Complex and detailed balancing of chemical reaction networks revisited

Transcription

1 J Math Chem (2015) 53: DOI /s ORIGINAL PAPER Complex and detailed balancing of chemical reaction networs revisited Arjan van der Schaft 1 Shodhan Rao 2 Bayu Jayawardhana 3 Received: 26 February 2015 / Accepted: 23 March 2015 / Published online: 7 April 2015 The Author(s) This article is published with open access at Springerlin.com Abstract The characterization of the notions of complex and detailed balancing for mass action inetics chemical reaction networs is revisited from the perspective of algebraic graph theory, in particular Kirchhoff s Matrix Tree theorem for directed weighted graphs. This yields an elucidation of previously obtained results, in particular with respect to the Wegscheider conditions, and a new necessary and sufficient condition for complex balancing, which can be verified constructively. Keywords Reaction networs Mass action inetics Complex balancing Detailed balancing Wegscheider conditions Stability 1 Introduction The notion of complex balancing of mass action inetics chemical reaction networs, generalizing the classical notion of detailed balancing, dates bac at least to the origin B Arjan van der Schaft A.J.van.der.Schaft@rug.nl Shodhan Rao shodhan.rao@ghent.ac.r Bayu Jayawardhana b.jayawardhana@rug.nl 1 Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands 2 Ghent University Global Campus, 119 Songdomunhwa-ro, Yeonsu-gu, Incheon , South Korea 3 Engineering and Technology Institute Groningen, University of Groningen, Groningen, The Netherlands

2 1446 J Math Chem (2015) 53: of chemical reaction networ (CRN) theory; see especially [6,11,12]. The assumption of existence of a complex-balanced equilibrium has powerful consequences for the dynamical behavior, precluding multi-stability and oscillations. In this note we will revisit the notion of complex balancing by a systematic use of notions and results from algebraic graph theory, in particular the Laplacian matrix and Kirchhoff s Matrix Tree theorem (see also [10,14] for other uses of this theorem in chemical reaction dynamics). This will result in a constructive necessary and sufficient condition for complex balancing. Furthermore, motivated by recent wor in [4] expanding on [7], we will provide a new perspective and results on the Wegscheider conditions for detailed balancing and formal balancing as introduced in [4]. The structure of this note is as follows. Section 2 gives a brief recap of the basic framewor of CRN theory from an algebraic graph theory perspective, based on [18, 21,23]. Section 3 introduces Kirchhoff s Matrix Tree theorem and shows how the application of this theorem leads to an improved, and more directly verifiable, condition for complex balancing as compared to [6,11]. Section 4 relates Kirchhoff s Matrix Tree theorem to the notion of formal balancing, cf. [4] and [7], and shows how this leads to an insightful graph-theoretic proof of the result obtained in [4] that complex balancing together with formal balancing implies detailed balancing and conversely. Notation: The space of n-dimensional real vectors consisting of all strictly positive entries is denoted by R n. The mapping Ln : Rn Rn, x Ln x, is the elementwise logarithm, and is defined as the mapping whose i-th component is given by ln(x i ). Similarly, Exp : R n R n is the mapping whose i-th component is given by exp x i. Furthermore, for two vectors x, y R n we let x y denote the vector in Rn with i-th component x i y i. Finally, 1 n denotes the n-dimensional vector with all entries equal to 1. Some graph-theoretic notions [3]: A directed graph 1 G with c vertices and r edges is characterized by its c r incidence matrix, denoted by D. Each column of D corresponds to an edge of the graph, and contains exactly one element 1 at the position of the head vertex of this edge and exactly one 1 at the position of its tail vertex; all other elements are zero. Clearly, 1c T D = 0. The graph is connected if any vertex can be reached from any other vertex by following a sequence of edges; direction not taing into account. It holds that ran D = c l, where l is the number of connected components of the graph. In particular, G is connected if and only if er D T = span 1 c. The graph is strongly connected if any vertex can be reached from any other vertex, following a sequence of directed edges. A subgraph of G is a directed graph whose vertex and edge set are subsets of the vertex and edge set of G. A graph is acyclic (or, does not contain cycles) if and only if er D = 0. A spanning tree of a directed graph G is a connected, acyclic subgraph of G that contains all vertices of G. 2 Recall of complex-balanced chemical reaction networs In this section, in order to set the stage, we will briefly recall the well-established framewor of (isothermal) CRN theory, originating in the wor of Horn, Jacson 1 Sometimes called a multigraph since we allow for multiple edges between vertices.

