EQUILIBRIA AND STABILITY ANALYSIS OF A BRANCHED METABOLIC NETWORK WITH FEEDBACK INHIBITION. F. Grognard Y. Chitour G. Bastin


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1 EQUILIBRIA AND STABILITY ANALYSIS OF A BRANCHED METABOLIC NETWORK WITH FEEDBACK INHIBITION F. Grognard Y. Chitour G. Bastin Projet COMORE. INRIA SophiaAntipolis. BP SophiaAntipolis Cedex, France Département de Mathématiques Université de Paris Sud Bâtiment Orsay, France CESAME Université Catholique de Louvain Bâtiment Euler,46, avenue G.Lemaitre, 1348 Louvain la Neuve, Belgium Astract: This paper deals with the analysis of a metaolic network with feedack inhiition. The considered system is an acyclic network of monomolecular enzymatic reactions in which metaolites can act as feedack regulators on preceding enzymes of their own pathway, and in which one metaolite is the root of the whole network. We show, under mild assumptions, the uniqueness of the equilirium. In the simplified case where inhiition only acts on the reactions having the root of the network as sustrate, we show that the equilirium is gloally attractive. This requires that we impose conditions on the kinetic parameters of the metaolic reactions. Keywords: Metaolic network, small gain, staility, limit cycle 1. INTRODUCTION The cellular metaolism is defined as the (huge) set of iochemical reactions that occur inside a living cell for growth and reproduction. It is usually represented y an intricate network connecting the involved iochemical species (called metaolites ). The pathways of the network are called metaolic pathways. In the metaolic engineering literature, it is widely accepted that despite their immense complexity, metaolic systems are characterized y their aility to reach stale steady states (quoted from Stephanopoulos, Aristidou & Nielsen (1997), Chapter 4). It should however e fair to recognize that a mathematical analysis of this fundamental staility property is a difficult question which was not much investigated. We shall limit ourselves to simple metaolic pathways which are made up of sequences of monomolecular enzymecatalysed reactions in the form X s X p. Those reactions can e inhiited y the presence of other metaolites in the network. Without this inhiition, the staility analysis is straightforward; we will then concentrate on networks with inhiition, like the one given y the aspartate aminoacid pathways (Umarger, 1978), see Figure 1. In this network, each produced aminoacid inhiits an enzyme of its own pathway. This action can e seen as a negative feedack, that regulates the ehavior of the network. Indeed, if we, for example, consider a large excess of isoleucine (X ), the reaction X 16 X 17 is shut down, so that the concentration of isoleucine is progressively reduced. In Section 2, a model of metaolic networks such as the one of Figure 1 will e presented. The equiliria of these models will then e studied in Section 3, followed y a staility analysis in Section 4, where gloal attractivity of a single equilirium is shown.
2 frag replacements x 1 x 2 In order to define the class of metaolic networks that we consider, we need the following definition from graph theory: x4 x 5 x 6 x 3 x 10 x 11 x 15 x 16 Definition 1. A directed graph is called an arorescence if, from a given node x, known as the root node, there is exactly one elementary path from this node to any other node y. x 7 x 8 x 9 x 12 x 13 x 14 x 17 x 18 x 19 x Fig. 1. Metaolic network representing the aspartate aminoacid pathways: the solid lines represent the reactions and the dashdotted lines the inhiition produced y the state at the start of the arrow onto the reaction that lies at the end of the arrow. The root of the metaolic pathway is x 1 (aspartate), and the products are the corresponding amino acids: lysine (x 9 ), methionine (x 14 ), threonine(x 16 ), and isoleucine (x ). The nongenericity of the staility of the equilirium is then illustrated in Section MODEL OF A METABOLIC NETWORK In our model of a metaolic network made of enzymecatalysed reactions in the form X s X p which are inhiited y other metaolites of the network, we will denote the inhiition factor y x [p], a n p dimensional vector containing the molar fractions of all the metaolites