What makes biochemical networks tick? A graphical tool for the identification of oscillophores Boris N. Goldstein 1 , Gennady Ermakov 1 , Josep J. Centelles 3 , Hans V. Westerhoff 2 and Marta Cascante 3 1 Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Moscow Region, Russia; 2 BioCentrum Amsterdam, Departments of Molecular Cell Physiology (IMC, VUA) and Mathematical Biochemistry (SILS, UvA), Amsterdam, the Netherlands; 3 Department of Biochemistry and Molecular Biology, Faculty of Chemistry and CeRQT at Barcelona Scientific Parc, University of Barcelona, Spain In view of the increasing number of reported concentration oscillations in living cells, methods are needed that can identify the causes of these oscillations. These causes always derive from the influences that concentrations have on reaction rates. The influences reach over many molecular reaction steps and are d efined by the d etailed molecular topology of the network. So-called Ôautoinfluence pathsÕ, which quantify the influence of one molecular species upon itself through a particular path through the network, can have positive or negative values. The former bring a ten- dency towards instability. In this molecular context a new graphical approach is presented that enables the classifica- tion of network topologies into oscillophoretic and non- oscillophoretic, i.e. into ones that can and ones that cannot induce concentration oscillations. The network topologies are formulated in terms of a set of uni-molecular and bi-molecular reactions, organized into branched cycles of directed reactions, and presented as graphs. Subgraphs of the n etwork topologies are then classified as negative ones (which can) and positive ones (which cannot) give rise to oscillations. A subgraph is oscillophoretic (negative) when it contains more positive than negative autoinfluence paths. Whether the former generates oscillations depends on the values of the other subgraphs, which again depend on the kinetic parameters. An example shows how this can be established. By following the rules of our new approach, various oscillatory kinetic models can be constructed and analyzed, starting from the classified simplest topologies and then working t owards desirable complications. Realistic biochemical examples are analyzed with the new method, illustrating two new main classes of oscillophore topologies. Keywords: graph-theoretic approach; kinetic mode lling; oscillations; s ystem identification; systems biology. Oscillatory biochemical networks have regained intensive interest during the past few years because of the importance of oscillatory signaling for various biological functions. Oscillations in glycolysis [1,2], oscillations of Ca 2+ concen- trations [3,4], and the cell cycle as such [5] are well known. Some of these have been predicted and analyzed by using mathematical models [6,7]. The need for such mathematical models is appreciated even more w hen studying biochemical oscillations and their synchronization [7–13]. The behavior of potential biochemical oscillators may depend on the kinetic properties of t heir surroundings, interacting with t he oscillator through c ommon metabolites (e.g [8,14]). Other systems, such as the cell cycle of tumor cells may be more autonomous [9]. Most intracellular oscillations involve more than five components that interact in a nonlinear man ner [8]. This makes them unsuitable for intuitive analysis, a phenomenon encountered more fre- quently in Systems Biology [8]. New theoretical approaches are needed that streamline the study of such cases of Systems Biology, dissecting the system into various inter- acting kinetic regimes, whilst relating to molecular mecha- nisms. Various types of approach can be helpful here. Graph- theoretic approaches can help dissect the dynamics of enzyme reactions [15,16] and this is what made others and ourselves [20,25,26] examine whether these approaches can also do this for networks. Earlier w e have applied graph theory in order t o simplify the King–Altman–Hill [15,16] analysis of steady-state enzyme reactions [17,18]. This approach was later extended to presteady-sta te enzyme kinetics [19], to s tability a nalysis o f enzyme systems [20], and to the analysis of concentration oscillations in enzyme cycles [21]. In this paper, the graph-theoretical stability analysis developed b y C larke [ 22] as modified b y I vanova [21,23,24] is the starting point for a more comprehensive approach to the an alysis of biochemical networks. It enables us to develop a graph-theoretical identification of networks that may, and of networks that cannot, serve as oscillophores (i.e. induce oscillations). In some aspects our approach is similar to that reported previously [25,26]. However, we use unimolecular a nd bimolecular steps and simple c atalytic cycles, rather than Correspondence to M. Cascante, Department of Biochemistry and Molecular Biology, Faculty of Chemistry and CERQT-Parc Scientific of Barcelona, University of Barcelona, c/Martı ´ iFranque ` s 1, 08028 Barcelona, Spain. Fax: +34 934021219; Tel.: +34 934021593; E-mail: Marta@bq.ub.es or Hans V. Westerhoff, Faculty of Earth and Life Sciences, Free University, De Boelelaan 1087, NL-1081 HV Amsterdam, the Netherlands. Fax: +31 204447229; E-mail: hw@bio.vu.nl (Received 6 July 2004, revised 30 J uly 2004, accepted 4 August 2004) Eur. J. Biochem. 