Global stability of complex balanced mechanisms

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  • Journal of Mathematical Chemistry 27 (2000) 89110 89

    Global stability of complex balanced mechanisms

    D. Siegel and D. MacLean Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

    Received 10 May 1999

    We prove that the -limit set of any solution of a complex balanced chemical reactionmechanism contains either a unique, positive complex balanced equilibrium point, or bound-ary complex balanced equilibrium points. Then, using this result, we are able to provideglobal stability results for three enzymatic mechanisms.

    1. Introduction

    The qualitative behavior of the solutions of chemical kinetic systems is studiedextensively in [7,10,11,15,17]. In particular, it is proved by Volpert in [15,17] thatthe -limit set of any solution of a detailed balanced chemical reaction mechanismconsists of a single positive point of detailed balanced equilibrium, or of boundarydetailed balanced equilibrium points (theorem 3 in [15], and the theorem in section 3.4in [17]). In this paper, we will present the equivalent result for complex balancedchemical reaction mechanisms. Complex balanced mechanisms were first studied byHorn, Jackson and Feinberg [7,10,11]. Furthermore, we will show that Volpertstheorem for detailed balanced mechanisms can be completely recovered by our result,after we prove a key relationship between detailed balanced mechanisms and complexbalanced mechanisms.

    In section 4, we will use our -limit set theorem to prove the global asymptoticstability of a subclass of complex balanced mechanisms. This is the group of complexbalanced mechanisms that do not admit boundary equilibrium points. Moreover, wewill show that three different enzymatic mechanisms, two of which involve inhibitors,are from this particular subclass, and hence, are globally asymptotically stable.

    2. Background

    The following is a short introduction to the terminology and notation of chemicalkinetics. For the most part, the notation to be used has been adopted from [1,2]. Supported by a Natural Sciences and Engineering Research Council of Canada Research Grant. Supported by a Natural Sciences and Engineering Research Council of Canada Post-Graduate

    Scholarship.

    J.C. Baltzer AG, Science Publishers

  • 90 D. Siegel, D. MacLean / Global stability of complex balanced mechanisms

    However, to concisely prove some results in this paper, we will also use the notationfrom [11]. Hence, in some instances below the same term will be defined with twodifferent notations. See [6,8,11] for a complete introduction to chemical kinetics.

    2.1. Chemical reaction mechanisms

    In general, a chemical reaction mechanism can be represented bymi=1

    zp(j)iAikj

    mi=1

    zp+(j)iAi for all j = 1, . . . , r, (1)

    where Ai, i = 1, . . . ,m, are the species involved in the r reactions of the mechanism.The linear combinations of species to the left and right of the reaction arrow are referredto as complexes, and zp(j)i (respectively, zp+(j)i) is the stoichiometric coefficient ofspecies Ai in the reactant complex (respectively, product complex) of the jth reactionof the mechanism. Finally, kj is rate constant of the jth reaction [1,2].

    If we assume a mechanism consists of n distinct complexes, (1) can also berepresented by

    Cik(i,j) Cj for all i, j = 1, . . . ,n, (2)

    where Cl =m

    k=1 zlkAk is a complex, and k(i, j) is the rate constant for the reactionwith reactant complex Ci and product complex Cj [11]. Note that k(i, j) = 0 if eitheri = j, or the mechanism does not contain a reaction with Ci as the reactant and Cj asthe product. Otherwise, k(i, j) > 0.

    A chemical reaction mechanism can also be expressed as a graph theoreticaldigraph. See [3,9] for the definition of a digraph, and the following two definitionsare from [7,10,11]:

    Definition 2.1. The HJF-graph of a chemical reaction mechanism is a digraph, wherethe vertices represent the distinct complexes of the mechanism and the arcs indicatethe reactions between the complexes.

    Definition 2.2. The linkage classes of a chemical reaction mechanism are the con-nected components of the HJF-graph. The number of linkage classes in a mechanismwill be denoted by `.

    If we assume the chemical reaction mechanism (1) is endowed with mass actionkinetics, the evolution of its species concentrations xi(t), i = 1, . . . ,m, can be modeledby the system of differential equations

    x = F (x) =rj=1Rjvj(x), (3)

  • D. Siegel, D. MacLean / Global stability of complex balanced mechanisms 91

    where

    Rj = zp+(j) zp(j) (4)is the reaction vector, and

    vj(x) = kjmi=1

    xzp(j)ii (5)

    is the rate function of the jth reaction of the mechanism.

    2.2. Equilibria

    Two special types of equilibria of (3) will be considered [7,10,11]:

    Definition 2.3. A concentration x will be called a complex balanced equilibrium pointof (3) if and only if for every complex p of the mechanism, the sum of the rates ofthe reactions with p as a reactant equals the sum of the rates of the reactions with pas a product. That is,

    p+(j)=pvj(x) =

    p(j)=p

    vj(x),

    where vj(x) is defined in (5). Alternatively,nj=1

    k(j, i)(x)(j) = (x)(i) nj=1

    k(i, j) for all i, . . . ,n,

    where (x)(k) = mi=1(xi )zki .Definition 2.4. A concentration x will be called a detailed balanced equilibrium pointof (3) if and only if the rate of the reaction p p equals the rate of the reactionp p. That is, vpp(x) = vpp(x), or alternatively,

    k(i, j)(x)(i) = k(j, i)(x)(j) for all i, j = 1, . . . ,n.Remark 2.5. We will say that a chemical reaction mechanism is complex balanced(respectively, detailed balanced) if it admits a positive complex balanced equilibriumpoint (respectively, positive detailed balanced equilibrium point) for at least one set ofpositive rate constants.

