arXiv:0903.5072v2 [physics.chem-ph] 14 Jun 2010 UK science ∗ of direction with this used for be “asymptotology” The developed. name should was culture terms special and “big” precautions, separation and simple “small” a on such that closer shows The small that them. consideration are neglect almost terms assumption or other neglect an to some enough and backgrounds big, are the terms models some in theoretical basic use the and of simplifications are Introduction 1. PACS: words: Key Applica examples. simple const on demonstrated kinetic rate is of reaction algorithms eigenvectors the separated and asymptoto eigenvalues well the of to with approximations extended networks is linear step limiting for the of concept The Abstract [email protected] [email protected] rpitsbitdt Elsevier to submitted Preprint mi addresses: Email 7RH, LE1 Leicester, of University author: Corresponding oto ahmtclmdl htral work really that models mathematical of Most 46.q 24.t 23.k 23.t8.5R,89.75.F 87.15.R-, 82.39.Rt 82.39.Fk, 82.40.Qt, 64.60.aq, ecinntok smttlg,dmnn ytm limiti system, dominant asymptotology, network, Reaction ntttCre 90ISR/ui/ie aiTc,2 rue 26 ParisTech, INSERM/Curie/Mines U900 Curie, Institut RA,UR62,Uiest fRne ,Cmu eBeaulieu, de Campus 1, Rennes of University 6625, UMR IRMAR, smttlg fCeia ecinNetworks Reaction Chemical of Asymptotology [email protected] A .Zinovyev). Y. (A. A .Gra ), Gorban N. (A. O Radulescu), (O. hmclEgneigSine6 21)2310-2324 (2010) 65 Science Engineering Chemical nvriyo ecse,UK Leicester, of University .N Gorban N. A. .Y Zinovyev Y. A. .Radulescu O. b arx cuayo siae spoe.Promneof Performance proven. is estimates of Accuracy matrix. ino loihst olna ytm sdiscussed. is systems nonlinear to algorithms of tion o Nwo oyern n ayohrthings). other many and polyhedron, New- (Newton I. ton by many developed and were older, approaches much fundamental are fundamental direction but this in (1963), research Kruskal by proposed was rlpstv variables positive eral upon.” decided descrip- been has its case of limiting and the once behavior tion, asymptotic the nonarbitrari- of the ness in resides el- asymptotology scientific in choosing The ement ... in examine partly to cases lies limiting asymptotology fruitful of case. limiting art a The in system ... a solution of specified solutions) of a family of (or behavior the describing of art Let nsi eeoe.W rsn loihsfrexplicit for algorithms present We developed. is ants oyo utsaerato ewrs opeetheory Complete networks. reaction multiscale of logy olwn rsa (1963), Kruskal Following smttcbhvo frtoa ucin fsev- of functions rational of behavior Asymptotic gse,mlicl smttc oe reduction model asymptotic, multiscale step, ng R ∗ ( k 1 k . . . , ’l,F54,Prs France Paris, F75248, d’Ulm, n = ) 54 ens France Rennes, 35042 P k ( i k 1 > k . . . , ie satoy-example. a us gives 0 asymptotology n ) /Q ( k 1 k . . . , n s“the is ) be such a function and P,Q be polynomials. To de- struct the explicit asymptotics of eigenvectors and rive fruitful limiting cases we consider logarithmic eigenvalues. All algorithms are represented topolog- straight lines ln ki = θiξ and study asymptotical be- ically by transformation of the graph of reaction (la- havior of R for ξ → ∞. In this asymptotics, for al- beled by reaction rate constants). The reaction rate most every vector (θi) (outside several hyperplanes) constants for dominant systems may not coincide there exists such a dominant monomial R∞(k) = with constant of original network. In general, they αi A i ki that R = R∞ + o(R∞). The function that are monomials of the original constants. associates a monomial with vector (θi) is piecewise This result fully supports the observation by constant:Q it is constant inside some polyhedral cones. Kruskal (1963): “And the answer quite generally Implicit functions given by equations which de- has the form of a new system (well posed prob- pend on parameters provide plenty of more inter- lem) for the solution to satisfy, although this is esting examples, especially in the case when the sometimes obscured because the new system is so implicit function theorem is not applicable. Some easily solved that one is led directly to the solution analytical examples are presented by Andrianov & without noticing the intermediate step.” Manevitch (2002) and White (2006). Introduction The dominant systems can be used for direct com- of algebraic backgrounds and special software is pro- putation of steady states and relaxation dynamics, vided by Greuel & Pfister (2002). especially when kinetic information is incomplete, For a difficult problem, analysis of eigenvalues for design of experiments and mining of experimen- and eigenvectors of non-symmetric matrices, Vishik tal data, and could serve as a robust first approxi- & Ljusternik (1960) studied asymptotic behavior of mation in perturbation theory or for precondition- spectra and spectral projectors along the logarith- ing. They can be used to answer an important ques- mic straight lines in the space of matrices. This anal- tion: given a network model, which are its critical ysis was continued by Lidskii (1965). parameters? Many of the parameters of the initial We study networks of linear reactions. For a linear model are no longer present in the dominant sys- system with reaction rate constants ki all the dy- tem: these parameters are non-critical. Parameters namical information is contained in eigenvalues and of dominant subsystems indicate putative targets to eigenvectors of the kinetic matrix or, more precisely, change the behavior of the large network. in its transformation to the Jordan normal form. It is Most of reaction networks are nonlinear, it is computationally expensive task to find this transfor- nevertheless useful to have an efficient algorithm for mation for a non-symmetric matrix which is usually solving linear problems. First, nonlinear systems stiff (Golub & Van Loan (1996)). Moreover, the an- often include linear subsystems, containing reac- swer could be verysensitiveto the errorsin constants tions that are (pseudo)monomolecular with respect ki. Nevertheless, it appears that stiffness can help to species internal to the subsystem (at most one us to find a robust approximation, and in the limit internal species is reactant and at most one is prod- when all constants are very different (well-separated uct). Second, for binary reactions A + B → ..., if constants) the asymptotical behavior of eigenvalues concentrations of species A and B (cA,cB) are well and eigenvectors follow simple explicit expressions. separated, say cA ≫ cB then we can consider this Analysis of this asymptotics is our main goal. reaction as B → ... with rate constant proportional In our approach, we study asymptotic behavior to cA which is practically constant, because its rel- of eigenvalues and eigenvectors of kinetic matrices ative changes are small in comparison to relative along logarithmic straight lines, ln ki = θiξ in the changes of cB. We can assume that this condition space of constants. We significantly use the graph is satisfied for all but a small fraction of genuinely representation of chemical reaction networks and nonlinear reactions (the set of nonlinear reactions demonstrate, that for almost every vector (θi) there changes in time but remains small). Under such exists a simple reaction network which describes an assumption, nonlinear behavior can be approxi- the dominant term of this asymptotic. Following mated as a sequence of such systems, followed one the asymptotology terminology (White (2006)), we each other in a sequence of “phase transitions”. In call this simple network the dominant system. For these transitions, the order relation between some of these dominant system there are explicit formulas species concentrations changes. Some applications for eigenvalues and eigenvectors. The topology of of this approach to systems biology are presented by dominant systems is rather simple: they are acyclic Radulescu, Gorban, Zinovyev & Lilienbaum (2008). networks without branching. This allows us to con- The idea of controllable linearization “by excess” of
2 some reagents is in the background of the efficient we introduce basic notions and notations. We con- experimental technique of Temporal Analysis of sider thermodynamic restrictions on the reaction Products (TAP), which allows to decipher detailed rate constants and demonstrate how appear systems mechanisms of catalytic reactions (Yablonsky, Olea, with arbitrary constants (as subsystems of more de- & Marin (2003)). tailed models). For linear networks, the main theo- In chemical kinetics various fundamental ideas rems which connect ergodic properties with topol- about asymptotical analysis were developed ogy of network, are reminded. Four basic ideas of (Klonowski (1983)): quasieqiulibrium asymptotic model reduction in chemical kinetics are described: (QE), quasi steady-state asymptotic (QSS), lump- QE, QSS, lumping analysis and limiting steps. ing, and the idea of limiting step. In Sec. 3, we introduce the dominant system for a Most of the works on nonequilibrium thermody- simple irreversible catalytic cycle with limiting step. namics deal with the QE approximations and correc- This is just a chain of reactions which appears after tions to them, or with applications of these approx- deletion the limiting step from the cycle. Even for imations (with or without corrections). There are such simple examples several new observation are two basic formulation of the QE approximation: the presented: thermodynamic approach, based on entropy max- – The relaxation time for a cycle with limiting step imum, or the kinetic formulation, based on selec- is inverse second reaction rate constant; tion of fast reversible reactions. The very first use of – For chains of reactions with well separated rate the entropy maximization dates back to the classical constants left eigenvectors have coordinates close work of Gibbs (1902), but it was first claimed for a to 0 or 1, and right eigenvectors have coordinates principle of informational statistical thermodynam- close to 0 or ±1. ics by Jaynes (1963). A very general discussion of For general reaction networks instead of linear the maximum entropy principle with applications to chains appear general acyclic non-branching net- dissipative kinetics is given in the review by Balian, works. For them we also provide explicit formulas Alhassid & Reinhardt (1986). Corrections of QE ap- for eigenvectors and their 0, ±1 asymptotics for proximation with applications to physical and chem- well-separated constants (Sec. 4). In (Sec. 5) the ical kinetics were developed by Gorban, Karlin, Ilg, main algorithm is presented. Sec. 6 is devoted to a & Ottinger¨ (2001); Gorban & Karlin (2005). simple demonstration of the algorithm application. QSS was proposed by Bodenstein (1913) and was In Sec. 7, we briefly discuss further corrections to elaborated into an important tool for analysis of dominant systems. The estimates of accuracy are chemical reaction mechanism and kinetics (Semenov given in Appendix. (1939); Christiansen (1953); Helfferich (1989)). The classical QSS is based on the relative smallness of 2. Main Asymptotic Ideas in Chemical concentrations of some of “active” reagents (radi- Kinetics cals, substrate-enzyme complexes or active compo- nents on the catalyst surface) (Aris (1965); Segel & Slemrod (1989)). 