271, 3877–3887 (2004) Ó FEBS 2004 doi:10.1111/j.1432-1033.2004.04324.x quadratic and cubic autocatalytic cycles. We identify subschemes of the specific biochemical network that induce instability. We then consider inte rconnections between such a subscheme a nd other parts of the kinetic scheme with or without eliminating the instability. The procedure o f this paper uses so-called dual graphs with two types of vertices, i.e. for both species and r eactions [21]. I n t his w ay all t ypes of reactions can be analyzed in a uniform manner. The procedure allows us to estimate the p arameter values for which oscillations occur. Presence or absence of steady states on the border of the phase space [21,23] then suffices to predict the occurrence of limit-cycle oscillations. We illustrate our method by applying it to two biochemical systems, which include oscillophores of two different classes. Results Paths: graphical representation of kinetic influences We represent kinetic schemes b y dual g raphs, combining reaction-centered and substance-centered graphs [21]. Accordingly, our kinetic schemes for biochemical networ ks have two kinds of vertices, i.e. one kind for species (here shown by open circles) a nd one kind for reactions (shown by closed circles). The circles are connected by arrows. For example, the reaction x i + x j fi x m is represented by the following reaction-centered graph: where x i , x j and x m are c hemical s pecies (substances ) and v r is the rate of the r th reaction. Graph 1 shows that two species x i and x j participate in the same r th reaction as substrates with corresponding stoichiometric coefficients a ir and a jr . The species x m is synthesized in this r th reaction at a stoichiometry b mr . Using the mass-action law, in which molecularity and k inetic order of reaction a re equal, we can write: v r ¼ k r Á x a ir i Á x a jr j ð1Þ where k r is the k inetic constant. T his implies t hat we do not dissect biochemical networks i nto t he net enzyme-catalyzed reactions, but into the unidirectional elementary r eaction steps underlying the enzyme kinetics. The terms x i , x j and x m include the concentrations of both m etabolites and enzyme- forms. Rates v r are always positive. As a consequence of the dissection down to t he molecular p rocesses, the stoichio- metric coefficients equal one or zero, i.e. a ir ; b ir ¼ 1or0 ð2Þ with 1 for participating and 0 for nonparticipating species. Reactions involving more than one molecule of a single species are described as a sequence of two independent reactions. Assuming s patial homogeneity and a sin gle compartment, the kinetic equations for t he reaction network are then written as follows: dx i dt ¼ X r ðÀa ir þ b ir Þv r ; ði ¼ 1; 2; :::; nÞð3Þ where summation is over all r ¼ 1, 2,…, R reactions. Species x i (i ¼ 1, 2,…, n) participate in these reactions as substrates and/or products, as illustrated graphically in the species-centered Graph 2: ðr À 1Þ ! b iðrÀ1Þ  x i À! a ir ðrÞ Graph 2: Similarly to the procedure developed b y Clarke [22], we linearize the system of Eqn (3) in the vicinity of the steady state. We do this to investigate the stability of this state. In this way we obtain the influence a small change in the concentration of substance j, i.e. Dx j ,hasonthetime displacement of t he conc entration of species i from its steady-state value: dDx i dt ¼ X r;j ðÀa ir þ b ir Þ @v r @x j Dx j ; ði ¼ 1; 2; :::; mÞð4Þ where the summation is over both all r ¼ 1, 2,…,R reactions and all j ¼ 1, 2,…, m<nindependent concen- trations. From t he law of m ass action ( Eqn 1) i t follows for the kinetic order of the re actions that: @v r @x j ¼ a jr v r x j ð5Þ Therefore, Eqn ( 4) can be rewritten as: dDx i dt ¼ X r;j ðÀa ir þ b ir Þa jr v r x j Dx j ¼ X j b ij Dx j ð6Þ Coefficients b ij are the elements of the Jacobian (matrix) B representing the direct influences of x j on x i : b ij ¼ X r ðÀa ir þ b ir Þa jr v r x j ð7Þ For small deviations from the steady state, the elements of b ij that multiply a b with an a characterize the sum of all reactions that convert x j to x i in a single step, i.e. all reactions directed as x j fi x i . Any reaction contributing to that overall reaction x j fi x i does so to the absolute extent v r /x j , t he sign o f its contribution depending on the direction of the reaction, as specified by Eqn (7). The terms that multiply an a and a b therewith represent the positive influence that a substrate of a reaction has on the product o f the r eaction. The terms of b ij that multiply two a’s, represent the negative influences of two substances on each other when both are consumed in that reaction. Indeed, each element of the Jacobian corr esponds to one or a number of such direct influences of one metabolite on another, direct in the sense that the influence is through s ingle reaction steps. A number of s uch reaction steps may operate in parallel (but not in series) for each element of the Jacobian. In addition, one reaction step may convey more than one influence. Graph 1. 3878 B. N. Goldstein et al.(Eur. J. Biochem. 