    Before theorems on the structural nature of complex balanced and detailed bal-anced mechanisms can be given, the following graph theoretical terminology and the-orem are required [3]:

    Definition 2.6. If a = (u, v) is an arc of a digraph D, we say u is adjacent to v and vis adjacent from u.

  • 92 D. Siegel, D. MacLean / Global stability of complex balanced mechanisms

    Definition 2.7. The outdegree od(v) of a vertex v of a digraph D is the number ofvertices of D that are adjacent from v.

    Definition 2.8. Let u and v be (not necessarily distinct) vertices of a digraph D. A uvwalk of D is a finite, alternating sequence

    u = u0, a1,u1, a2, . . . ,un1, an,un = v

    of vertices and arcs, beginning with u and ending with v, such that ai = (ui1,ui) fori = 1, . . . ,n.

    Definition 2.9. A vertex v is said to be reachable from a vertex u in a digraph D ifD contains a uv walk.

    Definition 2.10. A digraph D is strongly connected if for every two distinct verticesof D, each vertex is reachable from the other.

    Definition 2.11. If Di is a subdigraph of D and Di is strongly connected, then Di iscalled a strong component of D.

    Definition 2.12. Let D1,D2, . . . ,Dn be the strong components of D. Then, the con-densation D of D is that digraph whose vertices u1,u2, . . . ,un can be put in one-to-one correspondence with the strong components, where (ui,uj) is an arc of D, i 6= j,if and only if some vertex of Di is adjacent to at least one vertex of Dj .

    Theorem 2.13 (Theorems 15.4 and 15.8 in [3]).The condensation of every digraphcontains at least one vertex of outdegree zero.

    Additionally, the following chemical kinetics terminology is needed [8,10]:

    Definition 2.14. A chemical reaction mechanism is said to be weakly reversible if inits corresponding HJF-graph, every vertex is reachable from every other vertex.

    Definition 2.15. A chemical reaction mechanism is said to be reversible if in its cor-responding HJF-graph, the vertex v being adjacent to the vertex u implies that thevertex u is adjacent to the vertex v.

    We are now prepared to give two results for complex balanced and detailedbalanced mechanisms. Note, if x Rm, by x > 0 we mean that xi > 0 for alli = 1, . . . ,m.

    Theorem 2.16. Every complex balanced mechanism is weakly reversible.

  • D. Siegel, D. MacLean / Global stability of complex balanced mechanisms 93

    Proof. Let a be a positive, complex balanced equilibrium point of the mechanism.Hence, according to definition 2.3,

    nj=1

    k(j, i)a(j) = a(i)nj=1

    k(i, j)

    for all i = 1, . . . ,n, or equivalently,

    ni=1

    k(1, i) k(2, 1) . . . k(n, 1)

    k(1, 2) ni=1

    k(2, i) . . . k(n, 2).

    .

    .

    .

    .

    .

    ..

    .

    .

    .

    .

    k(1,n) k(2,n) . . . ni=1

    k(n, i)

    a(1)

    a(2).

    .

    .

    a(n)

    =

    00.

    .

    .

    0

    . (6)

    Notice that the sum of the entries in each column of the above matrix is zero.Now, suppose the mechanism is not weakly reversible. This implies that the

    HJF-graph of at least one linkage class in the mechanism is not strongly connected,and hence, the condensation, Dl , of that linkage class consists of more than one vertex.Furthermore, according to theorem 2.13, there exists at least one vertex, say u Dl ,such that od(u) = 0.

    Let Su be the indexing set of complexes in the strongly connected subdigraphcorresponding to u Dl , and let k = |Su|. If we let v1 be a k-vector with entriesa(i), i Su, and v2 be an (n k)-vector with entries a(j), j / Su, then (6) can berewritten as [

    A B0 C

    ] [v1v2

    ]=

    [00

    ], (7)

    where A is a kk matrix with form identical to the matrix in (6). It follows then that1 Av1 = 0, where 1 is a k-vector with each component equal to one. Furthermore,from (7) we have

    1 Bv2 = 0, (8)where v2 > 0. However, 1Bv2 > 0 due to the fact that B has nonnegative entries andat least one positive entry, since linkage classes are connected. Thus, a contradictionis reached and the proof is complete.

    Theorem 2.17. Every detailed balanced mechanism is reversible.

    Proof. Follows directly from definition 2.4.

  • 94 D. Siegel, D. MacLean / Global stability of complex balanced mechanisms

    2.3. Positivity and compatibility classes

    The solutions to (3) are not able to wander freely Rm, they are restricted in twodifferent ways.