2.1. Chemical Reaction Networks Lumping analysis aims to combine reagents into “quasicomponents” for dimension reduction (Wei & To define a chemical reaction network, we have to Kuo (1969); Kuo & Wei (1969); Li & Rabitz (1989); introduce: Toth, Li, Rabitz, & Tomlin (1997). – a list of components (species); The concept of limiting step gives the limit simpli- – a list of elementary reactions; fication: the whole network behaves as a single step. – a kinetic law of elementary reactions. This is the most popular approach for model simpli- The list of components is just a list of symbols (la- fication in chemical kinetics and in many areas be- bels) A1, ...An. Each elementary reaction is repre- yond kinetics. In the form of a bottleneck approach sented by its stoichiometric equation this approximation is very popular from traffic man- agement to computer programming and communi- αsiAi → βsiAi, (1) i si cation networks. The proposed asymptotic analysis X X can be considered as a wide extension of the classical where s enumerates the elementary reaction, and idea of limiting step (Gorban & Radulescu (2008)). the non-negative integers αsi, βsi are the stoichio- The structure of the paper is as follows. In Sec. 2 metric coefficients. A stoichiomentric vector γs with
3 coordinates γsi = βsi − αsi is associated with each Bokor, & Szederk´enyi (2004). In general reaction elementary reaction. network coefficients ks (3) or ϕs (4) are not inde- For analysis of closed chemical systems with de- pendent. In order to respect the second law of ther- tailed balance it is usual practice to group reactions modynamics, they should satisfy some equations in pairs, direct and inverse reactions together, but and inequalities. The most famous sufficient condi- in more general settings this is not convenient. tion gives the principle of detailed balance. Let us A non-negative real extensive variable Ni ≥ 0, group the elementary reactions in pairs, direct and amount of Ai, is associated with each component Ai. inverse reactions, and mark the variables for direct It measures “the number of particles of that species” reactions by superscript +, and for inverse reac- (in particles, or in moles). The concentration of Ai tions by −. Then the principle of detailed balance is an intensive variable: ci = Ni/V , where V is vol- for general kinetics (4) reads: ume. It is necessary to stress, that in many prac- ϕ+ = ϕ− (5) tically important cases the extensive variable V is s s neither constant, nor the same for all components (Feinberg (1972)). For the isothermal mass action Ai. For more details see, for example the book of law the principle of detailed balance can be formu- Yablonskii, Bykov, Gorban, & Elokhin (1991). For lated as follows: there exists a strictly positive point simplicity, we will consider systems with one con- c∗ of detailed balance, at this point stant volume and under constant temperature, but + ∗ − ∗ ws (c )= ws (c ) (6) it is necessary always keep in mind the possibility to return to general equations. For that conditions, for all s. This is, essentially, the same principle: if we the kinetic equations have the following form substitute in the general reaction rate (4) the frac- ∗ tion µi/RT by ln(ci/ci ), then we will get the mass dc + − = w (c,T )γ + υ, (2) action law, and ϕs = ϕs . The principle of detailed dt s s s balance is closely related to the microreversibility X where υ is the vector of external fluxes normalized and Onsager relations. to unit volume. It may be useful to represent exter- More general condition was invented by Stueck- nal fluxes as elementary reactions by introduction of elberg (1952) for the Boltzmann equation. He pro- new component ∅ together with incoming and out- duced them from the S-matrix unitarity (the quan- tum complete probability formula). For the general going reactions ∅ → Ai and Ai → ∅. The most popular kinetic law of elementary reac- law (4) without direct-inverse reactions grouping for tions is the mass action law for perfect systems: any state the following identity holds:
αsi 1 ws(c,T )= ks(T ) ci , (3) ϕ exp α µ s RT si i s i ! where “kinetic constant” ks(T )Y depends on temper- X X (7) 1 ature T . More general kinetic law, which can be used ≡ ϕ exp β µ . s RT si i for most of non-ideal (non-perfect) systems is s i ! X X 1 Even more general condition which guarantees the w (c,T )= ϕ exp α µ , (4) s s RT si i second law and has clear microscopicsense (the com- i ! X plete probability does not increase) was obtained by where R is the universal gas constant, µi is the chem- Gorban (1984): for any state ∂F (N,T,V ) ∂G(N,T,P ) ical potential, µi = = , F is the ∂Ni ∂Ni 1 Helmgoltz free energy, G is the Gibbs energy (free ϕ exp α µ s RT si i enthalpy), P is pressure and ϕ > 0 is an intensive s i ! s X X (8) variable, kinetic factor, which can depend on any set 1 ≥ ϕ exp β µ . of intensive variables, first of all, on T . s RT si i s i ! Chemical thermodynamics (Prigogine & Defay X X (1954)) provides tools of choice for stability analy- To obtain formulas for the isothermal mass action sis of reaction networks (Procaccia & Ross (1977)) law, it is sufficient just to apply the general law (4) and chemical reactors (Aris (1965)). The laws of with constant ϕs to the perfect free energy F = thermodynamics have been used for analyzing of RT i ci(ln ci + µi0) with constant µi0. More de- structural stability of process systems by Hangos, tailed analysis was presented, by Gorban (1984). P 4 In any case, reaction constants are dependent, finite Markov chains in continuous time (Meyn & and this dependence guarantees stability of equilib- Tweedie (2009); Meyn (2007)). rium and existence of global thermodynamic Lya- A linear conservation law is a linear function de- punov functions for closed systems (2) with υ = fined on the concentrations b(c) = i bici, whose 0. Nevertheless, we often study equations for such value is preserved by the dynamics (9). The conser- P systems with oscillations, bifurcations, chaos, and vation laws coefficient vectors bi are left eigenvectors other effects, which are impossible in systems with of the matrix K corresponding to the zero eigen- global Lyapunov function. Usually this means that value. The set of all the conservation laws forms the we study a subsystem of a large system, where some left kernel of the matrix K. Equation (9) always has 0 of concentrations do not change because they are a linear conservation law: b (c) = i ci = const. stabilized by external fluxes or by a large external If there is no other independent linear conservation reservoir. These constant (or very slow) concentra- law, then the system is weakly ergodicP. tions are included into new reaction constants, and A set E is positively invariant with respect to ki- after this redefinition they can loose any thermody- netic equations (9), if any solution c(t) that starts namic property. in E at time t0 (c(t0) ∈ E) belongs to E for t>t0 (c(t) ∈ E if t>t0). It is straightforward to check that the standard simplex Σ = {c | ci ≥ 0, i ci = 2.2. Linear Networks and Ergodicity 1} is positively invariant set for kinetic equation (9): P just to check that if ci = 0 for some i, and all cj ≥ 0 In this Sec., we consider a general network of thenc ˙i ≥ 0. This simple fact immediately implies linear reactions. This network is represented as a the following properties of K: directed graph (digraph) (Temkin, Zeigarnik, & – All eigenvalues λ of K have non-positive real Bonchev (1996)): vertices correspond to compo- parts, Reλ ≤ 0, because solutions cannot leave Σ nents Ai, edges correspond to reactions Ai → Aj in positive time; with kinetic constants kji > 0. For each vertex, Ai, – If Reλ = 0 then λ = 0, because intersection of a positive real variable ci (concentration) is defined. Σ with any plane is a polygon, and a polygon i A basis vector e corresponds to Ai with compo- cannot be invariant with respect to rotations to i nents ej = δij , where δij is the Kronecker delta. sufficiently small angles; The kinetic equation for the system is – The Jordan cell of K that corresponds to zero eigenvalue is diagonal – because all solutions dc i = (k c − k c ), (9) should be bounded in Σ for positive time. dt ij j ji i j – The shift in time operator exp(Kt) is a contrac- X tion in the l1 norm for t> 0. or in vector form:c ˙ = Kc. We don’t assume any spe- – For weakly ergodic systems there exists such a cial relation between constants, and consider them monotonically decreasing function δ(t) (t> 0,0 < as independent quantities. The thermodynamic re- δ(t) < 1, δ(t) → 0 when t → ∞) that for any two strictions on constants are not applicable here be- solutions of (9) c(t),c′(t) ∈ Σ cause, in general, we study pseudomonomolecular ′ ′ systems which are subsystems of larger nonlinear |ci(t) − ci(t)|≤ δ(t) |ci(0) − ci(0)| . (10) i i systems and don’t represent by themselves closed X X monomolecular systems. The ergodicity coefficient δ(t) was introduced by For any network of linear reactions the matrix of Dobrushin (1956)(see also a book by Seneta (1981)). kinetic coefficients K has the following properties: It can be estimated using the structure of the net- – non-diagonal elements of K are non-negative; work graph (Gorban, Bykov & Yablonskii (1986); – diagonal elements of K are non-positive; Meyn (2007)). – elements in each column of K have zero sum. Two vertices are called adjacent if they share a For any K with these properties there exists a net- common edge. A path is a sequence of adjacent ver- work of linear reactions with kinetic equationc ˙ = tices. A graph is connected if any two of its vertices Kc. This family of matrices coincide with the family are linked by a path. A maximal connected sub- of generators of finite Markov chains, and this class graph of graph G is called a connected component of of kinetic equations coincide with the class of inverse G. Every graph can be decomposed into connected Kolmogorov’s equations or master equations for the components.
