Vector-Space Methods and Kirchhoff Graphs for Reaction Networks

Transcription

1 Vector-Space Method and Kirchhoff Graph for Reaction Network Joeph D. Fehribach Fuel Cell Center WPI Mathematical Science and Chemical Engineering 00 Intitute Rd. Worceter, MA Thi article preent a vector pace formulation for contructing reaction route (reaction pathway) and Kirchhoff graph (reaction route graph) for reaction network. Specific example, many coming from fuel-cell electrochemitry, are included throughout to illutrate the more-general theoretical dicuion. The upport of the National Science Foundation under grant DMS and DMS i gratefully acknowledged.

2 Introduction For many year, reaction network have been widely tudied in the chemical and biological cience, and a number of graphical approache have been developed to help reearcher undertand thee network (cf. Fehribach [5] for an overview of thee approache). Among the mot intereting of thee are Kirchhoff graph which allow a reaction network to be identified with a circuit diagram that atifie the Kirchhoff law. Thi article preent a mathematical, vector-pace approach to undertanding how Kirchhoff graph correpond to reaction network and mot importantly, how to contruct Kirchhoff graph. Thee graph have been defined and extenively dicued by Fihtik, et al. [0,, 2, 2, 6, 7, 9, 8, 3, 22] where they are termed reaction route graph. The preent author prefer the term Kirchhoff graph a it eem more general and more ueful in helping the reader to undertand what the key propertie of thee graph are. The reaction network that we conider come from fuel-cell electrochemitry, although the reult preented here can be applied to any reaction network chemical, biochemical or biological provided the network can be thought of a reverible. For our purpoe, aume that a reaction network i a collection of reacting chemical pecie (e.g., H 2, OH, H 2 O, CO = 3, etc.) and a et of reverible reaction tep (including irreverible reaction tep may be poible, but it would complicate the dicuion, and o thi iue will not be conidered here). In Kirchhoff graph, directed edge repreent reaction tep, and vertice repreent combination of the reacting pecie in particular their component potential. Component potential are the um and difference of the electrochemical potential for the reacting pecie weighted by the appropriate toichiometric number. Thi implie that the potential difference between any two adjacent vertice i the affinity of the aociated tep. Indeed thi potential difference mut be the reaction tep affinity no matter where the aociated directed edge appear in the graph. The direction of each edge indicate the forward direction of the aociated tep. Vertice fall into two type: terminal vertice, correponding to combination of pecie which are either the product or the reactant for the overall reaction, and intermediate vertice, correponding to combination which contain (in part) intermediate pecie. A Kirchhoff graph for a given reaction network mut atify two fundamental defining propertie: Every equence of reaction tep which yield an overall reaction for the actual reaction network mut have a correponding equence of edge which connect the terminal vertice aociated with that overall reaction. The graph mut atify Kirchhoff law. In aying the a Kirchhoff graph atifie the Kirchhoff law, one mean that the graph atifie the following four condition: The um of the reaction rate for the tep incident on each terminal vertex mut be ome toichiometric multiple of the correponding overall reaction rate.

3 The um of the reaction rate for tep incident on each intermediate vertex mut be zero. The um of the affinitie (change in potential) around any cloed cycle mut be zero. The um of the affinitie for any route (trail, path, walk) between two terminal vertice mut be the affinity of the correponding overall reaction. The firt two of thee condition are equivalent to the Kirchhoff current law; the lat two are equivalent to the Kirchhoff potential law. One can then add edge repreenting the correct toichiometric multiplicity of each overall reaction to make every Kirchhoff graph a cyclic graph. Becaue Kirchhoff graph atify the Kirchhoff law, they allow one to tudy reaction network uing the tool that are traditionally ued to tudy electrical circuit. Kirchhoff graph and the aociated ytem of equation for the reaction rate and affinitie of the individual reaction tep can be ued to compute the overall rate and affinity for the reaction network. In addition, uing the graph, one can determine which route are mot ignificant (offer the leat reitance) and which can be eliminated a unneceary becaue their rate are too mall to be ignificant. Or the effect of, ay, temperature on the reaction network can be conidered ince the rate of reaction tep vary independently with temperature. One can alo define impler equivalent Kirchhoff graph which yield the ame overall reaction and overall reaction rate. But the key point i that baic linear algebra guarantee that in a ene that will be made clear below, a Kirchhoff graph for a reaction network completely characterize the reaction network. For a more-complete treatment of the application of Kirchhoff (reaction route) graph, ee Fihtik et al. [2, 7, 9, 22]. Before one can ue a Kirchhoff graph to tudy a reaction network, however, two baic quetion hould be conidered: () doe every reaction network have a correponding Kirchhoff graph?, and (2) if there i a Kirchhoff graph, how can one contruct it? The iue of exitence will not be addreed in detail here, although the author conjecture that a Kirchhoff graph exit for any given reaction network. Auming that a Kirchhoff graph doe exit, we will conider two approache for contructing it. In the firt, one tart by contructing an auxiliary graph called an overall reaction graph. Thi contruction i given in term of vector pace method in the next ection. The overall reaction graph can then act a a keleton in the contruction of a Kirchhoff graph. In a econd more general approach, a Kirchhoff graph i obtained from the contruction of an incidence matrix uing row operation on the row pace of the tranpoe of the toichimetric matrix for the reaction network. Thi incidence-matrix approach i preented in Section 4 below. The overall reaction() for a given reaction network may be known in advance, but a i dicued below, they are in general determined by the pecific reacting pecie and reaction tep. The next ection dicue the ue of vector pace in deciding which overall reaction are poible (achievable) for a given reaction network, then how how linear algebra can be ued to determine all equence of reaction tep which yield an overall reaction by connecting the initial tate repreenting the collection of reactant pecie to the final tate repreenting the collection of product 2

