Equilibria and stability analysis of a branched metabolic network with feedback inhibition


 Gwenda Walker
 1 years ago
 Views:
Transcription
1 Equiliria and staility analysis of a ranched metaolic network with feedack inhiition Frédéric Grognard, Yacine Chitour, Georges Bastin To cite this version: Frédéric Grognard, Yacine Chitour, Georges Bastin. Equiliria and staility analysis of a ranched metaolic network with feedack inhiition. 9th International Symposium on Computer Applications in Biotechnology, Mar 04, Nancy, France. <hal > HAL Id: hal https://hal.inria.fr/hal Sumitted on 5 Dec 14 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are pulished or not. The documents may come from teaching and research institutions in France or aroad, or from pulic or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, puliés ou non, émanant des étalissements d enseignement et de recherche français ou étrangers, des laoratoires pulics ou privés.
2 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.
3 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).
4 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:
5 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 }.
6 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.
7 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
Discussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski.
Discussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski. Fabienne Comte, Celine Duval, Valentine GenonCatalot To cite this version: Fabienne
More informationMobility management and vertical handover decision making in heterogeneous wireless networks
Mobility management and vertical handover decision making in heterogeneous wireless networks Mariem Zekri To cite this version: Mariem Zekri. Mobility management and vertical handover decision making in
More informationibalanceabf: a SmartphoneBased AudioBiofeedback Balance System
ibalanceabf: a SmartphoneBased AudioBiofeedback Balance System Céline Franco, Anthony Fleury, PierreYves Guméry, Bruno Diot, Jacques Demongeot, Nicolas Vuillerme To cite this version: Céline Franco,
More informationAnalyse échographique de la cicatrisation tendineuse après réparation arthroscopique de la coiffe des rotateurs
Analyse échographique de la cicatrisation tendineuse après réparation arthroscopique de la coiffe des rotateurs Martin Schramm To cite this version: Martin Schramm. Analyse échographique de la cicatrisation
More informationLa mort du narrateur et l interprétation du roman. L exemple de Pedro Páramo de Juan Rulfo
La mort du narrateur et l interprétation du roman. L exemple de Pedro Páramo de Juan Rulfo Sylvie Patron To cite this version: Sylvie Patron. La mort du narrateur et l interprétation du roman. L exemple
More informationVirginia Woolf, Comment devraiton lire un livre? ou le théoricien du commun
Virginia Woolf, Comment devraiton lire un livre? ou le théoricien du commun FrançoisRonan Dubois To cite this version: FrançoisRonan Dubois. Virginia Woolf, Comment devraiton lire un livre? ou le théoricien
More informationA usage coverage based approach for assessing product family design
A usage coverage based approach for assessing product family design Jiliang Wang To cite this version: Jiliang Wang. A usage coverage based approach for assessing product family design. Other. Ecole Centrale
More informationA graph based framework for the definition of tools dealing with sparse and irregular distributed datastructures
A graph based framework for the definition of tools dealing with sparse and irregular distributed datastructures Serge Chaumette, JeanMichel Lepine, Franck Rubi To cite this version: Serge Chaumette,
More informationANIMATED PHASE PORTRAITS OF NONLINEAR AND CHAOTIC DYNAMICAL SYSTEMS
ANIMATED PHASE PORTRAITS OF NONLINEAR AND CHAOTIC DYNAMICAL SYSTEMS JeanMarc Ginoux To cite this version: JeanMarc Ginoux. ANIMATED PHASE PORTRAITS OF NONLINEAR AND CHAOTIC DYNAMICAL SYSTEMS. A.H. Siddiqi,