5 A directed path is a sequence of adjacent edges tion laws (i.e. the geometric multiplicity of the zero where each step goes in direction of an edge. A ver- eigenvalue of the matrix K) is equal to the maximal tex A is reachable from a vertex B, if there exists a number of disjoint ergodic components (minimal directed path from B to A. sinks). A nonempty set V of graph vertices forms a sink, if there are no directed edges from Ai ∈ V to any Aj ∈/ 2.3. Quasi-equilibrium (QE) or Fast Equilibrium V . For example, in the reaction graph A1 ← A2 → A the one-vertex sets {A } and {A } are sinks. 3 1 3 Quasi-equilibrium approximation uses the as- A sink is minimal if it does not contain a strictly sumption that a group of reactions is much faster smaller sink. In the previous example, {A }, {A } 1 3 than other and goes fast to its equilibrium. We use are minimal sinks. Minimal sinks are also called er- below superscripts ‘f ’ and ‘s’ to distinguish fast and godic components. slow reactions. A small parameter appears in the A digraph is strongly connected, if every vertex A following form is reachable from any other vertex B. Ergodic com- dc 1 ponents are maximal strongly connected subgraphs = ws (c,T )γs + wf (c,T )γf , of the graph, but inverse is not true: there may ex- dt σ σ ε ς ς (11) σ, slow ς, fast ist maximal strongly connected subgraphs that have X X outgoing edges and, therefore, are not sinks. To separate variables, we have to study the spaces The weak ergodicity of the network follows from of linear conservation law of the initial system (11) its topological properties. and of the fast subsystem Theorem 1. The following properties are equiv- dc 1 = wf (c,T )γf alent (and each one of them can be used as an alter- dt ε ς ς ς, native definition of weak ergodicity): Xfast (i) There exist the only independent linear con- If they coincide, then the fast subsystem just dom- servation law for kinetic equations (9) (this is inates, and there is no fast-slow separation for 0 b (c)= i ci = const). variables (all variables are either fast, or constant). (ii) For any normalized initial state c(0) (b0(c) = But if there exist additional linearly independent 1) thereP exists a limit state linear conservation laws for the fast system, then c∗ = lim exp(Kt) c(0) let us introduce new variables: linear functions t→∞ b1(c), ...bn(c), where b1(c), ...bm(c) is the basis of the that is the same for all normalized initial con- linear conservation laws for the initial system, and ditions: For all c, b1(c), ...bm+l(c) is the basis of the linear conservation m+l+1 n 0 ∗ laws for the fast subsystem. Then b (c), ...b (c) lim exp(Kt) c = b (c)c . m m l t→∞ are fast variables, b +1(c), ...b + (c) are slow vari- 1 m (iii) For each two vertices Ai, Aj (i 6= j) we can find ables, and b (c), ...b (c) are constant. The quasi- such a vertex Ak that is reachable both from equilibrium manifold is given by the equations f f Ai and from Aj . This means that the following ς wς (c,T )γς = 0 and for small ε it serves as an structure exists: approximation to a slow manifold. In the old and Pstandard approach it is assumed that system (11) as Ai → ... → Ak ← ... ← Aj . well as system of fast reactions satisfies the thermo- One of the paths can be degenerated: it may dynamic restrictions, and the quasi-equilibrium is be i = k or j = k. just a partial thermodynamic equilibrium, and could (iv) The network has only one minimal sink (one be defined by conditional extremum of thermody- ergodic component). namic functions. This guarantees global stability of The proof of this theorem could be extracted from fast subsystems and all the classical singular per- detailed books about Markov chains and networks turbation theory like Tikhonov theorem could be (Meyn (2007); Van Mieghem (2006)). In its present applied. form it was published by Gorban, Bykov & Yablon- Recently, Vora & Daoutidis (2001) took notice skii (1986) with explicit estimations of ergodicity that this type of reasoning does not require clas- coefficients. sical thermodynamic restrictions on constants. For For every monomolecular kinetic system, the example, let us consider the mass action law ki- maximal number of independent linear conserva- netics and group the reactions in pairs, direct and
6 inverse reactions. If the set of stoichiometric vec- the same order”). We can apply the standard singu- tors for fast reactions is linearly independent, then lar perturbation techniques. If dynamics of fast com- for this system the detailed balance principle holds ponents under given values of slow concentrations (obviously), and it demonstrates the “thermody- is stable, then the slow attractive manifold exists, namic behaviour” without connection to classical and its zero approximation is given by the system thermodynamics. This case of “stoichiometrically of equations W f (cs,cf ) = 0. Bifurcations in fast sys- independent fast reactions” can be generalized for tem correspond to critical effects, including ignition irreversible reactions too (Vora & Daoutidis (2001)). and explosion. For such fast system the quasiequilbrium manifold This scheme was analyzed many times with plenty has the same nice properties as for thermodynamic of details, examples, and some complications. Ex- partial equilibrium, and approximates slow dynam- haustive case study of the simplest enzyme reaction ics for sufficiently small ε. was provided by Segel & Slemrod (1989) . For het- There are other classes of mass action law sub- erogenious catalytic reactions, the book by Yablon- systems with such a “quasi-thermodynamic” be- skii, Bykov, Gorban, & Elokhin (1991) gives analysis haviour, which depends on structure, but not on of scaling of fast intermediates (there are many kinds constants. For example, any system of reactions of possible scaling). In the context of the Computa- without interactions has such a property (Gorban, tional Singular Perturbation (CSP) approach, Lam Bykov, & Yablonskii (1986)). These reactions have (1993) and Lam & Goussis (1994) developed concept the form αAi → ...: any linear reaction are al- of the CSP radicals. Gorban & Karlin (2003, 2005) lowed, as well as reactions like 2Ai → Aj + Ak, considered QSS as initial approximation for slow in- P 3Ai → Aj + Ak + Al, etc. All such fast subsystems variant manifold. Analysis of the error of the QSS can serve for quasi-equilibrium approximation, be- was provided by Turanyi, Tomlin, & Pilling (1993). cause for them dynamics is globally stable. The QE approximation is also extremely popular Quasi-equilibrium manifold approximates expo- and useful. It has simpler dynamical properties (re- nentially attractive slow manifold and is used in spects thermodynamics, for example, and gives no many areas of kinetics either as initial approxima- critical effects in fast subsystems of closed systems). tion for slow motion, or just by itself (more discus- Nevertheless, neither radicals in combustion, nor in- sion and further references are presented by Gorban termediates in catalytic kinetics are, in general, close & Karlin (2005)). to quasi-equilibrium. They are just present in much smaller amount, and when this amount grows, then the QSS approximation fails. 2.4. Quasi Steady-State (QSS) or Fast Species The simplest demonstration of these two approx- imation gives the simple reaction: S + E ↔ SE → ± The quasi steady-state (or pseudo steady state) P + E with reaction rate constants k1 and k2. The assumption was invented in chemistry for descrip- only possible quasi-equilibrium appears when the ± ± tion of systems with radicals or catalysts. In the first equilibrium is fast: k1 = κ /ε. The correspond- s most usual version the species are split in two groups ing slow variable is C = cS +cSE, bE = cE +cSE = with concentration vectors cs (“slow” or basic com- const. For the QE manifold we get a quadratic equa- − f k ponents) and c (“fast intermediates”). For catalytic 1 s tion + c = c c = (C − c )(b − c ). This k SE S E SE E SE reactions there is additional balance for cf , amount 1 equation gives the explicit dependence c (Cs), and of catalyst, usually it is just a sum b = cf . The SE f i i the slow equation reads C˙ s = −k c (Cs), Cs + amount of the fast intermediates is assumed much 2 SE c = b = const. smaller than the amount of the basic components,P P S For the QSS approximation of this reaction ki- but the reaction rates are of the same order, or netics, under assumption b ≪ b , we have fast in- even the same (both intermediates and slow compo- E S termediates E and SE. For the QSS manifold there nents participate in the same reactions). This is the is a linear equation k+c c − k−c − k c = 0, source of a small parameter in the system. Let us 1 S E 1 SE 2 SE which gives us the explicit expression for c (c ): scale the concentrations cf and cs to the compatible SE S c = k+c b /(k+c + k− + k ) (the standard amounts. After that, the fast and slow time appear SE 1 S E 1 S 1 2 Michaelis–Menten formula). The slow kinetics reads and we could writec ˙s = W s(cs,cf ),c ˙f = 1 W f (cs,cf ), ε c˙ = −k+c (b −c (c ))+k−c (c ). The differ- where ε is small parameter, and functions W s, W f S 1 S E SE S 1 SE S ence between the QSS and the QE in this example are bounded and have bounded derivatives (are “of