4 pecie. Thee equence are known in the literature a overall reaction route, reaction pathway or reaction mechanim. Having found all overall reaction route, one can repreent them graphically in term of an overall reaction graph (mentioned above) which i a projection of the vector pace contruction jut developed. Such an overall reaction graph atifie the firt defining property for a Kirchhoff graph (that it repreent the overall reaction), and the Kirchhoff potential law, but not necearily the Kirchhoff current law. It can thu be tepping tone in the contruction of a Kirchhoff graph, and it implie that every Kirchhoff graph i in fact a geometric graph, i.e. it i baed on an underlying vector pace. Section 3 then dicue the contruction of a Kirchhoff graph itelf baed on thi overall reaction graph contruction. While Kirchhoff graph have important implication for the tudy of reaction network, they alo have an inherent mathematical beauty and therefore are intereting in their own right. Kirchhoff graph can be contructed for matrice in general, not jut thoe coming from reaction toichiometry. Finding a Kirchhoff graph i baically the invere problem to the contruction of the cycle pace and bond pace for a given graph []. In the concluding ection, Section 5, ome of the propertie of the relationhip between matrice and their aociated Kirchhoff graph are explored, independent of reaction network. But a final example then make clear how the tudy of a relatively complicated reaction network can be implied and organized through an appropriate Kirchhoff graph. A wa mentioned above, reaction route graph, which are eentially the ame a Kirchhoff graph, were defined by Fihtik, Datta et al. ( ) [6, 7, 8, 22]. Thee author contributed much to the undertanding of thi concept, particularly making clear it application to the tudy of the procee in reaction network. Their approach for contructing the graph, however, i baed on a form of Cramer rule (cf. Appendix A, []) and i thu computationally inefficient. It alo relie on lengthy lit or reaction route and rate condition, not taking advantage the concept of a vector pace bai. Both of thee iue are addre in the preent work. Similar graph have alo been ued by other author. Horn (973) [4, 5] proved a number of reult about reaction diagram and complex graph which are imilar to Kirchhoff graph, though he conidered reverible, weakly reverible and irreverible reaction. Feinberg et al. [20, 3, 4] have defined and tudied peciecomplex-linkage graph and pecie-reaction graph which again are both imilar to but ditinct from the Kirchhoff graph tudied here. Oter, Perelon & Katchalky (973) [7] referred to Kirchhoff graph a topological graph. Alo Robert (977) [9] ued a form of Kirchhoff graph to help in hi tudy of enzyme kinetic. More recently Qian, Beard & Liang (2003) [8] ued a imple Kirchhoff graph in their tudy of a three-tate kinetic cycle. None of thee author, however, conidered the vector pace apect of Kirchhoff graph, or tudied the Kirchhoff graph concept in detail. Finally, for reader le familiar with linear-algebraic concept ued here, a complete introduction to thee concept can be found in, e.g., Johnon & Rie [6] or Strang [2]. 3