    First of all, as proven by Volpert in [16,17], solutions are limited to Pm, wherePm = {x Rm | xi > 0 for i = 1, . . . ,m}. The statement and proof of their followingresult requires the use of the mechanisms corresponding Volpert graph. A Volpertgraph is a finite directed bipartite graph (see [9]), where the vertex ai represents thespecies Ai in the mechanism.

    Theorem 2.18 (Strict positiveness). If x(t) is the solution to (3), together with a setof nonnegative initial concentrations on the interval [0, ), then xi(t) > 0 (0 < t < )for all vertices ai that are reachable from A0, and xi(t) 0 for all vertices ai that arenonreachable from A0.

    Secondly, solutions to (3) are confined to stoichiometric compatibility classes [8,10]:

    Definition 2.19. The stoichiometric subspace for a chemical reaction mechanism isthe linear subspace S Rm defined by

    S := span{Rj Rm: j = 1, . . . , r},

    where Rj is defined in (4). The dimension of the stoichiometric subspace will bedenoted by dimS.

    Definition 2.20. The stoichiometric compatibility class (respectively, positive stoichio-metric compatibility class) containing the composition x0 Pm (respectively, x0 Pm)is the set (x0 + S) Pm.

    Positive stoichiometric compatibility classes can also be expressed as a systemof equations called conservation laws [5,14]:

    Definition 2.21. Conservation laws are equations of the form

    i x = i,where S = span{i: i = 1, . . . ,m dimS}, and i 6= 0 and i R for alli = 1, . . . ,m dimS.

    If ij > 0 and i > 0 for all i = 1, . . . ,m dimS and j = 1, . . . ,m, then thesystem of equations will be referred to as positive conservation laws.

    2.4. Asymptotic stability

    In [7,10,11], Horn, Jackson and Feinberg characterize a class of chemical reactionmechanisms that have a unique, positive, asymptotically stable equilibrium point in

  • D. Siegel, D. MacLean / Global stability of complex balanced mechanisms 95

    each compatibility class. Before we can state their result, the following definition isneeded:

    Definition 2.22. The deficiency of a chemical reaction mechanism, , is defined as = n ` dimS where n is the number of complexes, ` is the number of linkageclasses, and dimS is the dimension of the stoichiometric subspace.

    Theorem 2.23 (Deficiency Zero theorem). For any chemical reaction mechanism withzero deficiency, the following statements hold true:1. If the mechanism is not weakly reversible then, for arbitrary kinetics (not necessarily

    mass action), the differential equations for the corresponding reaction system cannotadmit either a positive equilibrium point or a cyclic composition trajectory alongwhich all the species concentrations are positive.

    2. If the mechanism is weakly reversible then, for mass action kinetics (but regardlessof the positive values the rate constants take), the differential equations for thecorresponding reaction system have the following properties: there exists within each positive stoichiometric compatibility class precisely one

    positive equilibrium point; that equilibrium point is asymptotically stable; and there is no nontrivial cyclic composition trajectory along which all species con-

    centrations are positive.

    The proof of the Deficiency Zero involves a Liapunov function. The Liapunovfunction used is

    H(x) =mi=1

    [xi(ln xi ln ai 1) + ai

    ], (9)

    with time derivativeH(x) = (ln x ln a) F (x), (10)

    where a Pm is a positive equilibrium point of the given mechanism. This Liapunovfunction is classical to chemical kinetics, and will be used in the proof of a latertheorem.

    In the next section we will present our major result, which provides new informa-tion on the qualitative nature of weakly reversible, deficiency zero mechanisms. It isimportant to note, however, that the result will hold true for the larger class of complexbalanced mechanisms, of which weakly reversible, deficiency zero mechanisms are asubclass [10]:

    Theorem 2.24. A necessary and sufficient condition for a mechanism to be complexbalanced for any set of positive rate constants is that the following two requirementsare met:

  • 96 D. Siegel, D. MacLean / Global stability of complex balanced mechanisms

    1. The mechanism is weakly reversible.2. = 0.

    3. -limit set theorem

    Given a weakly reversible, deficiency zero mechanism, it is known from theDeficiency Zero theorem that a solution, beginning at an initial condition sufficientlyclose to its uniquely compatible positive equilibrium point, will ultimately approachthat positive equilibrium point. What are the dynamics of the rest of the solutionsin the positive compatibility class? In this section, we will present a theorem whichgreatly limits the possibilities.

    Before we introduce the theorem, some terminology and a theorem from [13]must first be stated.

    For the general autonomous system

    x = f(x), (11)with f C1(E) where E is an open set of Rn, let t :E E, t R, define theflow of the system. If x0 E, let x(t) = t(x0) denote the solution of (11) satisfyingx(0) = x0.

    Theorem 3.1. For (11), x0 E, the -limit set, (x0), of t(x0) is a closed subsetof E, and if t(x0) is contained in a compact subset of Rm, then (x0) is a non-empty,connected, compact subset of E.

    Furthermore, the following notation related to the pth complex of...

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