7 is obvious. Rabitz (1989); Toth, Li, Rabitz, & Tomlin (1997)), The terminology is not rigorous, and often QSS is and more general nonlinear forms of lumped con- used for all singular perturbed systems, and QE is centrations are used (for example, concentration of applied only for the thermodynamic exclusion of fast quasicomponents could be rational function of c). variables by the maximum entropy (or minimum of Hutchinson & Luss (1970) studied lumping- free energy, or extremum of another relevant ther- analysis of mixtures with many parallel first order modynamic function) principle (MaxEnt). This ter- reactions. Farkas (1999) generalized these results minological convention may be convenient. Never- and characterized those lumping schemes which theless, without any relation to terminology, the dif- preserve the kinetic structure of the original system. ference between these two types of introduction of a Coxson & Bischoff (1987) placed lumping analysis small parameter is huge. There exists plenty of gen- in the linear systems theory and demonstrated the eralizations of these approaches, which aim to con- relationships between lumpability and the concepts struct a slow and (almost) invariant manifold, and to of observability, controllability and minimal real- approximate fast motion as well. The following ref- ization. Djouad & Sportisse (2002) considered the erences can give a first impression about these meth- lumping procedures as efficient techniques leading ods: Method of Invariant Manifolds (MIM) (Roussel to nonstiff systems and demonstrated efficiency & Fraser (1991); Gorban & Karlin (2005), Method of of developed algorithm on kinetic models of at- Invariant Grids (MIG), a discrete analogue of invari- mospheric chemistry. Lin, Leibovici & Jorgensen ant manifolds (Gorban, Karlin, & Zinovyev (2004)), (2008) formulated an optimal lumping problem as Computational Singular Perturbations (CSP) (Lam a mixed integer nonlinear programming (MINLP) (1993); Lam & Goussis (1994); Zagaris, Kaper, & and demonstrated that it can be efficiently solved Kaper (2004)) Intrinsic Low-Dimensional Manifolds with a stochastic optimization method, Tabu Search (ILDM) by Maas, & Pope (1992), developed further (TS) algorithm. in series of works by Bykov, Goldfarb, Gol’dshtein, & The power of lumping using a time-scale based Maas, U. (2006)), methods based on the Lyapunov approach was demonstrated by Whitehouse, Tom- auxiliary theorem (Kazantzis & Kravaris (2006)). lin, & Pilling (2004). This computationally cheap approach combines ideas of sensitivity analysis with simple and useful grouping of species with similar 2.5. Lumping Analysis lifetimes and similar topological properties caused by connections of the species in the reaction net- Wei & Prater (1962) demonstrated that for works. The lumped concentrations in this approach (pseudo)monomolecular systems there exist linear are simply sums of concentrations in groups. For ex- combinations of concentrations which evolve in time ample, species with similar composition and func- independently. These linear combinations (quasi- tionalities could be lumped into one single represen- components) correspond to the left eigenvectors of tative species (Pepiot-Desjardins & Pitsch (2008)). kinetic matrix: if lK = λl then d(l,c)/dt = (l,c)λ, Lumping analysis based both on mathematical where the standard inner product (l,c) is concentra- arguments and fundamental physical and chemi- tion of a quasicomponent. They also demonstrated cal properties of the components is now one of the how to find these quasicomponents in a properly main tools for model reduction in highly multicom- organized experiment. ponent systems, such as the hydrocarbon mixture This observation gave rise to a question: how in petroleum chemistry (Zavala & Rodriguez & to lump components into proper quasicomponents Vargas-Villamil (2004)) or biochemical networks to guarantee the autonomous dynamics of the in systems biology (Maria (2006)). The optimal quasicomponents with appropriate accuracy. Wei solution of lumping problem often requires the ex- and Kuo studied conditions for exact (Wei & Kuo haustive search, and instead of them various heuris- (1969)) and approximate (Kuo & Wei (1969)) lump- tics are used to avoid combinatorial explosion. For ing in monomolecular and pseudomonomolecular the lumping analysis of the systems biology mod- systems. They demonstrated that under certain els Dokoumetzidis & Aarons (2009) developed a conditions large monomolecular system could be heuristic greedy search strategy which allowed them well–modelled by lower–order system. to avoid the exhaustive search of proper lumped More recently, sensitivity analysis and Lie group components. approach were applied to lumping analysis (Li & Procedures of lumping analysisform a part of gen-
8 eral algebra of model building and model simplifi- IUPAC Compendium definition a rate-controlling cation transformations. Hangos & Cameron (2001) step always exists, because among the control func- applied formal methods of computer science and ar- tions generically exists the biggest one. On the con- tificial intelligence for analysis of this algebra. In trary, for the notion of limiting step that is used particular, a formal method for defining syntax and in practice, there exists a difference between sys- semantics of process models has been proposed. tems with limiting step and systems without limit- The modern systems and control theory provides ing step. efficient tools for lumping–analysis. The so-called During XX century, the concept of the limiting balanced model reduction was invented in late 1970s step was revised several times. First simple idea of a (Moore (1981)). For a linear system a set of “target “narrow place” (the least conductive step) could be variables” is selected. The dimension of the system n applied without adaptation only to a simple cycle is large, while the number of the target variables, for or a chain of irreversible steps that are of the first example, inputs m and outputs p, usually satisfies order (see Chap. 16 of the book Johnston (1966) or m,p ≪ n. The balanced model reduction problem can the paper by Boyd (1978)). When researchers try to be stated as follows (Gugercin & Antoulas (2004)): apply this idea in more general situations they meet find a reduced order system such that the following various difficulties such as: properties are satisfied: – Some reactions have to be “pseudomonomolecu- (i) The approximation error in the target vari- lar.” Their constants depend on concentrations ables is small, and there exists a global error of outer components, and are constant only un- bound. der condition that these outer components are (ii) System properties, like stability and passivity, present in constant concentrations, or change suf- are preserved. ficiently slow (i.e. are present in significantly big- (iii) The procedure is computationally efficient. ger amount). In large dimensions, special efforts are needed to re- – Even under fixed or slow outer components con- solve the accuracy/efficiency dilemma and to find centration, the simple “narrow place” behaviour efficiently the approximate solution of the model re- could be spoiled by branching or by reverse reac- duction problem (Antoulas & Sorensen (2002)). tions. The simplest example is given by the cycle: Various methods for balanced truncation are de- A1 ↔ A2 → A3 → A1. Even if the constant of veloped: Lyapunov balancing, stochastic balancing, the last step A3 → A1 is the smallest one, the bounded real balancing, positive real balancing, and stationary rate may be much smaller than k3b frequency weighted balancing (Gugercin & Antoulas (where b is the overall balance of concentrations, (2004)). Nonlinear generalizations are proposed as b = c1 + c2 + c3), if the constant of the reverse well (Lall, Marsden & Glavaki (2002); Condon & reaction A2 → A1 is sufficiently big. Ivanov (2004)). In a seriesof papers, Northrop (1981,2001) clearly explained these difficulties and suggested that the concept of rate–limiting step is “outmoded”. Nev- 2.6. Limiting Steps ertheless, the main idea of limiting is so attractive that Northrop’s arguments stimulated the search for In the IUPAC Compendium of Chemical Ter- modification and improvement of the main concept. minology (2007) one can find a definition of lim- Ray (1983) proposed the use of sensitivity analy- iting steps. Rate-controlling step (2007): “A rate- sis. He considered cycles of reversible reactions and controlling (rate-determining or rate-limiting) step suggested a definition: The rate–limiting step in a in a reaction occurring by a composite reaction se- reaction sequence is that forward step for which a quence is an elementary reaction the rate constant change of its rate constant produces the largest effect for which exerts a strong effect – stronger than that on the overall rate. of any other rate constant – on the overall rate.” Ray’s approach was revised by Brown & Cooper Let us complement this definition by additional (1993) from the system control analysis point of comment: usually when people are talking about view (see the book of Cornish-Bowden & Cardenas limiting step they expect significantly more: there (1990)). They stress again that there is no unique exists a rate constant which exerts such a strong ef- rate–limiting step specific for an enzyme, and this fect on the overallrate that the effect of all other rate step, even if it exists, depends on substrate, product constants together is significantly smaller. For the and effector concentrations.