inhiiting the reaction X s X p. This reaction is characterized y a velocity ϕ sp (x s, x [p] ). On the other hand, some metaolites of the considered network are used in stages of the metaolism that are not modeled in the considered network. Therefore, we must include consumption terms in the model for those reactions in the form X s... We will generically denote those terms ϕ s0 (x s ). We impose that these reaction velocities satisfy the following assumption: Assumption 1. For all s, p such that the reaction X s X p elongs to the metaolic network, the function ϕ sp (x s, x [p] ) satisfies ϕ sp (0) = 0, is nondecreasing in x s for x s 0 and nonincreasing in x [p] j. Note that this must also e valid when the value of p is 0. The metaolic networks that we consider in this paper then satisfy the following assumption Assumption 2. the involved species are denoted X 1, X 2,, X n the graphic representation of the network (with the different metaolites as nodes and the different reactions as oriented edges) is an arorescence with X 1 as root the inhiition acting on a reaction X s X p only results from the action of metaolites from the (su)arorescence rooted in X p Stemming from the definition and known properties of arorescences, Assumption 2 has the following consequence on the class of metaolic network that we consider: (i) Each metaolite is produced y a single other metaolite; (ii) There is no cycle of reactions; With these definitions and notations, we shall now define a massalance dynamical model in the form ẋ = Φ(x) µx + ce 1 where x = (x 1,, x n ) T IR n, and x i denotes the molar fraction of the metaolite X i inside the cell. The factor µ 0 represents the specific growth rate of the cell: we assume that the cell metaolism is analyzed during a period of exponential cell growth with a constant specific growth rate µ. The vector e 1 = (1, 0,, 0) T and the scalar c denote the constant supply rate of the metaolite X 1 at the root of the network. The function Φ includes all the reaction velocities, whether they correspond to reactions inside the network or reactions consuming the metaolites of the network for use in susequent stages of the metaolism. In order to specify Φ(x), we introduce the following notations: Notation 1. P(j) = {k the reaction X j X k elongs to the network }. P(j) defines the set of all metaolites that are produced y reactions having X j as sustrate. If there is a consumption term in the form ϕ j0 (x j ) in the derivative of x j, the index 0 is included in P(j). A(j) = {k X k elongs to the arorescence with its root in X j }. 0 is not included in A(j).
3 It can easily e seen that P(j) \ {0} is a susets of A(j) under Assumption 2. From the arorescence structure, it is clear that we can separate the metaolites into three different families: the root X 1 : we suppose that there is a constant supply rate c of X 1 so that the corresponding massalance equation is the following: ẋ 1 = c ϕ 1k (x 1, x [k] ) µx 1 (1) k P(1) the intermediate metaolite X j, which is the result of the reaction X i X j : ẋ j = ϕ ij (x i, x [j] ) ϕ jk (x j, x [k] ) µx j (2) k P(j) the oundary metaolite X j (such that P(j) = {0}), which is the product of a reaction X i X j : ẋ j = ϕ ij (x i, x [j] ) ϕ j0 (x j ) µx j (3) Under Assumption 2, we can only have x [j] = x j or x [j] =. A particular case of this network is the metaolic chain with sequential feedack inhiition (cf. Chitour, Grognard & Bastin (03)) X 1 X 2 X n. with the last metaolite X n acting as an inhiitor of the first reaction X 1 X 2. In this case, all x [j] are empty, except x [2] = x n. 3. EQUILIBRIUM OF A METABOLIC NETWORK Based on equations (1)(2)(3), we can now compute the massalance of the whole arorescence: n d dt ( n x l ) = c µ x l ϕ k0 (x k ) (4) {k 0 P(k)} and of the arorescence that has its root in X j d dt ( x l ) = ϕ ij (x i, x [j] ) µ l A(j) l A(j) {k 0 P(k) and k A(j)} x l ϕ k0 (x k ) (5) Those expressions will e critical in the proof of the following proposition: Proposition 1. If Assumptions 1 and 2 are satisfied, then: (A) the system (1)(2)(3) is positive; (B) if all ϕ s,p (x s, x [p] ) (where p can e 0) are increasing in x s then there is at most one equilirium x = ( x 1,, x n ) of (1)(2)(3) in IR n +; (C) if µ > 