More informationMinkowski Sum of Polytopes Defined by Their Vertices
Minkowski Sum of Polytopes Defined by Their Vertices Vincent Delos, Denis Teissandier To cite this version: Vincent Delos, Denis Teissandier. Minkowski Sum of Polytopes Defined by Their Vertices. Journal
More informationExpanding Renewable Energy by Implementing Demand Response
Expanding Renewable Energy by Implementing Demand Response Stéphanie Bouckaert, Vincent Mazauric, Nadia Maïzi To cite this version: Stéphanie Bouckaert, Vincent Mazauric, Nadia Maïzi. Expanding Renewable
More informationAutoScaling, Load Balancing and Monitoring in Commercial and OpenSource Clouds
AutoScaling, Load Balancing and Monitoring in Commercial and OpenSource Clouds Eddy Caron, Luis RoderoMerino, Frédéric Desprez, Adrian Muresan To cite this version: Eddy Caron, Luis RoderoMerino, Frédéric
More informationQASM: a Q&A Social Media System Based on Social Semantics
QASM: a Q&A Social Media System Based on Social Semantics Zide Meng, Fabien Gandon, Catherine FaronZucker To cite this version: Zide Meng, Fabien Gandon, Catherine FaronZucker. QASM: a Q&A Social Media
More informationManaging Risks at Runtime in VoIP Networks and Services
Managing Risks at Runtime in VoIP Networks and Services Oussema Dabbebi, Remi Badonnel, Olivier Festor To cite this version: Oussema Dabbebi, Remi Badonnel, Olivier Festor. Managing Risks at Runtime in
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationLes effets de la relaxation sur les apprentissages à l école
Les effets de la relaxation sur les apprentissages à l école Laurine Bouchez To cite this version: Laurine Bouchez. Les effets de la relaxation sur les apprentissages à l école. Éducation. 2012.
More informationArts et politique sous Louis XIV
Arts et politique sous Louis XIV Marie Tourneur To cite this version: Marie Tourneur. Arts et politique sous Louis XIV. Éducation. 2013. HAL Id: dumas00834060 http://dumas.ccsd.cnrs.fr/dumas00834060
More informationPrice of Anarchy in NonCooperative Load Balancing
Price of Anarchy in NonCooperative Load Balancing Urtzi Ayesta, Olivier Brun, Balakrishna Prabhu To cite this version: Urtzi Ayesta, Olivier Brun, Balakrishna Prabhu. Price of Anarchy in NonCooperative
More informationLes identités féminines dans le conte traditionnel occidental et ses réécritures : une perception actualisée
Les identités féminines dans le conte traditionnel occidental et ses réécritures : une perception actualisée des élèves Sarah Brasseur To cite this version: Sarah Brasseur. Les identités féminines dans
More informationFPHadoop: Efficient Execution of Parallel Jobs Over Skewed Data
FPHadoop: Efficient Execution of Parallel Jobs Over Skewed Data Miguel LirozGistau, Reza Akbarinia, Patrick Valduriez To cite this version: Miguel LirozGistau, Reza Akbarinia, Patrick Valduriez. FPHadoop:
More informationOptimization results for a generalized coupon collector problem
Optimization results for a generalized coupon collector problem Emmanuelle Anceaume, Yann Busnel, Ernst SchulteGeers, Bruno Sericola To cite this version: Emmanuelle Anceaume, Yann Busnel, Ernst SchulteGeers,
More informationKnowledge based tool for modeling dynamic systems in education
Knowledge based tool for modeling dynamic systems in education Krassen Stefanov, Galina Dimitrova To cite this version: Krassen Stefanov, Galina Dimitrova. Knowledge based tool for modeling dynamic systems
More informationThe truck scheduling problem at crossdocking terminals
The truck scheduling problem at crossdocking terminals Lotte Berghman,, Roel Leus, Pierre Lopez To cite this version: Lotte Berghman,, Roel Leus, Pierre Lopez. The truck scheduling problem at crossdocking
More informationFautil des cyberarchivistes, et quel doit être leur profil professionnel?
Fautil des cyberarchivistes, et quel doit être leur profil professionnel? JeanDaniel Zeller To cite this version: JeanDaniel Zeller. Fautil des cyberarchivistes, et quel doit être leur profil professionnel?.