5 2 Vector Space Method for Reaction Route Conider a reaction network compoed of the following n reaction tep: 0 α T + α 2 T α k T k + β I + β 2 I β l I l 0 α 2 T + α 22 T α 2k T k + β 2 I + β 22 I β 2l I l... () 0 α n T + α n2 T α nk T k + β n I + β n2 I β nl I l Here T j are the terminal pecie (pecie which are either produced or conumed by the reaction network, i.e., their net amount change), and I j are the intermediate pecie (pecie neither produced or conumed in the network). Even though they are not true intermediate pecie, reaction ite (often denoted S or M) are grouped with the intermediate when they appear in reaction tep becaue they are alo not produced or conumed in any overall reaction. By convention, the toichiometric coefficient (α ij and β ij ) are poitive integer if the correponding pecie i a product (produced by the i-th tep), negative integer if it i a reactant (conumed in the i-th tep), and zero if it i not preent in the i-th tep. Thi allow u to place all of the pecie on the right ide of the chemical reaction equation. Baed on thi reaction network, let u define vector repreenting the reaction tep: i := [α i, α i2,..., α ik, β i, β i2,..., β il ], and conider the vector pace repreenting all linear combination of the reaction tep: V := pan{, 2,..., n }. Clearly V i a ubpace of the entire toichiometric coefficient pace Q k+l where Q repreent the rational number. Since we are intereted in determining overall reaction and finding route that yield thee overall reaction, let u divide Q k+l into two ubpace: the firt ubpace aociated with the terminal pecie can be identified with Q k, while the econd aociated with the intermediate pecie can be identified with Q l. For an overall reaction the toichiometric coefficient for the intermediate pecie I j mut all be zero. So b Q k i the vector repreenting an achievable overall reaction if and only if it can be written a a linear combination of the reaction tep vector i with only integer toichiometric coefficient. Mathematically thi mean that b V Q k, i.e. b mut both be a linear combination of the i and have zero entrie aociated with the intermediate pecie. Thi arrangement i depicted in Figure. In the remainder of thi ection, the mot general full reaction route (combination of reaction tep) which reult in a given overall reaction b will be derived. From a mathematically view, thi amount to a claical problem in linear algebra, and the reult i well known. To accomplih our goal, we mut firt find all of the null or empty route (equivalent to finding the homogeneou olution) then find which overall reaction are achievable, and finally combined the null route with a full route (particular olution) to give a general repreentation of the full reaction route. The proce for finding the full reaction route will be illutrated uing the example of the hydrogen evolution reaction. 4

6 Q l p 2 2 p 7 7 p p n n b Q k Figure : Depiction of the terminal and intermediate ubpace. The intermediate ubpace Q l i repreented by the vertical axi, the terminal ubpace Q k i repreented by the horizontal axi, and b i a linear combination of the i lying in the terminal ubpace. The tep indice, 2, 7 and n are choen arbitrarily. 2. Null (Empty) Route The firt key quetion that mut be addreed i which route lead no where, i.e., which linear combination of reaction tep cancel themelve out leaving the amount of all pecie unchanged. Such a route i hown in Figure 2. Notice that the vector graph correponding to any null route i a cycle and that the coefficient a i mut be integer. Let A := [α ij β ij ] T be the tranpoe of the toichiometric matrix, i.e., A i the matrix whoe i-th column contain the toichiometric number from the i-th reaction. To find all poible null route, one mut find all poible olution of Av = 0 which mean that one mut compute Null(A), the null pace of A. For convenience let m be the dimenion of Null(A). Fortunately modern mathematical oftware uch a Maple, Mathematica or Matlab make thi computation relatively eay even when k, l, m and n are relatively large. Example a. A a relatively imple example of thi approach to finding reaction route, conider the HER (Hydrogen Evolution Reaction) network. HER i important in, among other place, certain corroion procee and it overall reaction (derived below) i well etablihed. Thi network i compoed of ix pecie (H 2, OH, H 2 O, e, S, H S) and three reaction tep. The firt five pecie are the terminal pecie, the lat (H S) i the only true intermediate pecie (S i treated a an intermediate pecie for our purpoe). The three reaction tep are T : 2H S 2S + H 2, V : S + H 2 O + e H S + OH, H : H S + H 2 O + e S + H 2 + OH. (2) 5

7 Q l a 7 7 a 3 3 a a 4 4 k Q Figure 2: Depiction of a null or empty route. Here the um of a, a 3 3, a 7 7 and a 4 4, end where it begin leaving the ytem unchanged. The ubcript which ditinguih the tep honor, repectively, Tafel, Volmer and Heyrovky. Moving the reactant to the righthand ide and giving them negative ign, one obtain the following toichiometric table: H 2 OH H 2 O e S H S T V 0 H Baed on thi table, the toichiometric matrix tranpoe i 0 0 A = 0 0. (3) 2 2 By direct calculation, one find that Null(A) = Span (4) which mean that T + V H = 0, i.e., that only multiple of thi combination of tep leave the ytem unchanged. 6

8 2.2 Overall Reaction Once the null route have been determined, the poible overall reaction() aociated with our reaction network can be found. In ome cae, thee may be known in advance, but in general the overall reaction hould be determined by the pecific reaction tep and reacting pecie. To determine what overall reaction are achievable, let B := [β ij ] T = [β ji ] be the tranpoe of the matrix of toichiometric coefficient for the n tep of our reaction network aociated with jut the reaction ite and the intermediate pecie. Becaue overall reaction mut be both nontrivial and free of intermediate pecie, overall reaction correpond to the portion of Null(B) which i perpendicular to Null(A). Mathematically thi relationhip can be expreed a Null(B) = Null(A) OR where OR i the ubpace correponding to the overall reaction. The dimenion of OR i the number of linearly independent overall reaction for our reaction network, and the overall reaction themelve can be determined from a bai for thi ubpace. Note that if Null(B) = Null(A), then OR i trivial and no overall reaction can be achieved. If OR i one dimenional, then there i a ingle bai vector and a ingle overall reaction. Becaue we are working with vector pace over the rational number, it i poible to chooe thi bai vector to have integer entrie with no common divior (probably the implet choice), keeping in mind that the entrie for product hould be poitive, while thoe for reactant hould be negative. If OR ha dimenion greater than one, then there are multiple overall reaction, and the ituation i more complicated. In particular, if the null pace i two dimenional, there are at leat three ditinct (though not linearly independent) overall reaction. Example b. Now returning to the HER network, and conidering only the intermediate pecie, one ee from Table 2. that the toichiometric matrix for intermediate pecie i [ ] 2 B =. (5) 2 Again by direct calculation, one find that Null(B) = Span, 0 Since the interection of Null(A) and Null(B) i panned by the vector [,, ] T, only the portion of the null pace of B perpendicular to [,, ] T repreent a nontrivial overall reaction. Since the two bai vector in (6) are perpendicular, OR = Span{[0,, ] T }. Thu the overall reaction for the HER reaction network i the one correponding to V + H, which written in term of pecie i the well-known overall reaction for the HER network: 2H 2 O + 2e H 2 + 2OH Note that the above reult guarantee that thi i the only poible overall reaction for the HER reaction network. (6) 7