9 Near critical conditions the critical simplification kmin, be much smaller than others (let it be kmin = appears, which is also a type of limitation, because kn): some reactions become critically important (Yablon- ki ≫ kmin if i 6= n . (14) sky, Mareels, & Lazman (2003)) In this case, in linear approximation w = knb, Two classical examples of limiting steps demon- strate us the chain of linear reaction and the linear kn kn c = b 1 − , and c = b for i 6= n . catalytic cycle, when they include a reaction which n k i k i
11 r zero-one asymptotic. In other words, approximation < k 1 k -1 < k of the left eigenvectors provides us with almost exact 1 1 1 lumping (for analysis of exact lumping see the paper by Li & Rabitz (1989)) . l
Fig. 1. Graphical representation of eigenvectors approxima- 4. Acyclic Non-branching Network: Explicit tion for the linear chain of reactions with well separated Formulas for Eigenvectors constants. To find the left (l) and right (r) eigenvectors for eigenvalue k it is necessary to delete from the chain all the reactions with the rate constants
12 < k For left eigenvectors, rows li, that correspond to r 1 nonzero eigenvalues we have the following asymp- 1 1 1 totics: 1 -1 < k 1 2 l 1 l ≈ (1, 0, 0, 0, 0, 0, 0, 0), l ≈ (0, 1, 0, 0, 0, 0, 0, 0), 1 l3 ≈ (0, 1, 1, 0, 0, 0, 0, 0),l4 ≈ (0, 0, 0, 1, 0, 0, 0, 0), < k l5 ≈ (0, 0, 0, 1, 1, 1, 1, 0), l6 ≈ (0, 0, 0, 0, 0, 1, 0, 0). l7 ≈ (0, 0, 0, 0, 0, 1, 1, 0) Fig. 2. Graphical representation of eigenvectors approxima- tion for the acyclic non-branching reaction network with well (29) separated constants (compare to Fig. 1). The eigenvalue −k For the corresponding right eigenvectors, columns corresponds to the reaction A → A (bold arrow). To the i φ(i) ri, we have the following asymptotics (we write right from Ai are vertices Aφq (i) and to the left are those q Aj , for which there exists such q that φ (j) = i. The reac- vector-columns in rows): tions with the rate constants
13 Att C1)( Att(C2) Att(Cq ) Aj
Ai k ji max {klil }
C1 C2 Cq
Fig. 3. Construction of the auxiliary reaction network by pruning. For every vertex, it is necessary to leave the out- going reaction with maximal reaction rate constant. Other reactions should be deleted.
can join all cones with the same dominant system. Fig. 4. Decomposition of a discrete dynamical system. We are interested in robust asymptotic and do not analyze directions (θij ) which belong to the bound- ary hyperplanes. This robust asymptotic with well separated constants and acyclic dominant systems scribed byc ˙ = Kc˜ , where K˜ = −κ δ + κ δ . is typical because the exclusive direction vectors be- ij j ij j iφ(j) lon to a finite number of hyperplanes. There may be other approaches based on (i) the Maslov dequantization and idempotent algebras 5.2.2. Decomposition of Discrete Dynamical (Litvinov & Maslov (2005)), (ii) the limit of log- Systems on Finite Sets uniform distributions in wide boxes of constants Discrete dynamical system on a finite set V = under some conditions (Feng, Hooshangi, Chen, {A , A ,...A } is a semigroup 1, φ, φ2, ..., where φ Li, Weiss, & Rabitz (2004); Gorban & Radulescu 1 2 n is a map φ : V → V . A ∈ V is a periodic point, (2008)), or (iii) on consideration of all possible or- i if φl(A ) = A for some l > 0; else A is a tran- derings of all monomials with integer exponents and i i i sient point. A cycle of period l is a sequence of l construction of correspondent dominant systems distinct periodic points A, φ(A), φ2(A),...φl−1(A) (Robbiano (1985) proved that there exists only a fi- with φl(A) = A. A cycle of period one consists of nal number of such orderings and enumerated all of one fixed point, φ(A)= A. Two cycles, C, C′ either them, see also the book by Greuel & Pfister (2002)). coincide or have empty intersection. They give the same final result but with different The set of periodic points, V p, is always intermediate steps. p nonempty. It is a union of cycles: V = ∪j Cj . For each point A ∈ V there exist such a positive integer q 5.2. Auxiliary Operations τ(A) and a cycle C(A) = Cj that φ (A) ∈ Cj for q ≥ τ(A). In that case we say that A belongs to basin 5.2.1. From Reaction Network to Auxiliary of attraction of cycle Cj and use notation Att(Cj )= Dynamical System {A | C(A)= Cj }. Of course, Cj ⊂ Att(Cj ). For dif- Let us consider a reaction network W with a given ferent cycles, Att(Cj )∩Att(Cl)= ∅. If A is periodic structure and fixed ordering of constants. The set of point then τ(A) = 0. For transient points τ(A) > 0. vertices of W is A and the set of elementary reactions So, the phase space V is divided onto subsets is R. Each reaction from R has the form Ai → Aj , Att(Cj ) (Fig. 4). Each of these subsets includes Ai, Aj ∈ A. The corresponding constant is kji. For one cycle (or a fixed point, that is a cycle of length each Ai ∈ A we define κi = maxj {kji} and φ(i) = 1). Sets Att(Cj ) are φ-invariant: φ(Att(Cj )) ⊂ arg maxj {kji}. In addition, φ(i)= i if kji = 0 for all Att(Cj ). The set Att(Cj ) \ Cj consist of transient j. points and there exists such positive integer τ that q The auxiliary discrete dynamical system for the re- φ (Att(Cj )) = Cj if q ≥ τ. action network W is the dynamical system Φ = ΦW Discrete dynamical systems on a finite sets corre- defined by the map φ on the finite set A. The auxil- spond to graphs without branching points. Notice iary reaction network (Fig. 3) V = VW has the same that for the graph that represents a discrete dy- set of vertices A and the set of reactions Ai → Aφ(i) namic system, attractors are ergodic components, with reaction constants κi. Auxiliary kinetics is de- while basins are connected components.
14 (to the whole glued cycle) with the same constant. QS Reaction of the third type changes into Ai → B with k jici i k ji Aj Aj the rate constant renormalization: let the cycle C 1 Ai A be the following sequence of reactions A1 → A2 →
C ...Aτi → A1, and the reaction rate constant for Ai →
§ · Ai+1 is ki (kτi for Aτi → A1). For the limiting reac- ¨ cQS 1¸ tion of the cycle C we use notation k . If A = A ¨ ¦ l ¸ i lim i j © Al C ¹ and k is the rate reaction for A → B, then the new i reaction A → B has the rate constant kklim i/kj. Fig. 5. Gluing a cycle with rate constants renormalization. This corresponds to a quasistationary distribution QS cl are the quasistationary concentrations on the cycle. Af- on the cycle (15). The new rate constant is smaller ter gluing, we have to leave the outgoing from A1 reaction than the initial one: kklim i/kj < k, because klim i < with the maximal renormalized rate constant, and delete others. kj due to definition of limiting constant. The same constant renormalization is necessary for reactions 5.3. Algorithm for Calculating the Dominant of the fourth type. These reactions transform into i j System A → A . Finally, reactions of the fifth type vanish. After we glue all the cycles (Fig. 5) of auxiliary dynamical system in the reaction network W, we get For this general case, the algorithm consists of W1. Let us assign W := W1, A := A1 and iterate two main procedures: (i) cycles gluing and (ii) cycles until we obtain an acyclic network and exit. This restoration and cutting. acyclic network is a “forest” and consists of trees oriented from leafs to a root. The number of such 5.3.1. Cycles Gluing trees coincide with the number of fixed points in the Let us start from a reaction network W with a final network. given structure and fixed ordering of constants. The After gluing we can identify the reactions, which set of vertices of W is A and the set of elementary will be included into the dominant system. Their reactions is R. constants are the critical parameters of the networks. If all attractors of the auxiliary dynamic system The list of these parameters, consists of all reac- ΦW are fixed points Af1, Af2, ... ∈ A, then the aux- tion rates of the final acyclic auxiliary network, and iliary reaction network is acyclic, and the auxiliary of the rate constants of the glued cycles, but with- kinetics approximates relaxation of the whole net- out their limiting steps. Some of these parameters work W. are rate constants of the initial network, other have In general case, let the system ΦW have sev- the monomial structure. Other constants and corre- eral attractors that are not fixed points, but cycles sponding reactions do not participate in the follow- C1, C2, ... with periods τ1, τ2, ... > 1. By gluing ing operations. To form the structure of the domi- these cycles in points, we transform the reaction nant network, we need one more procedure. 1 network W into W . The dynamical system ΦW is transformed into Φ1. For these new system and 1 1 network, the connection Φ = ΦW1 persists: Φ is 5.3.2. Cycles Restoration and Cutting the auxiliary discrete dynamical system for W1. We start the reverse process from the glued net- i m m m For each cycle, Ci, we introduce a new vertex A . work V on A . On a step back, from the set A to The new set of vertices, A1 = A∪{A1, A2, ...}\ Am−1 and so on, some of glued cycles should be re- i (∪iCi) (we delete cycles Ci and add vertices A ). stored and cut. On the qth step we build an acyclic All the reaction A → B from the initial set R, reaction network on Am−q, the final network is de- (A, B ∈ A) can be separated into 5 groups: fined on the initial vertex set and approximates re- (i) both A,B∈ / ∪iCi; laxation of W. m (ii) A/∈ ∪iCi, but B ∈ Ci; To make one step back from V let us select the m m−1 (iii) A ∈ Ci, but B/∈ ∪iCi; vertices of A that are glued cycles from V . Let m m m (iv) A ∈ Ci, B ∈ Cj , i 6= j; these vertices be A1 , A2 , .... Each Ai corresponds m−1 m−1 m−1 (v) A, B ∈ Ci. to a glued cycle from V , Ai1 → Ai2 → m−1 m−1 Reactions from the first group do not change. Reac- ...Aiτi → Ai1 , of the length τi. We assume that i m−1 m−1 tion from the second group transforms into A → A the limiting steps in these cycles are Aiτi → Ai1 .