More informationANALYSIS OF SNOEKKOSTER (H) RELAXATION IN IRON
ANALYSIS OF SNOEKKOSTER (H) RELAXATION IN IRON J. San Juan, G. Fantozzi, M. No, C. Esnouf, F. Vanoni To cite this version: J. San Juan, G. Fantozzi, M. No, C. Esnouf, F. Vanoni. ANALYSIS OF SNOEKKOSTER
More informationAdditional mechanisms for rewriting onthefly SPARQL queries proxy
Additional mechanisms for rewriting onthefly SPARQL queries proxy Arthur VaisseLesteven, Bruno Grilhères To cite this version: Arthur VaisseLesteven, Bruno Grilhères. Additional mechanisms for rewriting
More informationVR4D: An Immersive and Collaborative Experience to Improve the Interior Design Process
VR4D: An Immersive and Collaborative Experience to Improve the Interior Design Process Amine Chellali, Frederic Jourdan, Cédric Dumas To cite this version: Amine Chellali, Frederic Jourdan, Cédric Dumas.
More informationOverview of modelbuilding strategies in population PK/PD analyses: 20022004 literature survey.
Overview of modelbuilding strategies in population PK/PD analyses: 20022004 literature survey. Céline Dartois, Karl Brendel, Emmanuelle Comets, Céline Laffont, Christian Laveille, Brigitte Tranchand,
More informationOnline vehicle routing and scheduling with continuous vehicle tracking
Online vehicle routing and scheduling with continuous vehicle tracking Jean Respen, Nicolas Zufferey, JeanYves Potvin To cite this version: Jean Respen, Nicolas Zufferey, JeanYves Potvin. Online vehicle
More informationDistributed network topology reconstruction in presence of anonymous nodes
Distributed network topology reconstruction in presence of anonymous nodes ThiMinh Dung Tran, Alain Y Kibangou To cite this version: ThiMinh Dung Tran, Alain Y Kibangou Distributed network topology reconstruction
More informationAn Automatic Reversible Transformation from Composite to Visitor in Java
An Automatic Reversible Transformation from Composite to Visitor in Java Akram To cite this version: Akram. An Automatic Reversible Transformation from Composite to Visitor in Java. CIEL 2012, P. Collet,
More informationPricing access to the Internet with partial information
Pricing access to the Internet with partial information Ilaria Brunetti, Eitan Altman, Majed Haddad To cite this version: Ilaria Brunetti, Eitan Altman, Majed Haddad. Pricing access to the Internet with
More informationUnderstanding Big Data Spectral Clustering
Understanding Big Data Spectral Clustering Romain Couillet, Florent BenaychGeorges To cite this version: Romain Couillet, Florent BenaychGeorges Understanding Big Data Spectral Clustering 205 IEEE 6th
More informationInformation Technology Education in the Sri Lankan School System: Challenges and Perspectives
Information Technology Education in the Sri Lankan School System: Challenges and Perspectives Chandima H. De Silva To cite this version: Chandima H. De Silva. Information Technology Education in the Sri
More informationProcess Assessment Issues of the ISO/IEC emerging standard
Process Assessment Issues of the ISO/IEC 29110 emerging standard Vincent Ribaud, Philippe Saliou To cite this version: Vincent Ribaud, Philippe Saliou. Process Assessment Issues of the ISO/IEC 29110 emerging
More informationTerritorial Intelligence and Innovation for the SocioEcological Transition
Territorial Intelligence and Innovation for the SocioEcological Transition JeanJacques Girardot, Evelyne Brunau To cite this version: JeanJacques Girardot, Evelyne Brunau. Territorial Intelligence and
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationω h (t) = Ae t/τ. (3) + 1 = 0 τ =.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Lecture 2 Solving the Equation of Motion Goals for today Modeling of the 2.004 La s rotational
More informationDEM modeling of penetration test in static and dynamic conditions
DEM modeling of penetration test in static and dynamic conditions Quoc Anh Tran, Bastien Chevalier, Pierre Breul To cite this version: Quoc Anh Tran, Bastien Chevalier, Pierre Breul. DEM modeling of penetration
More informationUse of tabletop exercise in industrial training disaster.