3 J Math Chem (2015) 53: and Feinberg in the 1970s [6,9,11,12]. Consider a chemical reaction networ with m chemical species (metabolites) with concentrations x R m, among which r chemical reactions tae place. The left-hand sides of the chemical reactions are called substrate complexes and the right-hand sides the product complexes. To each chemical complex (substrate and/or product) of the reaction networ one can associate a vertex of a graph, and to every reaction (from substrate to product complex) a directed edge (with tail vertex the substrate and head vertex the product complex). Let c be the total number of complexes involved in the chemical reaction networ, then the resulting directed graph G with c vertices and r edges is called the graph of complexes 2, and is defined by its c r incidence matrix D. Furthermore we define the m c complex composition matrix 3 Z with non-negative integer elements expressing the composition of the complexes in terms of the chemical species: its -th column expresses the composition of the -th complex. The dynamics of the chemical reaction networ taes the well-nown form ẋ = Sv(x) = ZDv(x), (1) where v(x) is the vector of reaction rates, and S = ZDthe stoichiometric matrix. The most basic way to define v(x) is mass action inetics. For example, the mass action inetics reaction rate of the reaction X 1 2X 2 X 3 is given as v(x) = x 1 x 2 2 with > 0 a reaction constant. In general, for a single reaction with substrate complex S specified by its corresponding column Z S = [ Z S1...Z Sm ] T of the complex composition matrix Z, the mass action inetics reaction rate is given by x Z S1 1 x Z S2 2...x Z Sm m, which can be rewritten as exp(zs T Ln x), x Rm. Hence the reaction rates of the total reaction networ are given by ( ) v j (x) = j exp ZS T j Ln x, j = 1,...,r, where S j is the substrate complex of the j-th reaction with reaction constant j > 0. This yields the following compact description of the rate vector v(x). Define the r c matrix K as the matrix whose ( j,σ)-th element equals j if the σ -th complex is the substrate complex for the j-th reaction, and zero otherwise. Then v(x) = K Exp (Z T Ln x), x R m, and the dynamics of the mass action reaction networ taes the form ( ) ẋ = ZDKExp Z T Ln x, x R m (2) It can be easily verified that the c c matrix L := DK has nonnegative diagonal elements and nonpositive off-diagonal elements. Moreover, since 1c T D = 0also 1c T L = 0, i.e., the column sums of L are all zero. Hence L defines a weighted Laplacian matrix 4 for the graph of complexes G. 2 In the literature sometimes also referred to as reaction graphs. 3 In [11,12], it is called complex matrix and is denoted by Y. 4 In [11,12] (minus) this matrix is called the inetic matrix, and is denoted by A.