9 2.3 Overall Reaction Route Now that the achievable overall reaction have been identified, let u conider what route yield thee overall reaction. Following Fihtik, et al. [], for a given achievable overall reaction b, define an overall reaction route to be an integral linear combination of reaction tep which yield the overall reaction: b = x + x x n n (7) where each x i i an integer. Let u conider in general how to find all the poible choice for x i. The problem of finding overall reaction route in fact imply amount to finding the general olution of the ytem of linear equation Ax = b (8) where again A be the tranpoe of the matrix of toichiometric coefficient (the toichiometric matrix). The well-known general olution of (8) i the um of any particular olution p and an element of the null pace of A. Thu any overall reaction route aociated with b can be repreented a a vector in the form x = p + c v + c 2 v c m v m (9) where p i our particular olution, {v, v 2,..., v m } i a bai for Null(A) (found in ection 2. above) and the c i are arbitrary integer. Again finding a particular olution p i greatly implify through the ue of Maple, Mathematica or Matlab. If the i are in fact linearly independent, then thi null pace i trivial (contain only the zero vector), m = 0, there i only one particular olution, and the olution of (8) i unique: x = p, implying that there i only one reaction route that achieve the given overall reaction b. If, on the other hand, the i are not linearly independent, then thi null pace contain nonzero vector, and there are multiple overall reaction route that achieve the overall reaction aociated with b. Example c. Returning once more to the HER example, Figure 3 how two poible vector repreentation for the overall reaction route for HER. Thu whilt all overall reaction route x are of the form given in (9), the graphical repreentation depend on the choice of p and how the null cycle i adjoined to p, and therefore i not unique. The graph in Figure 3 are in fact both two-dimenional projection of the vector um for x given in (9), and directed graph (digraph) in the graph-theoretic ene. A uch, they repreent the connection between the vector-pace development in thi ection and the Kirchhoff graph of the next, and they give the tarting point for contructing thoe Kirchhoff graph. 2.4 Chooing a Bai for Null(A) Now uppoe that the null pace of A i multi-dimenional (m 2). Then the olution for (8) i not unique, and there are many poibilitie for how to chooe the bai vector v i for Null(A); what i the bet choice? Although the anwer to thi 8

10 Q0 l Q0 l T T T V V H H V V b Verion 0Q k b Verion 2 Figure 3: Two overall reaction graph for the HER reaction network. In both verion, Null(A) i one dimenional and v correpond to T + V H = 0 i the ole bai vector for thi null pace. In the firt verion, p correpond to H + V, while in the econd, p correpond to T + 2 V. Baed on the mallet total norm (defined below), the firt verion would be preferred. 0Q k quetion depend on what one mean by bet, it i poible to ee that there i a implet choice in the following ene. Let n v i := v ij (0) be the norm (length) of v i, and let T N := j= m v i () i= be the total norm for the bai. We take a the bet choice that which ha a mallet total norm. Since the entrie v ij mut be integer, a bai with minimal total norm mut exit, although it might not be unique. In ome cae finding exactly the bet choice may not be neceary; a convenient bai with total norm that i not too large may be deirable for ue in (9). A imilar minimal total norm criterion can alo be ued for determining a bai for the row pace below, and for that matter for determining a bet bai for any other finite vector pace over the rational. So in ummary, for a given reaction network, one can determine which overall reaction are achievable by finding all vector b Q k of the form given in (7). Then one can find all overall reaction route aociated with b by finding a particular olution p and a imple bai {v, v 2,..., v m } for the null pace of A and repreent the overall reaction route a x = p + c v + c 2 v c m v m. 3 Vector Space and Kirchhoff Graph The vector pace method ued in the previou ection to find reaction route and overall reaction graph can now be ued to contruct Kirchhoff graph. A dicued 9