15 A k2 1 In the simplest case, the dominant system is de- k1 A2 k Aj termined by the ordering of constants. But for suffi- k1 A2 Ai ciently complex systems we need to introduce aux- k2 A1 ki iliary elementary reactions. They appear after cycle klim k gluing and have monomial rate constants of the form W 1 ςi AW kς = i ki , where ςi are integers, but not manda- AW / tory positive. The dominant system depends on the kklim ki Q A place of these monomial values among the ordered ` j constants. For systems with well separated constants we can also assume that each of these new constants Fig. 6. The main operation of the cycle surgery: on a step will be well separated from other constants (Gorban back we get a cycle A1 → ... → Aτ → A1 with the limiting step Aτ → A1 and one outgoing reaction Ai → Aj . We & Radulescu (2008)). should delete the limiting step, reattach (“recharge”) the outgoing reaction Ai → Aj from Ai to Aτ and change its 5.4. Example rate constant k to the rate constant kklim/ki. The new value of reaction rate constant is always smaller than the initial one: kklim/ki < k if klim =6 ki. For this operation only one To demonstrate a possible branching of described condition k ≪ k is necessary (k should be small with respect i algorithm for cycles surgery (gluing, restoring and to reaction Ai → Ai+1 rate constant, and can exceed any other reaction rate constant). cutting) with necessity of additional orderings, let us consider the following system: m m Let us substitute each vertex Ai in V by τi ver- m−1 m−1 m−1 m tices Ai1 , Ai2 , ...Aiτi and add to V reactions 1 6 2 3 4 5 m−1 m−1 m−1 A1→A2→A3→A4→A5→A3, A4→A2, (31) Ai1 → Ai2 → ...Aiτi (that are the cycle reac- tions without the limiting step) with corresponding (where the upper index marks the order of rate con- constants from Vm−1. stants). The auxiliary discrete dynamical system for m If there exists an outgoing reaction Ai → B in reaction network (31) is Vm then we substitute it by the reaction Am−1 → iτi A →1 A →6 A →2 A →3 A →4 A . B with the same constant, i.e. outgoing reactions 1 2 3 4 5 3 m 2 3 4 Ai → ... are reattached to the heads of the limiting It has only one attractor, a cycle A3→A4→A5→A3. steps (Fig. 6). Let us rearrange reactions from Vm of This cycle is not a sink for the whole network (31) be- m 5 the form B → Ai . These reactions have prototypes cause reaction A4→A2 leads from that cycle. After m−1 1 in V (before the last gluing). We simply restore gluing the cycle into a vertex A3 we get the new net- m m 1 6 1 ? these reactions. If there exists a reaction Ai → Aj work A1→A2→A3→A2. The rate constant for the m−1 1 1 then we find the prototype in V , A → B, and reaction A3→A2 is k23 = k24k35/k54, where kij is m−1 substitute the reaction by Aiτi → B with the same the rate constant for the reaction Aj → Ai in the m m constant, as for Ai → Aj . initial network (k35 is the cycle limiting reaction). After that step is performed, the vertices set is The new network coincides with its auxiliary system m−1 6 1 ? A , but the reaction set differs from the reactions and has one cycle, A2→A3→A2. This cycle is a sink, of the network Vm−1: the limiting steps of cycles are hence, we can start the back process of cycles restor- excluded and the outgoing reactions of glued cycles ing and cutting. One question arises immediately: 1 are included (reattached to the heads of the limiting which constant is smaller, k32 or k23. The smallest steps). To make the next step, we select vertices of of them is the limiting constant, and the answer de- Am−1 that are glued cycles from Vm−2, substitute pends on this choice. Let us consider two possibili- 1 1 these vertices by vertices of cycles, delete the lim- ties separately: (1) k32 > k23 and (2) k32 < k23. 1 iting steps, attach outgoing reactions to the heads (1) Let as assume that k32 > k23. The final auxil- 1 6 1 ? of the limiting steps, and for incoming reactions re- iary system after gluing cycles is A1→A2→A3→A2. m−2 1 ? store their prototypes from V , and so on. Let us delete the limiting reaction A3→A2 from the 1 6 1 After all, we restore all the glued cycles, and con- cycle. We get an acyclic system A1→A2→A3. The 1 2 3 4 struct an acyclic reaction network on the set A. This component A3 is the glued cycle A3→A4→A5→A3. acyclic network approximates relaxation of the net- Let us restore this cycle and delete the limiting 4 work W. We call this system the dominant system reaction A5→A3. We get the dominant system 1 6 2 3 of W and use notation dom mod(W). A1→A2→A3→A4→A5. Relaxation of this system
16 approximates relaxation of the initial network (31) A1 A2 1 under additional condition k32 > k23. A 1 3 (2) Let as assume now that k32 < k23. The fi- nal auxiliary system after gluing cycles is the same, 1 6 1 ? A1 A2 A1 A2 A1 A2 A1 A2 A →A →A →A , but the limiting step in the cycle (c) 1 2 3 2 (a) (b) (d) 6 1 is different, A2→A3. After cutting this step, we get A3 A3 A3 A3 acyclic system A →1 A ←? A1, where the last reaction 1 2 3 Attractors 1 has rate constant k23. The component A1 is the glued cycle 3 Fig. 7. Four possible auxiliary dynamical systems for the re- 2 3 4 A3→A4→A5→A3 . versible triangle of reactions with k21 >kij for (i,j) =6 (2, 1): (a) k12 > k32, k23 > k13; (b) k12 > k32, k13 > k23; (c) Let us restore this cycle and delete the limiting re- k32 > k12, k23 > k13; (d) k32 > k12, k13 > k23. For each 4 action A5→A3. The connection from glued cycle vertex the outgoing reaction with the largest rate constant 1 ? 1 is represented by the solid bold arrow, and other reactions A3→A2 with constant k23 transforms into connec- tion A →? A with the same constant k1 . are represented by the dashed arrows. The digraphs formed 5 2 23 by solid bold arrows are the auxiliary discrete dynamical We get the dominant system: systems. Attractors of these systems are isolated in frames. 1 2 3 ? A1→A2 , A3→A4→A5→A2 . other vertices and, therefore, only four versions of The order of constants is now known: k21 > k43 > auxiliary dynamical systems for (32) (Fig. 7). Let 1 k54 > k23, and we can substitute the sign “?” by us analyze in detail case (a). For the cases (b) and 2 3 4 “4”: A3→A4→A5→A2. (c) the details of computations are similar. The ir- 1 1 For both cases, k32 > k23 (k23 = k24k35/k54) and reversible cycle (d) is even simpler and was already 1 k32 < k23 it is easy to find the eigenvectors explicitly discussed. and to write the solution to the kinetic equations in explicit form. 6.1. Auxiliary System (a): A1 ↔ A2 ← A3; k > k , k > k 6. The Reversible Triangle of Reactions 12 32 23 13 6.1.1. Gluing Cycles In this section, we illustrate the analysis of dom- The attractor is a cycle (with only two vertices) inant systems on a simple example, the reversible A ↔ A . This is not a sink, because two outgoing triangle of reactions. 1 2 reactions exist: A1 → A3 and A2 → A3. They are A1 ↔ A2 ↔ A3 ↔ A1 (32) relatively slow: k31 ≪ k21 and k32 ≪ k12. The limit- ing step in this cycle is A → A with the rate con- This triangle appeared in many works as an ideal 2 1 stant k . We have to glue the cycle A ↔ A into object for a case study. Our favorite example is the 12 1 2 one new component A1 and to add a new reaction work of Wei & Prater (1962). Now in our study the 1 A1 → A with the rate constant (see Fig. 5) triangle (32) is not necessarily a closed system. We 1 3 1 can assume that it is a subsystem of a larger sys- k31 = max{k32, k31k12/k21} . (33) tem, and any reaction A → A represents a reac- i j 1 tion of the form ...+A → A +..., where unknown As a result, we get a new system, A1 ↔ A3 with i j 1 1 but slow components are substituted by dots. This reaction rate constants k31 (for A1 → A3) and initial 1 means that there are no mandatory relations be- k23 (for A1 ← A3). This cycle is a sink, because tween reaction rate constants, and six reaction rate it has no outgoing reactions (the whole system is a constants are arbitrary nonnegative numbers. trivial example of a sink). Let the reaction rate constant k21 for the reaction A1 → A2 be the largest. 6.1.2. Dominant System Let us describe all possible auxiliary dynamical At the next step, we have to restore and cut the systems for the triangle (32). For each vertex, we cycles. First cycle to cut is the result of cycle gluing, 1 have to select the fastest outgoing reaction. For A1, A1 ↔ A3. It is necessary to delete the limiting step, it is always A1 → A2, because of our choice of enu- i.e. the reaction with the smallest rate constant. If 1 1 meration (the higher scheme in Fig. 7). There exist k31 > k23, then we get A1 → A3. If, inverse, k23 > 1 1 two choices of the fastest outgoing reaction for two k31, then we obtain A1 ← A3.