4 1448 J Math Chem (2015) 53: The aim of CRN theory, starting with [6,11,12], is to analyze the dynamical properties of(2), and in particular to derive conditions which ensure a dynamical behavior which is independent of the precise values of the reaction constants (which are often poorly nown or varying). This has culminated in the deficiency-zero and deficiencyone theorems (see e.g. [8]), while a somewhat complementary approach is based on the assumption of existence of a complex-balanced equilibrium [6,11], generalizing the classical notion of a detailed-balanced equilibrium. Definition 2.1 A chemical reaction networ (2) is called complex-balanced 5 if there exists an equilibrium x R m, called a complex-balanced equilibrium, satisfying Dv ( x ) ( = LExp Z T Ln x ) = 0 (3) Chemically (3) means that at the complex-balanced equilibrium x not only all the chemical species, but also the complexes remain constant; i.e., for each complex the total inflow (from the other complexes) equals the total outflow (to the other complexes). The assumption of complex balancing has been shown to have strong implications for the dynamical properties of (2); see in particular the classical papers [6,11,12]. As detailed in [18], expanding on [21], these properties can be easily proved by defining the diagonal matrix ( x ) := diag ( exp(z T i Ln x ) ) i=1,...,c, (4) and rewriting the dynamics (2) into the form ẋ = ZL ( x ) ( ( x )) Exp Z T Ln x, L ( x ) := L ( x ). (5) The ey point is that since L(x )Exp ( Z T Ln ( )) x x = 0 for x = x, and Exp (Z T Ln ( x x )) = 1 c, the transformed matrix L(x ) satisfies L ( x ) 1 c = 0, 1 T c L ( x ) = 0, (6) and thus is a balanced Laplacian matrix (column sums and row sums are zero). Together with convexity of the exponential function this implies the following ey fact. Proposition 2.2 γ T L(x )Exp (γ ) 0 for any γ R c, with equality if and only if D T γ = 0. First property which directly follows [18] from Proposition 2.2 is the classical result [6,11,12] that all positive equilibria are in fact complex-balanced, and that given one complex-balanced equilibrium x the set of all positive equilibria is given by 5 Note that in older references, e.g. [6,11], a reaction networ (or mechanism) is called complex-balanced if there exists a complex-balanced equilibrium for all positive reaction constants.

5 J Math Chem (2015) 53: E := { x R m ST Ln x = S T Ln x } (7) Furthermore, using an elegant result from [8], there exists for every initial condition x 0 R m a unique x E such that x x 0 im S. By using the Lyapunov function ( x ) G(x) = x T Ln x ( x x ) T 1m (8) Proposition 2.2 then implies that the vector of concentrations x(t) starting from x 0 will converge to x ; at least if the reaction networ is persistent 6. The chemical interpretation is that G is (up to a constant) the Gibbs free energy with gradient vector G x (x) = Ln ( ) x x being the chemical potentials. See e.g. [15,21,22] for further information 7. 3 A graph-theoretic characterization of complex-balancing In this section we will expand on earlier investigations to characterize the existence of a complex-balanced equilibrium (see in particular [4,6,11], and the references quoted therein), and derive a new necessary and sufficient condition for complex balancing which can be constructively verified. First note that by the definition of Ln : R m Rm the existence of an x R m such that LExp (Z T Ln x ) = 0 (i.e., x is a complex-balanced equilibrium) is equivalent to the existence of a vector μ R m such that LExp ( Z T μ ) = 0, (9) or equivalently, Exp (Z T μ ) er L. Furthermore, note that Exp (Z T μ ) R c. In case the graph G is connected the ernel of L is 1-dimensional, and a vector ρ R c (the closure of the positive orthant) with ρ er L can be computed by what is sometimes called Kirchhoff s Matrix Tree theorem 8, which for our purposes can be summarized as follows. Denote the (i, j)-th cofactor of L by C ij = ( 1) i j M i, j, where M i, j is the determinant of the (i, j)-th minor of L, which is the matrix obtained from L by deleting its i-th row and j-th column. Define the adjoint matrix adj(l) as the matrix with (i, j)-th element given by C ji. It is well-nown that 6 The reaction networ is called persistent if for every x 0 R m the ω-limit set of the dynamics (2) does not intersect the boundary of R m. It is generally believed that most reaction networs are persistent. However, up to now this persistence conjecture has been only proved in special cases (cf. [1,2,19] and the references quoted in there). 7 The form (5) also provides a useful starting point for structure-preserving model reduction [17,18,21]. 8 This theorem goes bac to the classical wor of Kirchhoff on resistive electrical circuits [13]; see [3]fora succinct treatment. Nice accounts of the Matrix Tree theorem in the context of chemical reaction networs can be found in [10,14]. In [4] Kirchhoff s Matrix Tree theorem is mentioned and exploited in the closely related, but different, context of investigating how far complex balancing is from detailed balancing; see the next section.