Use of tabletop exercise in industrial training disaster. Alexis Descatha, Thomas Loeb, François Dolveck, NathalieSybille Goddet, Valerie Poirier, Michel Baer To cite this version: Alexis Descatha, Thomas
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationFrom Microsoft Word 2003 to Microsoft Word 2007: Design Heuristics, Design Flaws and Lessons Learnt
From Microsoft Word 2003 to Microsoft Word 2007: Design Heuristics, Design Flaws and Lessons Learnt YinLeng Theng, Eng Kiat Ting, Xuehong Tao To cite this version: YinLeng Theng, Eng Kiat Ting, Xuehong
More informationNew implementions of predictive alternate analog/rf test with augmented model redundancy
New implementions of predictive alternate analog/rf test with augmented model redundancy Haithem Ayari, Florence Azais, Serge Bernard, Mariane Comte, Vincent Kerzerho, Michel Renovell To cite this version:
More informationGDS Resource Record: Generalization of the Delegation Signer Model
GDS Resource Record: Generalization of the Delegation Signer Model Gilles Guette, Bernard Cousin, David Fort To cite this version: Gilles Guette, Bernard Cousin, David Fort. GDS Resource Record: Generalization
More informationWhat Development for Bioenergy in Asia: A Longterm Analysis of the Effects of Policy Instruments using TIAMFR model
What Development for Bioenergy in Asia: A Longterm Analysis of the Effects of Policy Instruments using TIAMFR model Seungwoo Kang, Sandrine Selosse, Nadia Maïzi To cite this version: Seungwoo Kang, Sandrine
More informationPhysicians balance billing, supplemental insurance and access to health care
Physicians balance billing, supplemental insurance and access to health care Izabela Jelovac To cite this version: Izabela Jelovac. Physicians balance billing, supplemental insurance and access to health
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationCounting Primes whose Sum of Digits is Prime
2 3 47 6 23 Journal of Integer Sequences, Vol. 5 (202), Article 2.2.2 Counting Primes whose Sum of Digits is Prime Glyn Harman Department of Mathematics Royal Holloway, University of London Egham Surrey
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More informationSELECTIVELY ABSORBING COATINGS
SELECTIVELY ABSORBING COATINGS J. Vuletin, P. Kuli ik, M. Bosanac To cite this version: J. Vuletin, P. Kuli ik, M. Bosanac. SELECTIVELY ABSORBING COATINGS. Journal de Physique Colloques, 1981, 42 (C1),
More informationAn update on acoustics designs for HVAC (Engineering)
An update on acoustics designs for HVAC (Engineering) Ken MARRIOTT To cite this version: Ken MARRIOTT. An update on acoustics designs for HVAC (Engineering). Société Française d Acoustique. Acoustics 2012,
More informationELECTRONIC STRUCTURE OF AMORPHOUS SiOx
ELECTRONIC STRUCTURE OF AMORPHOUS SiOx E. Martínez, F. Ynduráin To cite this version: E. Martínez, F. Ynduráin. ELECTRONIC STRUCTURE OF AMORPHOUS SiOx. Journal de Physique Colloques, 1981, 42 (C4), pp.c41021c41024.