0, then system (1)(2)(3) has a unique equilirium x = ( x 1,, x n ) in IR n +. Moreover, the solutions of (1)(2)(3) are ounded for any initial condition in IR n +. Proof: (A) is easily seen y considering the system on the oundaries of the positive orthant. The proofs of (B) and (C) are very similar. We write the proof for (B) and highlight the differences that arise for the proof of (C). We will first consider system (2)(3) with x 1 = x 1 as constant input. For any value of x 1, we will denote y ( x 2,, x n ) the equilirium of (2)(3); this equilirium is a function of x 1, so that we will state that it is ( x 2,, x n )( x 1 ). We will now show, y induction, that every element x i is an increasing function of x 1 (resp. nondecreasing in case (C)). The initial step of the proof considers the equilirium of (3) with x i = x i as constant input ϕ ij ( x i, x [j] ) ϕ j0 ( x j ) µ x j = 0 where x [j] = x j or x [j] =. When x i = 0, x j = 0 is the only solution. Also, the lefthand side of this equation is an increasing function of x i (resp. nondecreasing in case (C)) and a decreasing function of x j. It is then easily seen that, if we increase x i, x j needs also to e increased (resp.increased or kept constant) to keep this equality satisfied. We then have that, in this case, x j ( x i ) is an increasing function such that x j (0) = 0 (resp. nondecreasing function such that x j (0) = 0). When µ = 0 (which can only happen in case (B)), the definition of x j could e limited to an interval [0, x m i ). Let us now make the following induction hypothesis: for a given j, the functions x k ( x j ) are increasing (resp. nondecreasing) functions for all k A(j) with x k (0) = 0. We then study the equilirium of the massalance of the arorescence that has its root in X j. From (5): ϕ ij ( x i, x [j] ( x j )) µ x l ( x j ) l A(j) {k 0 P(k) and k A(j)} ϕ k0 ( x k ( x j )) = 0 With a similar argument to that of the initial step, we see that x j is an increasing (resp. nondecreasing) function of x i and that x j (0) = 0. The same can e said for all x k with k A j ecause they are already increasing (resp. nondecreasing) functions of x j. By induction, we then have that every x k ( x 1 ) is an increasing function of x 1 defined on the interval [0, x m 1 ) (resp. non decreasing function defined for all x 1 ). An equilirium of the whole system then has to satisfy the equilirium of the total massalance. From (4), this comes to:
4 n µ x l ( x 1 ) + {k 0 P(k)} ϕ k0 ( x k ( x 1 )) = c The system admits as many equiliria as this equation has roots. In case (B), the lefthand side is increasing from 0 when x 1 increases from 0 to x m 1 (ecause of the second term). Therefore, if there exists an equilirium, it is unique. In case (C), the lefthand side is strictly increasing from 0 to + when x 1 increases from 0 to + (ecause of the µ x 1 term), so that the equilirium exists and it is unique. The final point of (C) is a direct consequence of (4); this implies d n dt ( n x l ) c µ x l which clearly implies oundedness of the solutions when µ > 0. Uniqueness of the equilirium, especially when it is coupled with oundedness of solutions, gives hope of the possiility of having some general result aout the structural gloal asymptotic staility of the equilirium. In the next section, we study a special case of feedack inhiition, in which we will e ale to prove gloal attractivity. 4. STABILITY OF A CLASS OF METABOLIC NETWORKS In this section, we will study the staility of the equilirium of a class of metaolic networks elonging to the family that was descried in Section 2. In order to do that, we will first present a technical lemma. 