6 1450 J Math Chem (2015) 53: L adj(l) = (det L)I c = 0 (10) Furthermore, since 1c T L = 0 the sum of the rows of L is zero, and hence by the properties of the determinant function it directly follows that C ij does not depend on i; implying that C ij = ρ j, j = 1,...,c. Therefore by defining ρ := (ρ 1,...,ρ c ) T, it follows from (10) that Lρ = 0. Furthermore, cf. [3, Theorem14 on p.58], ρ i is equal to the sum of the products of weights of all the spanning trees of G directed towards vertex i. In particular, it follows that ρ j 0, j = 1,...,c. In fact, ρ = 0 if and only if G has a spanning tree. Furthermore, since for every vertex i there exists at least one spanning tree directed towards i if and only if the graph is strongly connected, we may conclude that ρ R c if and only if the graph is strongly connected. Example 3.1 Consider the cyclic reaction networ C C 1 1 in three (unspecified) complexes C 1, C 2, C 3. The Laplacian matrix is given as 1 C L = By Kirchhoff s Matrix Tree theorem the corresponding vector ρ satisfying Lρ = 0 is given as ρ = , where each term corresponds to one of the three weighted spanning trees pointed towards the three vertices. In case the graph G is not connected the same analysis can be performed on any of its connected components. Remar 3.2 The existence (not the explicit construction)ofρ R c satisfying Lρ = 0 already follows from the Perron-Frobenius theorem [11], [20, LemmaV.2]; exploiting the fact that the off-diagonal elements of L := DK are all nonnegative 9. 9 This implies that there exists a real number α such that LαI m is a matrix with all elements nonnegative. Since the set of eigenvectors of L and LαI m are the same, and moreover by 1 T L = 0 there cannot exist a positive eigenvector of L corresponding to a non-zero eigenvalue, the application of Perron-Frobenius to L αi m yields the result; see [20, LemmaV.2] for details.

7 J Math Chem (2015) 53: Returning to the existence of μ R m satisfying LExp (Z T μ ) = 0 this implies the following. Let G j, j = 1,...,l, be the connected components of the graph of complexes G. For each connected component, define the vectors ρ 1,...,ρ l as above by Kirchhoff s Matrix Tree theorem (i.e., as cofactors of the corresponding diagonal sub-blocs of L or as sums of products of weights along spanning trees). Define the total vector ρ as the staced column vector ρ := col(ρ 1,...,ρ l ). Partition correspondingly the composition matrix Z as Z = [Z 1...Z l ]. Then there exists μ R m satisfying LExp (Z T μ ) = 0 if and only if each connected component G j, j = 1,...,l,is strongly connected and Exp ( Z T j μ ) = β j ρ j, β j > 0. (11) This in its turn is equivalent to strong connectedness of each connected component G j and the existence of constants β j such that Z T j μ = Ln ρ j β j1, j = 1,...,l. Furthermore, this is equivalent to strong connectedness of each connected component of G, and Ln ρ im Z T er D T (12) Finally, (12) is equivalent to Summarizing we have obtained D T Ln ρ im D T Z T = im S T (13) Theorem 3.3 The reaction networ dynamics ẋ = ZLExp (Z T Ln x) on the graph of complexes G is complex-balanced if and only if each connected component of G is strongly connected (or, equivalently, ρ R c ) and (13) is satisfied, where the elements of the sub-vectors ρ j of ρ are obtained by Kirchhoff s Matrix Tree theorem applied to L for each j-th connected component of G. Furthermore, for a complex-balanced reaction networ a balanced Laplacian matrix 10 L(x ) defined in (5) is given as L ( x ) = Ldiag (ρ 1,...,ρ c ) (14) Remar 3.4 The above theorem is a restatement of Theorem 3C in [11]; the main difference being that in [11] the positive vector ρ er L remains unspecified, while in our case it is explicitly given by Kirchhoff s Matrix Tree theorem. We directly obtain the following corollary stated before in [11, Eq. (3.21)]: Corollary 3.5 The reaction networ dynamics ẋ = ZLExp (Z T Ln x) is complexbalanced if and only if ρ R c and ρ σ 1 1 ρσ ρσ c c = 1, (15) 10 It can be easily seen that the balanced Laplacian matrices L(x ) for different equilibria x just differ from each other by a positive multiplicative constant for each connected component of G.