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationCracks detection by a moving photothermal probe
Cracks detection by a moving photothermal probe J. Bodnar, M. Egée, C. Menu, R. Besnard, A. Le Blanc, M. Pigeon, J. Sellier To cite this version: J. Bodnar, M. Egée, C. Menu, R. Besnard, A. Le Blanc, et
More informationThe CHI 2013 Interactive Schedule
The CHI 2013 Interactive Schedule Arvind Satyanarayan, Daniel Strazzulla, Clemens Klokmose, Michel BeaudouinLafon, Wendy E. Mackay To cite this version: Arvind Satyanarayan, Daniel Strazzulla, Clemens
More informationGlobal Identity Management of Virtual Machines Based on Remote Secure Elements
Global Identity Management of Virtual Machines Based on Remote Secure Elements Hassane Aissaoui, P. Urien, Guy Pujolle To cite this version: Hassane Aissaoui, P. Urien, Guy Pujolle. Global Identity Management
More informationAn aggregator for distributed energy storage units under multiple constraints in the nice grid demonstrator
An aggregator for distributed energy storage units under multiple constraints in the nice grid demonstrator Andrea Michiorri, Georges Kariniotakis, Fiona Foucault To cite this version: Andrea Michiorri,
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationLoad Balancing and Switch Scheduling
EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load
More informationMathilde Patteyn. HAL Id: dumas
L exercice du théâtre à l école Mathilde Patteyn To cite this version: Mathilde Patteyn. L exercice du théâtre à l école. Éducation. 2013. HAL Id: dumas00861849 http://dumas.ccsd.cnrs.fr/dumas00861849
More informationAligning subjective tests using a low cost common set
Aligning subjective tests using a low cost common set Yohann Pitrey, Ulrich Engelke, Marcus Barkowsky, Romuald Pépion, Patrick Le Callet To cite this version: Yohann Pitrey, Ulrich Engelke, Marcus Barkowsky,
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationOligopoly Games under Asymmetric Costs and an Application to Energy Production
Oligopoly Games under Asymmetric Costs and an Application to Energy Production Andrew Ledvina Ronnie Sircar First version: July 20; revised January 202 and March 202 Astract Oligopolies in which firms
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationA Passivity Measure Of Systems In Cascade Based On Passivity Indices
49th IEEE Conference on Decision and Control December 57, Hilton Atlanta Hotel, Atlanta, GA, USA A Passivity Measure Of Systems In Cascade Based On Passivity Indices Han Yu and Panos J Antsaklis Abstract
More informationThe Basics of Graphical Models
The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures
More informationAdvantages and disadvantages of elearning at the technical university
Advantages and disadvantages of elearning at the technical university Olga Sheypak, Galina Artyushina, Anna Artyushina To cite this version: Olga Sheypak, Galina Artyushina, Anna Artyushina. Advantages
More informationRunning an HCI Experiment in Multiple Parallel Universes
Running an HCI Experiment in Multiple Parallel Universes,, To cite this version:,,. Running an HCI Experiment in Multiple Parallel Universes. CHI 14 Extended Abstracts on Human Factors in Computing Systems.
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationSynthesis of Microwave filters: a novel approach based on computer algebra
Synthesis of Microwave filters: a novel approach based on computer algebra Fabien Seyfert To cite this version: Fabien Seyfert. Synthesis of Microwave filters: a novel approach based on computer algebra.