11 above, an overall reaction graph atifie Kirchhoff potential law; it i frequently poible to extend an overall reaction graph o that the extenion alo atifie Kirchhoff current law, i.e. that the rate aociated with all reaction tep tarting and ending at each vertex mut um to zero (once the overall reaction vector have been included). l Q0 b V H T T H V b 0Q k Figure 4: A planar Kirchhoff graph for the HER reaction network baed on the overall reaction graph in Figure 3. The central vertex i a null vertex. A wa the cae for the potential law in the previou ection, the current law here come directly from condition on the intermediate and terminal pecie in the reaction network. In a reaction network, the rate at which intermediate pecie are produced and conumed mut balance, meaning that there can be no net gain or lo in the network for any intermediate. For the terminal pecie, on the other hand, the rate of production and conumption are determined by the overall reaction rate. Thee condition can be expreed in term of an augmented verion of the tranpoe of the toichiometric matrix A. Suppoe there are µ linearly independent overall reaction. Let à := A b b 2... b µ, i.e., the matrix A with the overall reaction vector b i adjoined to it. A Kirchhoff graph will then atify the current condition if and only if the combination of edge incident on each vertex of the graph lie in Row(Ã), the row pace of Ã. The meaning of thi row-pace requirement i bet undertood in term of our HER example: Example d: One lat time, let u return to the HER reaction network. For thi reaction network, à = (2) So Row(Ã) = Span{w, w 2 }, (3) 0

12 where w := [, 0,, ], w 2 := [,, 0, ] atify the minimum norm requirement dicued above. One now contruct an HER Kirchhoff graph more-or-le by inpection. Since each of the overall reaction graph in Figure 3 already atify the potential law, either provide a tarting point for contructing a Kirchhoff graph for HER. The goal i to check the exiting vertice to ee which atify the row pace condition, then create new null route (cycle) by adding edge and vertice, including edge correponding to overall reaction, to achieve a graph that doe atify all the Kirchhoff condition. Starting with the firt verion, notice that the vertex at the origin already atifie a current condition (lie in the row pace). The vertex at the top of the graph, on the other hand, doe not lie in the row pace, o one or more additional directed edge mut be added to atify a current condition. The implet way to do thi i to add a copy of the overall reaction vector b tarting from the top vertex; then thi top vertex doe lie in the row pace ince it can be repreented by w 2. Notice that adding thi edge alo create a new vertex. From here, the implet way to complete the proce and achieve a Kirchhoff graph i to add two more edge: a copy of T on the right ide of the graph and a copy of H heading from the center to the upper right hand vertex of the graph. The reulting Kirchhoff graph i hown in Figure 4. A imple check confirm that the reulting graph atifie all the current and potential law. Notice that the center vertex i a null vertex in the ene that V and H both head in and out. Such null vertice alway lie in the row pace (the zero vector mut be in any vector pace). Remark. (4). Becaue Kirchhoff graph are geometric graph (i.e., the edge of our Kirchhoff graph are alo projection of vector in a vector pace), the length and direction of all edge aociated with a given reaction tep are the ame in Figure 4. So other verion of thi Kirchhoff graph can be drawn by changing the angle of projection. 2. In ome ene, the HER Kirchhoff graph in Figure 4 i the union of the two overall graph in Figure 3. But not all Kirchhoff graph can be eaily contructed a the extenion of a given overall reaction graph. In particular, Kirchhoff graph are not unique. Thi can be een when a Kirchhoff graph i a cycle ince the order in which the edge appear around the cycle i arbitrary. For a more intereting example, conider an alternate Kirchhoff graph for the HER network preented by Fihtik et al. [9] hown in Figure 5. Notice that thi econd Kirchhoff graph genuinely ditinct, not jut a different projection of the firt: it ha one le vertex than doe the firt and no null vertex. Which of thee two Kirchhoff graph one prefer depend on what ue one wihe to make of the graph.

13 Q0 l H T V b V H Q 0 k Figure 5: A econd Kirchhoff graph for the HER reaction network. It i a Kirchhoff graph ince it atifie both the current and potential condition, but it i ditinct from from the previou Kirchhoff graph ince it ha a different number of vertice, and the vertice are aociated with differing edge. In particular, thi graph ha no null vertex. It i a multigraph ince multiple edge are needed to achieve the current balance at each of the vertice. Alo there i no planar repreentation for thi graph becaue it mut be a geometric graph. 4 Kirchhoff Graph from an Incidence Matrix The previou ection preented the contruction of a Kirchhoff graph for a reaction network by inpection baed on the overall reaction graph and the Kirchhoff current requirement. Thi approach work well when a Kirchhoff graph i cloe to the overall reaction graph, but it may not work well when the Kirchhoff graph i more complicated. The current ection addree thi iue, preenting an approach which i baed on the vector-pace requirement, but not on the overall reaction graph contruction. Thi dicuion alo make clear that the Kirchhoff graph concept i in fact a property of a matrix whether or not the matrix i the tranpoe of the toichiometric matrix of ome reaction network. Example 2a: Conider the following toy (model) reaction network made up of four intermediate pecie and four reaction tep 2 : : CH 4 + C 3 H 8 2C 2 H 6, 2 : 2CH 4 + C 4 H 0 3C 2 H 6, 3 : CH 4 + 2C 4 H 0 3C 3 H 8, 4 : C 2 H 6 + C 4 H 0 2C 3 H 8. (5) Thi network ha no terminal pecie and thu no overall reaction. Nonethele, it demontrate a nontrivial ituation where the Kirchhoff graph i urpriingly com- 2 Thi network wa preented to the author by Ilie Fihtik a an example of a mall but difficult network for which to contruct a Kirchoff or reaction route graph. 2