8 1452 J Math Chem (2015) 53: for all vectors σ = (σ 1,σ 2,...,σ c ) T er Z im D. In particular, if er Z im D = 0 (zero-deficiency [8,9,11]) then ẋ = ZLExp (Z T Ln x) is complex-balanced. Proof Ln ρ im Z T er D T if and only if σ T Ln ρ = 0 for all σ (im Z T er D T ) = er Z im D, or equivalently 0 = σ 1 ln ρ 1 σ c ln ρ c = ln ρ σ 1 1 ln ρσ c c = ln ( ρ σ ) ρσ c c for all σ er Z im D. Example 3.6 Consider Example 3.1 for the special case 1 = 2 = 3 = 0 (irreversible reactions). Then the vector ρ reduces to ρ = The reaction networ with complex composition matrix Z is complex-balanced if and only i > 0, i = 1, 2, 3, and D T Ln ρ im D T Z T, D = , This last condition can be further written out as ln 1 2 ln 2 3 im D T Z T ln 3 1 [ ] 102 As a mathematical example, tae the complex composition matrix Z = 111 (corresponding to the complexes X 1 X 2, X 2, 2X 1 X 2 ). In this case the networ is complex-balanced if and only if 1 > 0, 2 > 0, 3 > 0, and ( 1 )2 = Relation with the Wegscheider conditions and detailed balancing In this section we will relate the conditions for complex balancing as obtained in the previous section to Wegscheider-type conditions. This will also relate complex balancing to the classical concept of detailed balancing. Throughout this section we will consider reversible chemical reaction networs, in which case the edges of G come in pairs: if there is a directed edge from vertex i to j

9 J Math Chem (2015) 53: then there also is a directed edge from j to i (and in the case of multiple edges from i to j there are as many edges from i to j as edges from j to i). This means that the connected components of G are always strongly connected, or equivalently, that ρ as obtained from Kirchhoff s Matrix Tree theorem is in R c. Define the undirected graph Ḡ as having the same vertices as G but half its number of edges, by replacing every pair of oppositely directed edges of G by one undirected edge of Ḡ. Denote the number of edges of Ḡ by r = 2 1 r. Endow subsequently Ḡ with an arbitrary orientation (all results in the sequel will be independent of this orientation), and denote the resulting incidence matrix by D. Clearly, after possible reordering of the edges, D is related to the incidence matrix D of G as D = [ D D ] (16) To the j-th edge of Ḡ there now correspond two reaction constants j, j (the forward and reverse reaction constants with respect to the chosen orientation of Ḡ). Then define the equilibrium constants K eq j (K eq 1,...,K eq r )T. := j, j = 1,..., r, and the vector K eq := j Recall [7,24] that the reaction networ is called detailed-balanced 11 if and only if it satisfies Ln K eq im S T, (17) where S := Z D is the stoichiometric matrix of the reversible networ with graph Ḡ. This is equivalent to σ 1 lnk eq 1 σ r lnk eq r = 0 for all σ = (σ 1,...,σ r ) such that σ T S T = 0. Writing out K eq j = j j this is seen to be equivalent to ( ) σ1 1 ( ) σ r r = ( 1 ) σ1 ( r ) σ r (18) for all σ such that Sσ = 0, nown as the (generalized) Wegscheider conditions. Recently in [4] the notion of formally balanced was introduced, based on Feinberg s circuit conditions in [7], and weaening the above Wegscheider conditions. In our setup this notion is defined as follows. Definition 4.1 The reversible reaction networ Ḡ with incidence matrix D, and vector of equilibrium constants K eq is called formally balanced if Ln K eq im D T Since S = Z D formally balanced is trivially implied by detailed-balanced, while if im S T = im D T (zero-deficiency) the reverse holds. Furthermore, any reaction networ with acyclic Ḡ (and thus er D = 0) is automatically formally balanced. 11 This means, see e.g. [12,15,21], that there exists an equilibrium for which every forward reaction is balanced by its reverse reaction.