17 A1 1 7. Corrections to Dominant Dynamics 1 k21 k31 1 A A A 1 k31 1 2 3 A1 1 A if k31 ! k23 1 2 A1 2 1 A3 The hierarchy of systems W, W , W , ... can be (a) k31 k23 A1 used for multigrid correction of the dominant dy- 1 k21 k23 A3 A3 k23 A1 A2 A3 namics. The simple example of multigrid approach 1 A3 if k23 ! k31 gives the algorithm of steady state approximation (Gorban & Radulescu (2008)). For this purpose, on Fig. 8. Dominant systems for case (a) (defined in Fig. 7) the way up (cycle restoration and cutting, Sec. 5.3.2) we calculate distribution in restoring cycles with higher accuracy, by exact formula (13), or in linear After that, we have to restore and cut the cycle approximation (15) instead of the simplest zero-one which was glued into the vertex A1. This is the two- 1 asymptotic (16). Essentially, the way up remains the vertices cycle A ↔ A . The limiting step for this 1 2 same. cycle is A ← A , because k ≫ k . If k1 > k , 1 2 21 12 31 23 After termination of the gluing process, we can then following the rule visualized by Fig. 6, we get find all steady state distributions by restoring cy- the dominant system A → A → A with reaction 1 2 3 cles in the auxiliary reaction network Vm. Let rate constants k for A → A and k1 for A → 21 1 2 31 2 Am , Am , ... be fixed points of Φm. The set of steady A . If k > k1 then we obtain A → A ← A with f1 f2 3 23 31 1 2 3 states for Vm is the set of all distributions on the reaction rate constants k21 for A1 → A2 and k23 for m m set of fixed points {Af1, Af2, ...}. A2 ← A3. All the procedure is illustrated by Fig. 8. m Let us take one of the basis distributions, cfi = 1, m m other ci = 0on V . If the vertex Afi is a glued cycle, 6.1.3. Eigenvalues and Eigenvectors then we substitute them by all the vertices of this cy- m The eigenvalues and the corresponding eigenvec- cle. Redistribute the concentration cfi between the tors for dominant systems in case (a) are represented vertices of the corresponding cycle by the rule (13) below in zero-one asymptotic. (or by an approximation). As a result, we get a set 1 of vertices and a distribution on this set of vertices. (i) k31 > k23, the dominant system A1 → A2 → A3, If among these vertices there are glued cycles, then we repeat the procedure of cycle restoration. Termi- 0 0 nate when there is no glued cycles in the support of λ0 =0 , r ≈ (0, 0, 1) , l = (1, 1, 1) ; the distribution. 1 1 λ1 ≈−k21 , r ≈ (1, −1, 0) , l ≈ (1, 0, 0) ; The resulting distribution is the approximation to 1 2 2 a steady state of W, and the basis of steady states λ2 ≈−k , r ≈ (0, 1, −1) , l ≈ (1, 1, 0) ; 31 for W can be approximated by this method. (34) For example, for the system Fig. 8 we have, first (ii) k > k1 , 23 31 of all, to compute the stationary distribution in the the dominant system A1 → A2 ← A3, 1 1 cycle A1 ↔ A3, c1 and c3. On the base of the general 0 0 formula for a simple cycle (13) we obtain: λ0 =0 , r ≈ (0, 1, 0) , l = (1, 1, 1) ; 1 1 1 1 w w λ1 ≈−k21 , r ≈ (1, −1, 0) , l ≈ (1, 0, 0) ; w = 1 1 , c1 = 1 , c3 = . (36) 1 + k k23 k k23 31 2 2 31 λ2 ≈−k23 , r ≈ (0, −1, 1) , l ≈ (0, 0, 1) . After that, we have to restore the cycle glued into (35) 1 1 A1. This means to calculate the concentrations of Here, the value of k31 is given by formula (33). 1 Analysis of examples provided us by an impor- A1 and A2 with normalization c1 +c2 = c1. Formula tant conclusion: the number of different dominant (13) gives: systems in examples was less than the number of 1 ′ ′ ′ c1 w w all possible orderings. For many pairs of constants w = 1 1 , c1 = , c2 = . (37) + k21 k12 k21 k12 kij , klr it is not important which of them is larger. There is no need to consider all orderings of mono- For eigenvectors, there appear two operations of mials. We have to consider only those inequalities corrections: (i) correction for an acyclic network between constants and monomials that appear in without branching (43), (45), and (ii) corrections for the construction of the dominant systems. a cycle with relatively slow outgoing reactions (49).
18 These corrections are by-products of the accuracy works may change in time (this is the significant dif- estimates given in Appendix. ference from the linear case) but remain relatively simple.
