of y (cid:48) are equal to those of y as k y (cid:48) x = k yx , d y (cid:48) = d y , and ¯ a y (cid:48) = ¯ a y .By solving Eq. (3), we can obtain the following decomposition of thefluctuation of y as σ x = ( I ) (cid:122) (cid:125)(cid:124) (cid:123) G xx ¯ x + ( IV ) (cid:122) (cid:125)(cid:124) (cid:123) G xy (cid:48) G y (cid:48) y (cid:48) ¯ y (cid:48) ,σ y = G yy ¯ y (cid:124) (cid:123)(cid:122) (cid:125) ( II ) + G yx G xx ¯ x (cid:124) (cid:123)(cid:122) (cid:125) ( III ) + [ G yy (cid:48) + G yxy (cid:48) ] G y (cid:48) y (cid:48) ¯ y (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) ( V ) , (22)where G xy (cid:48) = G xy , G yxy (cid:48) := G yx G xy (cid:48) = G yx G xy = L x L y , G yy (cid:48) := L y / . (23)The term ( 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 fluctuationfrom the replica y (cid:48) in this opened FF network. The new term ( V ) accountsfor the propagation of the fluctuation from y (cid:48) down to y . The gain of thispropagation has two terms, G yxy (cid:48) and G yy (cid:48) . The first term, G yxy (cid:48) , is the totalgain of the fluctuation propagation from y (cid:48) to x and from x to y (cid:48) because G yxy (cid:48) = G yx G xy (cid:48) holds. In order to see the meaning of the second term G yy (cid:48) ,we need to rearrange Eq. (22) as σ y = G yy ¯ y + G yx σ x + G yy (cid:48) G y (cid:48) y (cid:48) ¯ y (cid:48) . (24) This representation clarifies that G yy (cid:48) describes the propagation of the fluc-tuation from y (cid:48) to y that cannot be reflected to the fluctuation of the inter-mediate component, x . In addition, by solving Eq. (3), we can see that thegain G yy (cid:48) is directly related to the covariance between y and y (cid:48) as σ y (cid:48) ,y = (cid:112) G yy (cid:48) ¯ y (cid:48) = L y y (cid:48) . (25)This implies that we have at least two types of propagation of the fluctua-tion. One described by G yxy (cid:48) is that the fluctuation of the upstream, i.e., y (cid:48) ,is absorbed by the intermediate component, i.e., x , and then the absorbedfluctuation propagates to the downstream, i.e., y . This component does notconvey the information of the upstream because that does not affect the co-variance between the upstream and the downstream. The other described by G yy (cid:48) is that the fluctuation of the upstream propagates to the downstreamwithout affecting the intermediate component. This fluctuation conveys theinformation on the upstream to the downstream because it is directly linkedto their covariance. The fact that L y is related to the latter indicates thatthe FB efficiency is directly linked to the information transfer in the openedloop from y (cid:48) to y . By considering another opened loop network where thereplica of x is introduced as in Fig. 3 (C), we can obtain the following result: σ y = ( II ) (cid:122) (cid:125)(cid:124) (cid:123) G yy ¯ y + ( III ) (cid:122) (cid:125)(cid:124) (cid:123) G yx (cid:48) G x (cid:48) x (cid:48) ¯ x (cid:48) ,σ x = G xx ¯ x (cid:124) (cid:123)(cid:122) (cid:125) ( I ) + G xy G yy ¯ y (cid:124) (cid:123)(cid:122) (cid:125) ( IV ) + [ G xx (cid:48) + G xyx (cid:48) ] G x (cid:48) x (cid:48) ¯ x (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) ( V I ) , (26)where G xx (cid:48) := L x /