10 1454 J Math Chem (2015) 53: As above, the notion of formally balanced is seen to be equivalent to σ 1 lnk eq 1 σ r lnk eq r = 0 for all σ = (σ 1,...,σ r ) T such that σ T D T = 0, which in turn is equivalent to ( 1 ) σ1 ( r ) σ r = ( 1 ) σ1 ( r ) σ r (19) for all σ such that Dσ = 0 (that is, for all cycles σ ). We will refer to (19)asthewea Wegscheider conditions. Note that the wea Wegscheider conditions only depend on the structure of the graph Ḡ (i.e., its cycles) and the equilibrium constants, and not on the complex composition matrix Z as in the case of the strong Wegscheider conditions (18). Theorem 4.2 Consider a reversible chemical reaction networ given by the graph Ḡ with incidence matrix D, and with ρ determined by L. The following statements are equivalent 1. Ldiag (ρ 1,...,ρ c ) is symmetric 2. Ln (K eq ) = D T Ln ρ 3. Ln (K eq ) im D T (formally balanced) Proof (1) (2) Let us first prove the equivalence between (1) and (2). Consider the i-th and j-th vertex of Ḡ, and suppose that the orientation has been taen such that the α-th edge between i and j is such that i is the tail vertex and j is the head vertex. Then the (i, j)-th element of Ldiag (ρ 1,...,ρ c ) is given by α ρ j, while the ( j, i)-th element equals α ρ i. Symmetry of Ldiag (ρ 1,...,ρ c ) thus amounts to α ρ j = α ρ i for all pairs of vertices i, j. On the other hand, the α-th element of the vector D T Ln ρ is given by ln ρ j ln ρ i while the α-th element of Ln (K eq ) is given by ln α α = ln α ln α Equality of ln ρ j ln ρ i and ln α ln α is thus equivalent to ln ρ j ln α = ln ρ i ln α which in its turn is equivalent to α ρ i = α ρ j as above. (2) (3)

11 J Math Chem (2015) 53: Obviously (2) implies (3). For the reverse implication, consider any pair of vertices lined by an edge of the graph Ḡ. Depending on the orientation of Ḡ refer to one vertex as the tail vertex t and the other vertex as the head vertex h. Refer to the positive reaction constant from t to h by and to the negative reaction constant by. Now consider a spanning tree directed towards t with product of weights denoted by τ t. In case the edge between t and h is part of this spanning tree then it follows that by reversing the orientation of this edge it defines a spanning tree directed towards to h with product of weights denoted by τ h. It follows directly that τ t = τ h (20) In case the edge between t and h is not part of this spanning tree then divide the edges of the spanning tree into two sets; the set E 1 containing the edges of the part of the spanning tree from t to h (containing say l edges) and the set E 2 containing the remaining edges of the spanning tree. Observe that E 1 together with the edge from t to h forms a cycle of the graph Ḡ. Since Ln (K eq ) im D T it follows that for the reaction constants along this cycle (choosing an appropriate orientation), 1... l = 1... l. (21) Now within the spanning tree directed towards t, if the orientation of each of the edges of E 1 is reversed, we obtain another spanning tree directed towards h with product of weights denoted again by τ h.byusing(21) it is readily verified that also in this case we obtain the same relation (20). Summing up over all spanning trees we thus obtain the equality ρ t = ρ h, (22) which can be equivalently written as ln = ln ρ h ln ρ t. Doing this for all adjacent vertices t and h this exactly amounts to the required equality Ln (K eq ) = D T Ln ρ. Example 4.3 Consider again the reaction networ described in Example 3.1 (without specifying the complexes C 1, C 2, C 3 ). The transformed Laplacian matrix is computed as L = = ( 1 2 )( 2 3 ) ( 1 3 ) ( 2 3 ) ( )( 1 3 ) ( 2 3 ) which is symmetric if and only if 3 ( 1 2 ) ( 1 2 ) ( 1 3 ) ( )( ) = (23)