14 plicated. Since there are no overall reaction, all the cycle in the Kirchhoff graph lie in Null(A) which in thi cae i Null(A) = Span 0 2, 2 0. (6) Again ince there are no overall reaction, the vertex current balance condition are given by the row pace of A (for hort, the vertice mut lie in the the row pace of A). Row(A) = Span{[, 2,, 0], [0,, 2, ]}, (7) Contruction of a Kirchhoff graph can be baed a the tranformation by elementary row operation of the tranpoe of the toichiometric matrix for thi network to a incidence matrix for the Kirchhoff graph. An incidence matrix for a graph ha a poitive integral entry a ij in the i, j-th poition when an edge repreenting tep i i baed at vertex j with multiplicity a ij. A negative entry repreent an edge ending at (heading into) a given vertex. The tranformation begin with the reduction of the toichiometric tranpoe to a matrix form of the row pace bae which atifie our minimal norm criteria from Section 2.4. Such a bae for Example 2 i given in (7). All zero row may be dicarded. One mut then add new row which are linear combination of exiting row and which have entrie which pair with exiting entrie to form incidence pair repreenting the beginning and ending vertice of an edge (or multiple of an edge) in the Kirchhoff graph. Thee incidence pair mut alo be conitent with the null pace requirement for the cycle in the Kirchhoff graph. Again thi proce i probably bet een in term of an example: Example 2b: Let u begin with the firt bai vector in (7) for the row pace and contruct the firt cycle in (6). The firt entry in the firt row pace bai vector i, o there mut be a correponding in the (2,) entry of the incidence matrix. Alo ince 2 i not in the firt cycle, one can try to contruct the econd vertex without thi tep. Thi implie a 0 in the (2,2) poition. To achieve both of thee reult, the econd row mut be form adding twice the econd row pace bai vector to the negative of the firt bai vector. The reulting econd row ha a 2 in the (2,4) poition. Since our cycle ha two 4 tep, it i reaonable to place thee tep in erie with a null-vertex between them. It i often ueful to place uch a null-vertex at the center of the graph. For the fourth and final vertex in thi cycle, it i again reaonable to try to avoid 2 ; thi mean that for the fourth row in the incidence matrix, the (4,4) entry mut be 2 and the (4,2) entry hould be 0. Thi row can only be achieved by ubtracting twice the econd bai vector in (7) from the firt bai vector. The reulting row ha a 3 in the (4,3) poition, and the null-pace requirement now force the (,3) entry to be it incidence pair and thu be 3. Thi of coure doe not match the initial firt row where the (,3) entry i. If one add the econd bai vector in (7) to the initial firt row, however, one obtain a revied firt row that doe atify the requirement for the firt cycle in (6). Thi reviion lead to a 3 in the (,2) poition which i unavoidable given the 3

15 other requirement o far. The partial incidence matrix contructed o far i (8) The entrie which have been paired o far are in bold. Now the firt cycle from the null pace bai (6) mut be added to the graph and in particular to the partial incidence matrix (8). Starting with the currently unpaired 3 in the (,2) poition in (8), one mut contruct a fifth row with a 3 in the (5,2) poition. In addition, ince the firt cycle doe not contain 3, one can try to avoid thi tep in the new row and correponding vertex by placing a 0 in the (5,3) poition. Both of thee requirement can be achieved by adding the econd bai vector from (7) to the negative of the firt bai vector. The reulting row ha a 2 in the (5,) poition. Since the firt vector in the null pace bai (6) contain 2 copie of the firt reaction tep, it again make ene to try thee in erie with a null vertex between them. Thi may be (but doe not necearily have to be) the ame null vertex a the one in the previou cycle. For the moment, let u try to achieve a Kirchhoff graph with only one null vertex. To complete thi cycle, one need a ixth row (and ixth vertex) with a 2 in the (6,) poition and a - in the (6,4) poition. Thee reult can be achieved if the ixth row i the negative of the fifth. With all of thee additional row, the partial incidence matrix become (9) The entrie that are paired to form the econd cycle are in italic. Of coure the matrix in (9) i not yet an incidence matrix; there are till unpaired entrie. But reviewing the thee unpaired entrie, one find that they can be paired through a ingle additional row and the addition of thi row i conitent with the null pace cycle requirement. One can alo note that the row of the incidence matrix can be permuted o that the reulting matrix i antiymetric about the null row: (20) The entrie in the row that correpond to the null vertex have been changed in (20) to zero. Uing thi incidence matrix, one can contruct a Kirchhoff graph, one 4