4.1 Technical lemma In order to analyze the staility of the considered metaolic networks, we will use the smallgain theorem of De Leenheer, Angeli & Sontag (03) for the interconnection of monotone systems. Due to space limitation, the details of this theory are omitted here, and we only give two lemmas that we will use in the following sections (and which are proven in Grognard, Chitour & Bastin (03). Let us consider the staility of a system ẋ = f(x) x IR n (6) which is not cooperative, ut which satisfies the following assumption: Assumption 3. Each offdiagonal element of the Jacoian J of f(x) is signdefinite (independent of x). i, if J ij < 0, then replace x j in f i (x) with the constant v j. Define u 1 as the vector that contains all the constants v j that are necessary for the preceding construction (not necessarily all j have een required); we denote those constants y v s1,, v sl. From this construction, it directly appears that the system ẋ = F (x, u 1 ) (7) is cooperative (in the sense defined in De Leenheer et al. (03)) We then define the output of (7) as y 1 = (x s1,, x sl ) T. The interconnection of system (7) with the trivial system y 2 = u 2 (8) through the feedack connections u 1 = y 2 and u 2 = y 1 results in the original system (6). We can then show the following lemma Lemma 1. Suppose that the solutions of (6) are ounded and that system (7) has a unique gloally attractive equilirium for any constant input u 1 (this equilirium defines an input output characteristic ȳ 1 = k y1 (u 1 ) for system (7)). Suppose that the expression of y 1 = k y1 (u 1 ) is unknown, ut that it is known that it satisfies equation y 1 = G(y 1, u 1 ) then system (6) has a unique gloally attractive equilirium if there exists a norm. such that sup G + G < 1 (9) u 1, y 1 IR m y 1 u 1 and if the unique equilirium of system (7) with u 1 constant is gloally attractive for any u 1. Proof: see Grognard et al. (03) 4.2 Feedack inhiition The metaolic networks that we will consider for our staility study are made up of an arorescence of monomolecular enzymecatalysed reactions which satisfy the assumption that only one type of inhiition is present: the inhiition of the reactions that use X 1 as sustrate. Also, a metaolite X m can only inhiit the reaction X 1 X p, which is the first reaction of the unique elementary path linking X 1 to X m (this is a consequence of the third point of Assumption 2). This translates into Assumption 4. For all k P(1), x [k] =. We then uild a new system ẋ = F (x, v) according to the following construction: for all i, j with j For simplification of notations, we will denote the elements of P(1) as {k 1,, k r }.
5 A massalance model for such a network fits in the structure that was descried in the previous section, so that we know that there is a single equilirium. The model can e written as (1) (2)(3), ut is particularized due to the presence of Assumption 4: ẋ 1 = c r ϕ 1ki (x 1, x [ki] ) µx 1 (10) i=1 ẋ ki = ϕ 1ki (x 1, x [ki] ) ẋ k = ϕ lk (x l ) j P(k) j P(k i) ϕ kij(x ki ) µx ki (11) ϕ kj (x k ) µx k (12) where x ki is a product of x 1 and x k of x l x 1. We impose the oundedness of the partial derivatives of ϕ ij in the following assumption: Assumption 5. There exist d ij 0, α [kj] 0 such that 0 ϕ ij x i d ij for all i, j α [kj] ϕ 1k j x [kj] We now define a new notation 0 for all j r, n j Notation 2. From the arorescence structure, we know that there exists a unique path from X 1 to any metaolite X s. This path takes the form X 1 X kj X k X l X w X s ; if X s is an aritrary metaolite, we will store the indices of this path (without 1 and s) in C s ; alternatively, if x s corresponds to some x [kj], we will also denote this path C [kj]. Similarly, we denote g s (k) or g [kj] (k) the index of the metaolite that follows X k in the path that connects X 1 to X s. This allows for the following theorem Theorem 2. If Assumptions 1, 2, 4 and 5 are satisfied and n r kj ( ) d1,kj d [k µ + 1 kg j ] (k) µ + d [k j=1 =1 kg j ] (k) k C [k j ] < µ max ks,c α [ks] c (13) and ϕ 1ki is ounded for all i {1,, r} (0 ϕ 1ki B ki ), then the equilirium of system (10) (11)(12) is gloally attractive in the positive orthant. Proof: The proof of this Theorem is an application of Lemma 1 (see Grognard et al. (03)). The decomposition of (10)(11)(12) as in (7) is done y replacing all the inhiiting terms x [p] with the constant vector u 1. We see that, despite the fact that the inhiition is classically presented as a negative feedack that regulates the system, we have only een ale to prove gloal attractivity under the restrictive condition (13), while staility is easily seen in the asence of inhiition. This condition is very strong, especially if the specific growth rate is small. We will analyze this condition further in the next section. 