12 1456 J Math Chem (2015) 53: On the other hand, Ln K eq D T amounts to ln ln 2 2 im ln 3 3 which reduces to ln 1 1 ln 2 2 ln 3 3 = 0, and hence to the same condition (23). Thus the reaction networ is formally balanced if and only if (23) holds. Now let us relate all this to the necessary and sufficient conditions for complex balancing obtained before, cf. (13). Note that by (16) D T Ln ρ im D T Z T D T Ln ρ im D T Z T Hence a reversible reaction networ is complex-balanced if and only if D T Ln ρ im D T Z T = im S T. We directly obtain the following corollary proved by other methods in [4] 12. Corollary 4.4 A reversible reaction networ is detailed-balanced if and only if it is formally balanced as well as complex-balanced. Proof We have seen before that detailed-balanced implies complex-balanced as well as formally balanced. For the converse we note that formally balanced implies that Ln (K eq ) im D T. Hence by Theorem 4.2 Ln (K eq ) = D T Ln ρ. Since furthermore the networ is complex-balanced D T Ln ρ im S T. Hence Ln (K eq ) = D T Ln ρ im S T, i.e., the reaction networ is detailed-balanced. In case the reversible reaction networ is formally balanced the symmetric matrix L(x ) = Ldiag (ρ 1,...,ρ c ) can be written as Ldiag (ρ 1,...,ρ c ) = DK D T where K is the r r diagonal matrix, with α-th diagonal element given by κ α := α ρ j = α ρ i where the α-th edge of Ḡ corresponds to the reversible reaction between the i-th and the j-th complex. For the interpretation of the positive constants κ α as conductances of the reversible reactions please refer to [5,22]. If additionally the formally balanced reaction networ is complex-balanced (and thus, cf. Corollary 4.4, detailed-balanced), then its dynamics thus taes the form ẋ = Z DK D T Exp ( ( x )) Z T Ln x 12 The result is formulated in [4] as the equivalence between complex-balanced and detailed-balanced under the assumption of formally balanced.

13 J Math Chem (2015) 53: In this case, see e.g. [12,21], all equilibria x are in fact detailed-balanced equilibria, that is, D T Z T Ln x = Ln (K eq )(= D T Ln ρ). 5 Conclusions By a systematic use of notions from algebraic graph theory, in particular the Laplacian matrix and Kirchhoff s Matrix Tree theorem, previously derived results on complex, detailed and formal balancing have been proved in a simple manner. Furthermore, it has resulted in a new necessary and sufficient condition for complex balancing, which can be verified constructively. The results obtained in this note can be immediately extended to mass action inetic reaction networs with constant inflows and mass action outflow exploiting the classical idea of adding a zero complex ; see [9] and [23] for further details. Current research is concerned with the application of the developed framewor to questions of occurrence of multi-stability and structure-preserving model reduction; see for the latter also [17,18,21]. Conflict of interest The authors declare that they have no conflict of interest. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. References 1. D.F. Anderson, A proof of the global attractor conjecture in the single linage class case. SIAM J. Appl. Math. 71(4), (2011) 2. D. Angeli, P. De Leenheer, E.D. Sontag, Persistence results for chemical reaction networs with timedependent inetics and no global conservation laws. SIAM J. Appl. Math. 71, (2011) 3. B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics 184 (Springer, New Yor, 1998) 4. A. Dicenstein, M. Perez Millan, How far is complex balancing from detailed balancing? Bull. Math. Biol. 73, (2011) 5. M. Ederer, E.D. Gilles, Thermodynamically feasible inetic models of reaction networs. Biophys. J. 92, (2007) 6. M. Feinberg, Complex balancing in chemical inetics. Arch. Rational Mech. Anal. 49, (1972) 7. M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity. Chem. Eng. Sci. 44(9), (1989) 8. M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networs. Arch. Ration. Mech. Anal. 132, (1995) 9. M. Feinberg, F.J.M. Horn, Dynamics of open chemical systems and the algebraic structure of the underlying reaction networ. Chem. Eng. Sc. 29, (1974) 10. J. Gunawardena, Time-scale separation Michaelis and Menten s old idea, still bearing fruit. FEBS J. 281(2), (2013) 11. F.J.M. Horn, Necessary and sufficient conditions for complex balancing in chemical inetics. Arch. Ration. Mech. Anal. 49, (1972) 12. F. Horn, R. Jacson, General mass action inetics. Arch. Ration. Mech. Anal. 47, (1972) 13. G. Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der Linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72, (1847) 14. I. Mirzaev, J. Gunawardena, Laplacian dynamics on general graphs. Bull. Math. Biol. 75, (2013)

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