More informationOn the DStability of Linear and Nonlinear Positive Switched Systems
On the DStability of Linear and Nonlinear Positive Switched Systems V. S. Bokharaie, O. Mason and F. Wirth Abstract We present a number of results on Dstability of positive switched systems. Different
More informationVirtual plants in high school informatics Lsystems
Virtual plants in high school informatics Lsystems Janka Majherov To cite this version: Janka Majherov. Virtual plants in high school informatics Lsystems. Michael E. Auer. Conference ICL2007, September
More informationApplicationAware Protection in DWDM Optical Networks
ApplicationAware Protection in DWDM Optical Networks Hamza Drid, Bernard Cousin, Nasir Ghani To cite this version: Hamza Drid, Bernard Cousin, Nasir Ghani. ApplicationAware Protection in DWDM Optical
More informationCobi: Communitysourcing LargeScale Conference Scheduling
Cobi: Communitysourcing LargeScale Conference Scheduling Haoqi Zhang, Paul André, Lydia Chilton, Juho Kim, Steven Dow, Robert Miller, Wendy E. Mackay, Michel BeaudouinLafon To cite this version: Haoqi
More informationDetermination of the normalization level of database schemas through equivalence classes of attributes
Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,
More informationStudy on Cloud Service Mode of Agricultural Information Institutions
Study on Cloud Service Mode of Agricultural Information Institutions Xiaorong Yang, Nengfu Xie, Dan Wang, Lihua Jiang To cite this version: Xiaorong Yang, Nengfu Xie, Dan Wang, Lihua Jiang. Study on Cloud
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationTreerepresentation of set families and applications to combinatorial decompositions
Treerepresentation of set families and applications to combinatorial decompositions BinhMinh BuiXuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationEffets du monoxyde d azote inhalé sur le cerveau en développement chez le raton
Effets du monoxyde d azote inhalé sur le cerveau en développement chez le raton Gauthier Loron To cite this version: Gauthier Loron. Effets du monoxyde d azote inhalé sur le cerveau en développement chez
More informationMinimumDensity Identifying Codes in Square Grids
MinimumDensity Identifying Codes in Square Grids Marwane Bouznif, Frédéric Havet, Myriam Preissmann To cite this version: Marwane Bouznif, Frédéric Havet, Myriam Preissmann. MinimumDensity Identifying
More informationComputation of Switch Time Distributions in Stochastic Gene Regulatory Networks
Computation of Switch Time Distributions in Stochastic Gene Regulatory Networks Brian Munsky and Mustafa Khammash Center for Control, Dynamical Systems and Computation University of California Santa Barbara,
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationTowards Unified Tag Data Translation for the Internet of Things
Towards Unified Tag Data Translation for the Internet of Things Loïc Schmidt, Nathalie Mitton, David SimplotRyl To cite this version: Loïc Schmidt, Nathalie Mitton, David SimplotRyl. Towards Unified
More informationInformation behaviour and practices of PhD students
Information behaviour and practices of PhD students Thea Marie Drachen, Asger Væring Larsen, Eystein Gullbekk, Hilde Westbye, Karin Lach To cite this version: Thea Marie Drachen, Asger Væring Larsen, Eystein
More informationAn integrated planningsimulationarchitecture approach for logistics sharing management: A case study in Northern Thailand and Southern China
An integrated planningsimulationarchitecture approach for logistics sharing management: A case study in Northern Thailand and Southern China Pree Thiengburanathum, Jesus GonzalezFeliu, Yacine Ouzrout,
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationThe binomial interpolated lattice method fro step double barrier options
The binomial interpolated lattice method fro step double barrier options Elisa Appolloni, Gaudenzi Marcellino, Antonino Zanette To cite this version: Elisa Appolloni, Gaudenzi Marcellino, Antonino Zanette.
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationClass notes Program Analysis course given by Prof. Mooly Sagiv Computer Science Department, Tel Aviv University second lecture 8/3/2007
Constant Propagation Class notes Program Analysis course given by Prof. Mooly Sagiv Computer Science Department, Tel Aviv University second lecture 8/3/2007 Osnat Minz and Mati Shomrat Introduction This
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationInterlibrary loan and document supply in France the Montpellier meeting
Interlibrary loan and document supply in France the Montpellier meeting Joachim Schöpfel To cite this version: Joachim Schöpfel. Interlibrary loan and document supply in France the Montpellier meeting.
More informationGenerating models of a matched formula with a polynomial delay
Generating models of a matched formula with a polynomial delay Petr Savicky Institute of Computer Science, Academy of Sciences of Czech Republic, Pod Vodárenskou Věží 2, 182 07 Praha 8, Czech Republic
More informationGuillaume et Rainouart : figures du guerrier démesuré
Guillaume et Rainouart : figures du guerrier démesuré Florian Jingand To cite this version: Florian Jingand. Guillaume et Rainouart : figures du guerrier démesuré. Linguistique. Université Paul Valéry
More information