2, and we also have σ x (cid:48) ,x = (cid:112) G xx (cid:48) ¯ x (cid:48) = L x x (cid:48) . (27)By combining Eq. (25) and Eq. (27), we can estimate the loop gains, L x and L y , by only measuring the averages and covariances of the opened networks as follows: L x σ x (cid:48) ,x (cid:104) x (cid:48) (cid:105) , L y σ y (cid:48) ,y (cid:104) y (cid:48) (cid:105) . (28)While this strategy sounds eligible theoretically, it has an experimental diffi-culty in constructing the opened networks. In the opened network, the replica,e.g., y (cid:48) , must be designed so that it is free from the regulation of x by keepingall the other properties and kinetic parameters the same as those of y . Forexample, if x and y are regulatory proteins and if they regulate each other astranscription factors as Fig. 2 (B), the replica, y (cid:48) , must not be regulated by x but its expression rate must be equal to the average expression rate of y underthe regulation of x as in Fig. 3 (B). This requires fine-tuning of the expressionrate of y (cid:48) by modifying the DNA sequences relevant for the rate. In order toconduct this tuning, we have to measure several kinetic parameters of theoriginal FB networks that undermines the advantage of the opened networkthat measurements of the kinetic parameters are unnecessary to estimate L x and L y via Eq. (28). As a more promising strategy for the measurement of the loop gains, we pro-pose a conjugate FB and FF network that is an extension of the dual reportersystem for the estimation of the intrinsic and the extrinsic components. Inthe conjugate network, we couple the original FB network with a replica thatis opened as in Fig. 4 (A). x and y are the same as the original FB network.The replica, y (cid:48) , is regulated by x as y is but does not regulate x back. Thus, x and the replica, y (cid:48) , form an FF network. If x and y are regulatory proteinsas in Fig. 2 (B), the replica y (cid:48) can be engineered by duplicating the gene y with its promoter site, and by modifying the coding region of the replica (A) XY (B) Y XY’ Y’
No interaction
XYX’ (C) (D)
Y XX’
No interaction
FFFB FBFF
Fig. 4 (A) The structure of the conjugate FB and FF network obtained by repli-cating y in the FB network. (B) A schematic diagram of the network in (A) forgene regulation. (C) The structure of the conjugate FB and FF network obtainedby replicating x in the FB network. (D) A schematic diagram of the network in(C) for gene regulation. so that y (cid:48) looses the affinity for binding to the regulatory region of x as inFig. 4 (B). This engineering is much easier than that required for designingthe opened network.Next, we show how to use this conjugate network to measure the loopgains. For this network, K and D in Eq. (3) become K = − d x k xy k yx − d y k y (cid:48) x − d y (cid:48) , D = ¯ a x a y
00 0 ¯ a y (cid:48) . (29)Because the replica y (cid:48) affects neither x nor y , the fluctuation of x and thatof y are the same as those of the FB network in Eq. (19). The variance of the replica y (cid:48) can be decomposed as σ y (cid:48) = ( II ) (cid:122) (cid:125)(cid:124) (cid:123) G y (cid:48) y (cid:48) ¯ y (cid:48) + (1 − L y ) + (1 + L y (cid:48) )(1 − L x − L y )(1 − L y (cid:48) ) ( III ) (cid:122) (cid:125)(cid:124) (cid:123) G y (cid:48) x G xx ¯ x + 1(1 − L x − L y )(1 − L y (cid:48) ) ( G y (cid:48) y + G y (cid:48) xy ) G yy ¯ y (cid:124) (cid:123)(cid:122) (cid:125) ( V ) , (30)where L y (cid:48) = L y , G y (cid:48) y (cid:48) = G yy , G y (cid:48) x = G yx , G y (cid:48) y = G yy (cid:48) , G y (cid:48) xy = G yx G xy . (31)Rearrangement of this equation leads to σ y (cid:48) = σ y − L y − L y / G yy ¯ y. (32)In addition, we have the following expression for the covariance between y and y (cid:48) as σ y,y (cid:48) = 1(1 − L x − L y ) G yx G xx ¯ x + (1 + L x ) L y (cid:48) (1 − L x − L y )(1 − L y (cid:48) ) G yy ¯ y, = σ y − − L y / G yy ¯ y. (33)A similar result can be obtained by replicating x as in Fig. 4 (C) and (D).By using these relations, we have L x = ( a ) (cid:122) (cid:125)(cid:124) (cid:123) σ x (cid:48) − σ x σ x,x (cid:48) − σ x = ( b ) (cid:122) (cid:125)(cid:124) (cid:123) / G xx / ( F x − F x (cid:48) ) = ( c ) (cid:122) (cid:125)(cid:124) (cid:123) (cid:18) − G xx F x − F x,x (cid:48) (cid:19) ,L