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Seneta, E. (1981). Nonnegative Matrices and Markov Chains, Springer, New York. The famous Gerschgorin theorem (Marcus & Stueckelberg, E.C.G. (1952). Theoreme H et unitarite de S, Minc (1992), Varga (2004)) gives estimates of Helv. Phys. Acta 25 (5), 577–580. eigenvalues. We need also estimates of eigenvec- Temkin, O.N., Zeigarnik, A.V., & Bonchev, D.G. (1996). Chemical Reaction Networks: A Graph-Theoretical Ap- tors. Below A = (aij ) is a complex n × n matrix, proach, Boca Raton, FL: CRC Press. Qi = j,j6=i |aji| (sums of non-diagonal elements Toth, J., Li, G., Rabitz, H., & Tomlin, A. S. (1997). The in columns). Effect of Lumping and Expanding on Kinetic Differential GerschgorinP theorem (Marcus & Minc (1992), Equations, SIAM J. Appl. Math. 57, 1531–1556. p. 146): The characteristic roots of A lie in the closed Turanyi, T., Tomlin, A.S., & Pilling, M.J. (1993). On the Q error of the quasi-steady-state approximation, J. Phys. region G of the z-plane Chem. 97 (1), 163–172. Q Q Q G = G (G = {z |z − aii|≤ Qi}. (38) Van Mieghem, P. (2006). Performance Analysis of Commu- i i i nications Networks and Systems, Cambridge University [ Press, Cambridge. Q Areas Gi are the Gerschgorin discs. (The same esti- Varga, R.S. (2004). Gerschgorin and His Circles, Springer mate are valid for sums in rows, Pi. Here and below series in computational Mathematics, 36, Berlin – Hei- delberg – New York: Springer. we don’t duplicate the estimates.) Q Vishik, M. I., & Ljusternik, L. A. (1960). Solution of some Gerschgorin disks Gi (i = 1,...n) are isolated, perturbation problems in the case of matrices and self- Q Q ∅ P if Gi ∩ Gj = for i 6= j. If disks Gi (i =1,...n) adjoint or non-selfadjoint differential equations. I, Russian are isolated, then the spectrum of A is simple, and Math. Surveys, 15, 1–73. Q Vora, N., & Daoutidis, P. (2001). Nonlinear Model Reduction each Gerschgorin disk Gi contains one and only one of Chemical Reaction Systems AIChE Journal, 47 (10), eigenvalue of A (Marcus & Minc (1992), p. 147). 2320–2332. Q We assume that Gerschgorin disks Gi (i = Wei, J., & Prater, C. (1962). The structure and analysis of 1,...n) are isolated: for all i, j (i 6= j) complex reaction systems. Adv. Catalysis, 13, 203–393. Wei, J., & Kuo, J. C. (1969). A lumping analysis in |aii − ajj | >Qi + Qj . (39) monomolecular reaction systems: Analysis of the exactly lumpable system, Ind. Eng. Chem. Fundam., 8, 114–123. Let us introduce the following notations: White, R. B. (2006). Asymptotic Analysis of Differential Qi |aij | Equations, London: Imperial College Press & World Sci- = εi, ε = max εi, = χij , χ = max χij , |a | i |a | i,j,i6=j entific. ii jj Whitehouse, L. E., Tomlin, A. S., & Pilling, M. J. (2004). |aii − ajj | gi = min , g = min gi. Systematic reduction of complex tropospheric chemical j,j6=i |aii| i mechanisms, Part II: Lumping using a time-scale based (40) approach, Atmos. Chem. Phys., 4, 2057–2081. Yablonskii, G. S., Bykov, V. I., Gorban, A. N., & Elokhin, Usually, we consider εi and χij as sufficiently small V. I. (1991). Kinetic models of catalytic reactions. Com- numbers. In contrary, the diagonal gap g should not
21 be small, (this is the gap condition). For example, if Direct use of this theorem and estimates for a kinetic for any two diagonal elements aii, ajj either aii ≫ matrix K in the standard basis is impossible, the ajj or aii ≪ ajj , then gi & 1 for all i. diagonal dominance in this coordinate system is not Q Let λi ∈ Gi be the eigenvalue of A (|λi − a11| < large, and sums of elements in columns are zero. To Q1). Let us estimate the corresponding right eigen- apply this theorem we need two lemmas. (i) i vector r . We take ri = 1 and for j 6= i introduce Let W be a reaction network without branching i i i a (n − 1)-dimensional vectorx ˜ :x ˜j = rj (ajj − aii) (a finite dynamical system) with n vertices. Then (i 6= j). Forx ˜i we get equation the number of reactions in W is n − f, where f is the number of fixed points (the vertices without out- (i) i i (1 − B )˜x = −a˜ (41) going reactions). Let Γ be the set of stoichiometric wherea ˜i is a vector of the non-diagonal elements vectors for W. i Lemma 1. Γ forms a basis in the subspace of the ith column of A (˜aj = aij , j 6= i), and the (n − 1) × (n − 1) matrix Bi has matrix elements {c | i ci = 0} if and only if the reaction network (j, l 6= i) W is acyclic and connected (has only one fixed point).P (i) λi − aii (i) ajl Let us consider a general reaction network on the bjj = , bjl = (l 6= j) (42) ajj − aii all − aii set A1, ...An. For stoichiometric vector of reaction A → A we use notation γ . Assume that the auxil- Due to the Gerschgorin estimate, |b(i)| < Qi . i l li jj |ajj −aii| iary dynamical system i 7→ φ(i) for a given reaction From Eq. (41) we obtain: network is acyclic and has only one attractor, a fixed x˜i = −a˜i − B(i)(1 − B(i))−1a˜i. (43) point. For this auxiliary network, we use notation: κi = kji for the only reaction Ai → Aj , or κi = 0. From this definition and simple estimates in l1 norm, For every reaction of the initial network, Ai → Al, we get the following estimate of eigenvectors. a linear operators Qil can be defined by its action Theorem 2. Let the Gerschgoring disks be iso- on the basis vectors, γφ(i) i: lated, and the diagonal gap be big enough: g >nε. Then for the ith eigenvector of A the following uni- Qil(γφ(i) i)= γli,Qil(γφ(p) p)=0for p 6= i. (46) form estimate holds: Lemma 2. The kinetic equation for the whole χ nε2 reaction network (9) could be transformed to the |ri |≤ + (j 6=1, ri = 1). (44) j g g(g − nε) i form So, if the matrix A is diagonally dominant and the dc kli = 1+ Qil γφ(i) iκici diagonal gap g is big enough, then the eigenvectors dt κi i l, l6=φ(i) are proven to be close to the standard basis vectors X X with explicit evaluation of accuracy. klj = 1+ Qjl γ κici (47) The first correction to eigenvectors is also given by κ φ(i) i j i Eq. (43). If for the iteration we use the Gerschgorin j,l (Xl6=φ(j)) X estimates for eigenvalue λi ≈ aii, then we can write k i lj ˜ in the next approximation for eigenvectors (ri = = 1+ Qjl Kc, κj 1, j 6= i): j,l (Xl6=φ(j)) (i) (i) −1 i where K˜ is kinetic matrix of the kinetic equation for i aji (Bnd (1 − Bnd ) a˜ )j rj = − − (45) the auxiliary network. a − a a − a jj ii jj ii By construction of auxiliary dynamical system, (i) (i) k < κ if l 6= φ(i), and for reaction networks with where Bnd is the non-diagonal part of B : it has li i the same non-diagonal elements and zeros on diago- well separated constants kli ≪ κi. Notice also that nal. There exists plenty of further simplifications for the matrix Qjl does not depend on rate constants this iteration formula. For example, one can leave values. just the first term, that gives the first order approx- For matrix K˜ we have the eigenbasis in explicit imation in the power of ε (χ ≤ ε). form. Let us represent system (47) in this eigenbasis To apply these estimates to an acyclic network of K˜ . Any matrix B in this eigenbasis has the form ˜ ˜ i j i j supplemented by additional reactions, we have to B = (bij ), bij = l Br = qs lqbqsrs, where (bqs) is use the eigenbasis of this acyclic network (Sec. 4). matrix B in the initial basis, li and rj are left and P 22 right eigenvectors of K˜ (27), (28). In eigenbasis of K˜ This representation allows us to produce useful the estimates of eigenvalues and estimates of eigen- estimates, for example, when the unperturbed sys- ∗ ∗ ∗ vectors are much more efficient than in original coor- tem is a cycle, we find |ri − ci | < 3ε|ci | under con- dinates: the system is strongly diagonally dominant. dition ε < 0.25, where ε = εi. Formula for the ∗ ∗ ∗ Transformation to this basis is an effective precondi- first correction gives (r = c + δri, w = kic ): iP i tioning for the perturbation theory that uses auxil- i iary kinetics as a first approximation to the kinetics vi ∗ δri = , vi = v + w (εcj − εj ), ki of the whole system. j=1 Xn (49) w v = i(εc∗ − ε ). Estimates for Perturbed Ergodic Systems n i i i=1 X Let us consider a strongly connected network For more complex networks, the explicit formulas with kinetic matrix K. The corresponding kinet- for corrections could be produced on the base of the ics is ergodic and there exists unique normalized network graphs, similar to the steady-state formu- steady state c∗ > 0, c∗ = 1. For each i we define las, presented, for example, by Yablonskii, Bykov, i i i Gorban, & Elokhin (1991). κi = j kji. The number −κi is the iith diagonal element of unperturbedP kinetic matrix K. So, the asymptotic analysis gives good approxi- LetP this network be perturbed by outgoing reac- mation of eigenvectors and eigenvalues for kinetic matrix. The condition number is big (unbounded) tions Ai → 0. The perturbation has the “loss form”: but these estimates work even better when the con- the perturbed matrix is K−diag(εiκi), perturbation of each diagonal element is relatively small (diag is stants become more separated. Nevertheless, some the diagonal matrix). caution is needed: the error is proven to be small, but the residuals (the values kKr − λrk for approx- The perturbations εiκi are relatively small with imations of r and λ) may be not small (Gorban & respect to κi, but not obligatory small with respect to other rate constants. Radulescu (2008)). First, we do not assume anything about value of εi ≥ 0 and make the following transformation. For an arbitrary normalized vector r (ri ≥ 0, i ri = 1) we add to the network reactions Ai → Aj with reac- P tion rates qji = rj εiκi. We use Q(r) for the kinetic matrix of this additional network. Simple algebra gives
Q(r) + diag(εiκi) = [ε κ r, ε κ r, ...εnκnr] 1 1 2 2 (48) = r(ε1κ1,ε2κ2, ...εnκn). Here, in the right hand side we have a matrix, all columns of which are proportional to the vector r, this is a product of r on the vector-rawof coefficients. We represent the perturbed matrix in the form K − diag(εiκi)= K + Q(r) − (Q(r) + diag(εiκi)). Theorem 3. There exists such normalized posi- tive r∗ that (K + Q(r∗))r∗ = 0. This r∗ is an eigen- vector of the perturbed network with the eigenvalue ∗ λ = i ri εiκi, and, at the same time, it is a steady- state for the network with kinetic matrix K +Q(r∗). ToP prove existence it is sufficient to mention, that for any r the network with kinetic matrix K + Q(r) has unique positive normalized steady state c∗(r), which depends continuously on r. The map r 7→ c∗(r) has a fixed point r∗ (the Brouwer fixed point theorem).
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