5. LIMIT CYCLES IN METABOLIC NETWORKS Having otained the sufficient result for gloal attractivity of Theorem 2, it is relevant to ask two questions: is condition (13) necessary and sufficient, on the one hand, and is Theorem 2 still valid without condition (13) on the other hand? The answer to the first question is easily seen to e no : the staility of the equilirium is retained even if condition (13) is slightly violated (a few simulations of simple systems is already convincing). In this section, we will show that the answer to the second question is also no : without condition (13), the staility can e lost, so that we see that the staility of the metaolic networks is not a simple consequence of the structure of the models. Indeed, in this section, we shall exhiit an example where the equilirium ecomes unstale with a limit cycle (Hopf ifurcation) when condition (13) is not satisfied. We will concentrate on the staility of the equilirium of a simple sequential pathway of four metaolites without ranching and with sequential feedack inhiition (that was presented at the end of Section 2). Each metaolite produces a single other metaolite, and X 1 X 2 is inhiited y the last metaolite, X 4. We can directly apply Theorem 2 to this system: 1 3.2x 1 ẋ 1 = (x 4 /19) p 0.01x x x 1 ẋ 2 = 1 + (x 4 /19) p 1.4x x x x 2 ẋ 3 = 1.4x x 2 1.2x x x 3 ẋ 4 = 1.2x x 3 x x x 4 (14) 1 3.2x where we take ϕ 1 (x 1, x 4 ) = 1 1+(x 4/19) p 1+x 1. The parameter p mainly influences the maximal slope of 1 the inhiiting factor 1+(x 4/19) (which takes place in p x 4 = 19). Condition (13) ecomes ( ) α d1 µ µ + 1 d2 µ + d 2 d 3 µ + d 3 < 1 p < which is a condition very similar to what was otained in Chitour et al. (03). This condition is very strong ecause it is ased on a smallgain analysis. System (14) has a single equilirium in x = (19, 19, 19, 19) T.
6 At the equilirium, we can see that the characteristics polynomial of the Jacoian matrix has the form (s )(s )(s )(s ) p(s ) which is Hurwitz for p < , and is not Hurwitz for p larger than that value. This transition from a stale to an unstale equilirium is illustrated on Figure 2, where the time responses of the four states is illustrated for the value of p = 0 (no inhiition), p = 10 (weak inhiition), and p = 60 (strong inhiition). In the latter case, oscillations appear. This corresponds to a limit cycle in the statespace. A Hopf ifurcation has taken place in p = Despite this oscillation, the reaction rates for the three reactions X 2 X 3, X 3 X 4, and X 4... are close to their maximum after the transient. The only limiting reaction is X 1 X 2 which, due to the inhiition is far from its maximum reaction rate. (POSTA 03), edited y Luca Benvenuti, Alerto De Santis and Lorenzo Farina, Lecture Notes on Control and Information Sciences vol. 294, SpringerVerlag, Heidelerg, 03. De Leenheer P., Angeli D. and Sontag E.D., Smallgain theorems for predatorprey systems, in Positive Systems and Applications... (see Reference Chitour et al.) Grognard F., Chitour Y., Bastin G., Equiliria and staility analysis of a ranched metaolic network with feedack inhiition, sumitted to Mathematical Biosciences. Stephanopoulos G., Aristidou A. and Nielsen J., Metaolic Engineering: Principles and Methodologies, Academic Press, Umarger H.E., Amino acid iosynthesis and regulation, Ann. Rev. Biochem., vol. 47, , x1 50 x x3 x Fig. 2. Evolution of the states of system (14) for p = 0 (dotted line), p=10 (dashdotted line) and p = 60 (solid line). 6. CONCLUSION In this paper, we have shown that a large class of models of metaolic system only has a single equilirium. We then have proven that, under a small gain condition, this equilirium is gloally attractive in the particular case where inhiition only acts on reaction having the root as sustrate. Finally, we have shown that staility of this equilirium is not a generic property of the metaolic systems: a condition needs to e imposed on the parameters to have staility (similar to the small gain condition that we found). 7. REFERENCES Chitour Y., Grognard F. and Bastin G., Staility analysis of a metaolic model with sequential feedack inhiition, in Positive Systems and Applications. Proceedings of the First Multidisciplinary Symposium on Positive Systems
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