16 4 2 3 Figure 6: Kirchhoff graph for the methane-ethane-butane-propane toy reaction network. To implify the diagram, only one vector of each reaction-tep type i labeled; other of the ame type have the ame length and direction. Alo multiple edge are now indicated by hah mark, for example three hah mark indicate that three edge hould connect the vertice at each end of the edge hown. verion of which i hown in Figure 6. Thi Kirchhoff graph i inherently nonlinear. It ha 7 vertice and 22 edge. Nonethele it appear to be (up to projection orientation) the implet Kirchhoff graph for thi reaction network. Of coure in the contruction above, there were a number of free choice where one could have taken another route and thereby contructed a different matrix or reached an impa indicating that the choen route did not lead to an valid incidence matrix. If one had reached an impa, it would have been neceary to go back and change one or more of the free choice until a valid incidence matrix i reached. While there i no guarentee that thi will alway work, experience indicate that at leat it often doe. Thi proce can be programed o that a computer could accomplih the earch. In ummary, Example 2 ugget a general method for contructing Kirchhoff graph. Let A now denote the augmented tranpoe of the toichiometric matrix (which wa denoted a à in Example above). The tep for contructing an incidence matrix and thereby a Kirchhoff graph a follow: Step : Contruct all the element of a cycle bai (a bai for Null(A) ) which atifie the minimal total norm requirement. Step 2: Contruct a minimal total norm bai for the row pace of A. Step 3: Starting from the matrix whoe row are the bai in Step 2 ue row operation to contruct incidence pair, adding new row whenever neceary. The incidence pair mut combine to yield element of the cycle pace from Step. A in both Example and Example 2, it may be neceary to introduce one or more null vertice (a row of zero) to form an incidence matrix which i conitent with the cycle bai. 5

17 Step 4: If all vertice now atify Kirchhoff law, the contruction i complete. If not, return to Step 3 and re-pair the element of the cycle bai (changing which element are adjacent to each other). If an incidence matrix till can not be achieved, return to Step and 2 and conider another bai which atifie (or nearly atifie) the minimal norm requirement. 5 Concluion Perhap the mot intereting outcome of thi work i the conjecture that at leat one Kirchhoff graph exit for every matrix, not jut thoe aociated with reaction network. While there i no attempt here to fully tudy matrice and their Kirchhoff graph, a number of concluding obervation eem to be in order. There are two extreme cae (which do not correpond to reaction network) to conider firt, followed by a number of other intereting cae: For the zero matrix (any dimenion), the Kirchhoff graph i a ingle vertix with no edge. Similarly if A i a matrix with only one linearly independent row, then the null pace condition imply that all the edge are calar multiple of a ingle edge. Since every Kirchhoff graph i a cycle, the only poibility i again the degenerate cae with one vertix and no edge. At the other extreme, uppoe Null(A) =. Then the only poible cycle i a null cycle, and uch a cycle can alway be contructed. To ee thi, let A be an n m matrix; then Null(A) = implie n m and there mut be m linearly independent row. Row operation can reduce thi to an m m diagonal block, and from thi block, row operation can generate the incidence matrix for a ingle null cycle where all of the tep cancel. A alway, there i no claim that thi Kirchhoff graph i unique. Example 3: A = (2) The incidence pair are each diagonal entry in each 3 3 block and the oppoite igned entry jut below. The correponding null cycle i preented in Figure 7. The example o far might make one think that Kirchhoff graph are alway ymmetric; thi i not the cae. Example 4: Suppoe A = (22)

18 2 3 Figure 7: A null cycle Kirchhoff graph for a 5 3 matrix whoe null pace i empty baed on the incidence matrix in (2). Then the implet Kirchhoff graph correponding to thi matrix i 3 2 Figure 8: A nonymmetric Kirchhoff graph for (22). It i worth noting that a relatively imple matrix may not have in any ene a imple Kirchhoff graph. Example 5: For the Then A = (23) Null(A) = Span{[ 3, 2,, 8, 3, 0], [,,, 2, 3, ]} (24) and the row-reduced form i (25) Therefore a Kirchhoff graph for thi matrix would be compoed of two cycle one of which ha length 27. The Kirchhoff current balence condition would alo be complicated. So there i no guarentee that a relatively mall matrix (or a relatively mall reaction network) will have a imple Kirchhoff graph. 7