y = σ y (cid:48) − σ y σ y,y (cid:48) − σ y (cid:124) (cid:123)(cid:122) (cid:125) ( a ) = 11 / G yy / ( F y − F y (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) ( b ) = 2 (cid:18) − G yy F y − F y,y (cid:48) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ( c ) , (34)where F x := σ x / (cid:104) x (cid:105) , F y := σ y / (cid:104) y (cid:105) , F x,x (cid:48) := σ x,x (cid:48) / (cid:104) x (cid:105) , and F y,y (cid:48) := σ y,y (cid:48) / (cid:104) y (cid:105) are the Fano factors of x and y and normalized covariances. This result (A)(B) x y K x = ● :200 K x = ▲ :400 K x = ■ :800 K x = ◯ :1600 K x = △ :3200 x x x x Y X Y’X’ (C) F x + F x F x,x F y + F y F y , y (D) Equation (34a) Equation (34b) Equation (34c) ● :200 ▲ :400 ■ :800 ◯ :1600 △ :3200 □ :6400 K x = L x L x L x L y Fig. 5 (A) A schematic diagram of the conjugate FB and FF network used for thenumerical simulation. (B) The distributions of p ( x, y ) , p ( x, y (cid:48) ), and p ( x (cid:48) , y ) sampledfrom the simulation for different parameter values of K x . The other parameters are f = g = 400, K y = 100, and d x = d y = 1. The blue, red, and green dots are thedistributions of p ( x, y ) , p ( x, y (cid:48) ), and p ( x (cid:48) , y ), respectively. Solid and dashed lines arethe nullclines of Eq. (2) defined as d x/ d t = 0 and d y/ d t = 0 . (C) ( L x , L y ) estimatedwith the relations, (a), (b), and (c), in Eq. (34) from the numerical simulation. Theparameter values used were the same as in (B). For each estimation, the means andthe covariances required in Eq. (34) were calculated from 9 × samplings. Foreach parameter value, we calculated the estimates 10 times to see their variation.Black markers show the analytically obtained values of ( L x , L y ) for each parametervalue. (D) A plot of the estimates of F x + F x (cid:48) − F x,x (cid:48) and F y + F y (cid:48) − F y,y (cid:48) derivedin Eq. (35). The parameter values used were the same as those in (C). indicate 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 afluctuation relation that holds between the statistics as F x + F x (cid:48) − F x,x (cid:48) = 2 G xx , F y + F y (cid:48) − F y,y (cid:48) = 2 G yy , (35)which is the generalization of Eq. (16) for the dual reporter system. x and y are involved to measure L x and L y , simultaneously as inFig. 5(A). For the variable n = ( x, y, x (cid:48) , y (cid:48) ) T , the stoichiometric matrix andthe propensity function are S = − − − − , a ( x ) = f ( y ) − d x xg ( x ) − d y yf ( y ) − d x x (cid:48) g ( x ) − d y y (cid:48) . (36)The simulation is conducted by the Gillespie’s next reaction algorithm [13].First, we test linear negative feedback regulation defined by f ( y ) := max (cid:104) f [1 − yK y ] , (cid:105) ,and g ( x ) := max (cid:104) g xK x , (cid:105) under which the LNA holds exactly as long as itstrajectory has sufficiently small probability to reach the boundaries of x = 0or y = 0. In Fig. 5(B), the distributions of p ( x, y ), p ( x, y (cid:48) ), and p ( x (cid:48) , y ) areplotted 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. (34) for the param-eter values are plotted in Fig. 5(C). The three estimators, (a), (b), and (c) inEq. (34), are used for comparison. For this simulation, all the estimators workwell, 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 verysmall, i.e. much less than 1, the estimators show relatively larger variabilityand bias, suggesting that the estimation of very weak FB efficiency requires (A) x y x x x x (B) F x + F x F x,x F y + F y F y , y (C) Equation (34a) Equation (34b) Equation (34c) | n | = ● :2 ▲ : 2 ■ : 2 ◯ : 2 △ : 2 □ : 2 ☓ : 2 L x L x L x L y | n | = ★ : 2 ★ : 2 □ : 2 ◯ : 2 ▲ : 2 ● :2 | n | = | n | = | n | = | n | = Fig. 6 (A) The distributions of p ( x, y ) , p ( x, y (cid:48) ), and p ( x (cid:48) , y ) sampled from thesimulation for different parameter values of n x and n y where n x = n y = | n | .The other parameters are f = g = 300, K x = K y = 150, and d x = d y = 1.The blue, red, and green dots are the distributions of p ( x, y ) , p ( x, y (cid:48) ), and p ( x (cid:48) , y ),respectively. Solid and dashed curves are the nullclines of Eq. (2) defined as d x/ d t =0 and d y/ d t = 0 . (B) ( L x , L y ) estimated with the relations, (a), (b), and (c), inEq. (34) from the numerical simulation. The same parameter values were used asin (A). For each estimation, the means and the covariances required in Eq. (34)were calculated from 9 × samplings. For each parameter value, we calculatedthe estimates 10 times to see their variation. Black markers show the analyticallyobtained values of ( L x , L y ) for each parameter value. (C) A plot of the estimatesof F x + F x (cid:48) − F x,x (cid:48) and F y + F y (cid:48) − F y,y (cid:48) shown in Eq. (35). The parameter valuesused were the same as those in (B). more sampling. For the same parameter values, we also check Eq. (35) inFig. 5(D). Both F x + F x (cid:48) − F x,x (cid:48) and F y + F y (cid:48) − F y,y (cid:48) localize near 2 ir-respective of the parameter values. As Fig. 5(C) and (D) demonstrate, theestimators 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 investi-gated non-linear negative feedback regulation defined by f ( y ) := f y/K y ) ny ,and g ( x ) := g x/K x ) nx x/K x ) nx . We change the Hill coefficients, n x and n y , bykeeping the fixed point unchanged as in Fig. 6(A). Compared with the linearcase, the variability of the estimators is almost similar even though the feed-back regulation is nonlinear (Fig. 6(B)). 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. (34) works as good estimators when thetrajectories of the system are localized sufficiently near the fixed point as inFig. 6(A). For the very large value of | L x | and | L y | where | n | = 2 / , all theestimators are slightly biased towards smaller values, and (b) in Eq. (34) haslarger variance than the others. In addition, similarly to the linear case, theestimators require larger sampling when | L x | and | L y | are much less than 1.Even with the nonlinear regulation, the fluctuation relation holds robustlyas shown in Fig. 6(C), which is consistent with the good agreement of theestimators of L x and L y with their analytical values as in Fig. 6(B).All these results indicate that the estimators obtained in Eq. (34) can beused to estimate the FB efficiency as long as the efficiency is moderate andthe trajectories of the system are localized near the fixed point. In this work, we extended the fluctuation decomposition formula obtainedfor the FF network to the FB network. In this extension, the FB loop gainsare naturally derived as the measure to quantify the efficiency of the FB. Byconsidering the opened FF network obtained by opening the loop of the FBnetwork, the relation between the loop gains and the fluctuation propagationin the FF network was clarified. In addition, we proposed the conjugate FBand FF network as a methodology to quantify the loop gains by showingthat the loop gains are estimated only from the statistics of the conjugatenetwork. By using numerical simulation, we demonstrated that the loop gainscan actually be estimated by the conjugate network while we need moreinvestigation on the bias and variance of the estimators. Furthermore, thefluctuation relation that holds in the conjugate network was also verified.We think that our work gives a theoretical basis for the conjugate networkas a scheme for experimental estimation of the FB loop gains. As for the further problems on the efficiency of FB and its quantification,the