19 b 3 Figure 9: A Kirchhoff graph for (26). Although the network i rather complicated, thi Kirchhoff graph i make it much eaier to undertand. Hah mark again indicate edge multicity. The perpective i choen for ethetic conideration; horizontal tep do not necearily indicate terminal pecie in thi perpective. Returning to the connection between reaction network and Kirchhoff graph, it i worth preenting a omewhat complicated reaction network which i more eaily undertood in term of a Kirchhoff graph. In ome ene thi i the invere of Example 5 where the Kirchhoff graph i if anything more complicated than the matrix it i baed on. Conider the following reaction network for methanol decompoition on Pt() in a direct methanol fuel cell [22]: : CH 3 OH + S CH 3 OH S, 2 : CH 3 OH S + S CH 3 O S + H S, 3 : CH 3 O S + 2S CH 3 O S 2 + H S, 4 : CH 3 O S 2 CHO S + H S, 5 : CHO S CO S + H S, 6 : CH 3 OH S + S CH 2 OH S + H S, 7 : CH 2 OH S + S CHOH S + H S, 8 : CHOH S + S COH S + H S, 9 : COH S + S CO S + H S, 0 : CHOH S + 2S CO S + 2H S, : CH 2 OH S + 2S CH 2 O S 2 + H S, 2 : CO S CO + S, 3 : H S + H S H 2 + 2S, (26) Vilekar, Fihtik & Datta (2007) have preented a Kirchhoff graph for thi reaction network and then ue the graph to calculate an overall reaction rate and to find a impler equivalent Kirchhoff graph and correponding reaction network. Depite the apparent complexity of (26), one find that there i a ingle overall reaction, and after thi overall reaction i augmented to the network, the null pace conitent of 8

20 four cycle which can be joined together to produce the Kirchhoff graph in Figure 9. 6 Acknowledgment The author wihe to thank Ravi Datta, Ilie Fihtik and Caitlin Callaghan for introducing him to the concept of reaction route graph, and Bill Martin and Pete Chritopher for many ueful graph-theoretic dicuion. Reference [] J. A. Bondy and U. S. R. Mutty, Graph theory with application, North-Holland, New York, 976, Chapter 2 deal with cycle and bond pace. [2] C. A. Callaghan, I. Fihtik, R. Datta, M. Carpenter, M. Chmielewki, and A. Lugo, An improved microkinetic model for the water ga hift reaction on copper, Surf. Sci. 54 (2003), [3] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction network: I. the injectivity property, SIAM J. Appl. Math. 65 (2005), [4] G. Craciun, Y. Tang, and M. Feinberg, Undertanding bitability in complex enzyme-driven reaction network, PNAS 03 (2006), [5] J. D. Fehribach, The ue of graph in the tudy of electrochemical procee, Modern Apect of Electrochemitry, vol. 40, 2006, pp. xxx xxx. [6] I. Fihtik, C. A. Callaghan, and R. Datta, Reaction route graph. i. theory and algorithm, J. Phy. Chem. B 08 (2004), [7], Reaction route graph. ii. example of enzyme and urface catalyzed ingle overall reaction, J. Phy. Chem. B 08 (2004), [8], Reaction route graph. iii. non-minimal kinetic mechanim, J. Phy. Chem. B 09 (2005), [9] I. Fihtik, C. A. Callaghan, J. D. Fehribach, and R. Datta, A reaction route graph analyi of the electrochemical hydrogen oxidation and evolution reaction, J. Electroanal. Chem. 576 (2005), [0] I. Fihtik and R. Datta, A thermodynamic approach to the ytematic elucidation of unique reaction route in catalytic reaction, Chem. Eng. Sci. 55 (2000), [], De donder relation in mechanitic and kinetic analyi of heterogeneou catalytic reaction, Ind. Eng. Chem. Re. 40 (200), [2], A UBI-QEP microkinetic model for the water-ga hift reaction on Cu( ), Surf. Sci. 52 (2002),

21 [3], New network architecture for toichiometrically, thermodynamically and kinetically balanced metabolic reaction ytem, Phyica A 378 (2007), [4] F. Horn, On a connexion between tability and graph in chemical kinetic. I. Stability and the reaction diagram, Proc. R. Soc. Lond. A. 334 (973), [5], On a connexion between tability and graph in chemical kinetic. II. Stability and the complex graph, Proc. R. Soc. Lond. A. 334 (973), [6] L. W. Johnon and R. D. Rie, Introduction to linear algebra, Addion-Weley, Reading, Maachett, 98. [7] G. F. Oter, A. S. Perelon, and A. Katchalky, Network thermodynamic: dynamic modelling of biophyical ytem, Q. Review Biophy. 6 (973), 34. [8] H. Qian, D. A. Beard, and S d. Liang, Stoichiometric network theory for nonequilibrium biochemical ytem, European J. Biochem. 270 (2003), [9] D. V. Robert, Enzyme kinetic, Cambridge Univ. Pre, Cambridge, 977. [20] P. Schloer and M. Feinberg, A theory of multiple teady tate in iothermal homogeneou CFSTR with many reaction, Chem. Engrg. Sci. 49 (994), [2] G. Strang, Linear algebra and it application, 4 th ed., Academic Pre, New York, [22] S. A. Vilekar, I. Fihtik, and R. Datta, Topological analyi of catalytic reaction network: Methanol decompoition on pt(), J. Catalyi 252 (2007),