generality of the FB loop gain should be clarified. In the case of theFF network, the fluctuation decomposition proposed initially by using theLNA was generalized as the variance decomposition formula with respectto the conditioning of the history of the upstream fluctuation [4, 17]. In thecase of the FB network, similarly, the decomposition and the FB loop gainswere obtained and defined via the LNA. However, its generalization is notstraightforward because the contribution of the interlocked components can-not be dissected in the FB network because of the circulative flow of thefluctuation. The previous work on the relation between the fluctuation de-composition and the information transfer for the FF network [4] suggeststhat the FB loop gains can also be interpreted in terms of the informationtransfer. As a candidate for such information measure, we illustrate a con-nection of the conjugate FB and FF network with the directed informationand Kramer’s causal conditioning. Let us consider the joint probability of thehistories of x ( t ) and y ( t ), P [ X ( t ) , Y ( t )]. From the definition of the conditionalprobability, we can decompose this joint probability as P [ X ( t ) , Y ( t )] = P [ X ( t ) |Y ( t )] P [ Y ( t )] = P [ Y ( t ) |X ( t )] P [ X ( t )] . (37)However, when x ( t ) and y ( t ) are causally interacting, we can decompose thejoint probability differently as P [ X ( t ) , Y ( t )] = (cid:34) t (cid:89) i =1 P [ x i |X i − , Y i − ] (cid:35) (cid:34) t (cid:89) i =1 P [ y i |X i , Y i − ] (cid:35) (38)= P y || x [ Y ( t ) ||X ( t )] × P x || y [ X ( t ) ||Y ( t − , (39)where P y || x [ Y ( t ) ||X ( t )] and P x || y [ X ( t ) ||Y ( t − y back to x , then this decom- position is reduced to P [ X ( t ) , Y ( t )] = P y || x [ Y ( t ) ||X ( t )] × P [ X ( t )] . (40)The directed information from y to x is defined as I [ Y ( t ) → X ( t )] := (cid:28) ln P [ X ( t ) , Y ( t )] P y || x [ Y ( t ) ||X ( t )] P [ X ( t )] (cid:29) P [ X ( t ) , Y ( t )] , (41)where the joint probability of X ( t ) and Y (t) is compared with the distribu-tion, P y || x [ Y ( t ) ||X ( t )] × P [ X ( t )] [30]. Thus, I [ Y ( t ) → X ( t )] is zero when noFB exists from y to x , and measures the directional flow of information from y back to x . In the conjugate network, the replica y (cid:48) is driven only by x .Thus, the joint probability between X ( t ) and Y (cid:48) ( t ) becomes P [ X ( t ) , Y (cid:48) ( t )] = P y || x [ Y (cid:48) ( t ) ||X ( t )] P [ X ( t )] . (42)Thereby, in principle , the directed information can be calculated by obtain-ing the joint distributions, P [ X ( t ) , Y ( t )] and P [ X ( t ) , Y (cid:48) ( t )], of the conjugatenetwork. This relation of the conjugate network with the directed informa-tion strongly suggests that the directed information and the causal decom-position are relevant for the loop gains. Resolving this problem will lead tomore fundamental understanding of the FB in biochemical networks becausethe directed information is found fundamental in various problems such asthe information transmission with FB [48], gambling with causal side infor-mation [30], population dynamics with environmental sensing [36], and theinformation thermodynamics with FB [38]. This problem will be addressedin our future work. Acknowledgements
We thank Yoshihiro Morishita, Ryota Tomioka, Yoichi Wakamoto,and Yuki Sughiyama for discussion. This research is supported partially by Plat- In practice, measuring the joint probability of histories is almost impossible.8form for Dynamic Approaches to Living System from MEXT, Japan, the AiharaInnovative Mathematical Modelling Project, JSPS through the FIRST Program,CSTP, Japan, and the JST PRESTO program.
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