271) Ó FEBS 2004 To predict t he dynamics of the system it is i ndeed helpful to classify these influences into positive and negative ones. Again, depending on whether the reaction stoichiometry is an a or a b, two types of influence are seen in Eqn (7). They are shown graphically in Eqns (8) and (9) together with their corresponding contributions to b ij : x j À! a jr  v r À! b ir x i ; b ij ¼ a jr Áþb ir Á v r x j ¼þ v r x j ð8Þ x j À! a jr  v r À a ir x i ; b ij ¼ a jr ÁÀa ir Á v r x j ¼À v r x j ð9aÞ Although they are similar to Graph (1), Eqns (8) and (9a) have meanings that differ from t he meaning of Graph (1): They do not represent chemical conversions but rather the influences of one substance on another. For this reason w e shall c all t hem one-step influences or one-reaction (influ- ence) steps: they are not branched and correspond to any step between two substances in graphs such as Graph (1), i.e. any path that involves a single chemical reaction. One actual chemical reaction may effect a number of such one- step influences, typically from any s ubstrates onto any of its products, b etween its substrates and of a substrate on itself. Equations (8) a nd (9a) should be read as follows. They indicate the influence the (production rate of the) substance on the right may experience through reaction r,fromthe substance o n the left, which is a substrate of that reaction r if the l eft h and factor a equals 1 (and not zero, as other wise). Such an influence exists a nd is positive if the substance on the r ight is the product of that reaction (then there is a factor b equal to 1) a nd the right hand arrows points towards that substance. Such a n influence also exists but is negative when the s ubstance on the right i s a substrate o f the reaction. Then the a rrow points b ackward, i.e. a way from t he substance and there is a right-hand factor – a equal to )1. Influences of the type in Eqn (8) contribute positive values to b ij and are called positive one-reaction (influence) paths, or positive (influence) steps. This is the influence that a substrate has on the product of a reaction. If i ¼ j,this positive path becomes a positive loop (see bel ow). Influences of the type in Eqn (9a) are designated as negative one-reaction (influence) paths or negative i nfluence steps [21] because they contribute negative values to b ij . They correspond to the influence of a substrate on a nother substrate o f the same reaction r. If i ¼ j, Eqn (9a) defines a so-called negative half-step instead of a negative step (we omit ÔinfluenceÕ for brevity): x i À! a ir  v r À a ir x i ; b ii ¼ a ir ÁÀa ir Á v r x i ¼À v r x i ð9bÞ which could also have b een symbolized as: x i À! ða ir Þ 2  v r ; b ii ¼ a ir ÁÀa ir Á v r x i ¼À v r x i ð9cÞ hence its name Ôhalf-stepÕ. This is the (negative) influence a substrate has on its own removal. It is obtained for all substrates of any elementary reaction. The main point of the present section is that any Jacobian matrix element equals the sum of a number of direct parallel influence steps (one-step influence paths) in the kinetic scheme, i.e. the sum of paths through reaction-centered graphs of the type o f Graph 1 (these paths may contain parallel and antiparallel arrows). The sign of that element therefore depends on the both the sign and the magnitudes of these influen ce paths (see below). If all its i nfluence paths are positive, the Jacobian matrix element will be positive and for the J acobian matrix eleme nt to b e negative at l east one influence path must be negative. These are properties that we shall use below. How graphical structures relate to instability For the linear system given in Eqn (6) the so-called characteristic polynomial p(k)is: pðkÞ¼detðB À kIÞ¼0 ð10Þ Here B is again the Jacobian with elements b ij and I is the unit matrix. The polynomial Eqn (10) can be expanded as follows pðkÞ¼k m þ a 1 k mÀ1 þ a 2 k mÀ2 þ ::: þ a m ¼ 0 ð11Þ where m<n continues to refer to the number of inde- pendent concentration variables. The coefficient a i is related to the element b ij of the Jacobian by: a 1 ¼ðÀ1Þ 1 Á X i b ii ; a 2 ¼ðÀ1Þ 2 Á  X i;j b ii b jj À X i;j b ij b ji  ; a 3 ¼ðÀ1Þ 3 Á  X i;j;k b ii b jj b kk À X i;j;k b ii b jk b kj þ X i;j;k b ij b jk b ki  ; ; a m ¼ðÀ1Þ m Á det B ð12Þ Each coefficient of the characteristic equation hereby is a ÔsumÕ of products (with various signs, see below) of elements of the Jacobian. In graphic al terms the coefficient a p equals the sum of all possible Ôp th order simplest combinations of minus auto- influence pathsÕ. An autoinfluence path (or, shorter, a cycle) is defined as a cyclic path of any length through the diagram, such that any reactio n and any species occurs only once on that path. Autoinfluence paths of lengths 1, 2 , 3, etc. correspond to the terms b ii , b ij b ji , b ij b jk b ki , etc., respectively, inEqn(12).Theycontain1,2,3,etc.speciesand1,2,3,etc. influence steps, respectively. Autoinflu ence paths of length 1 are h alf-steps, graphically represented as i n Eqn (9c). An autoinfluence path runs from some species k back to species k an d travels through positive i nfluence steps [reaction nodes with equally directed arrows, as in E qn (8)] or negative influnce s teps [reaction nodes with oppositely directed arrows, as in E qn (9a)]. Consequently, a Ôminus autoinfluence pathÕ is negative (Ôeven cycleÕ) if t he number of its negative s teps as shown in Eqn (9a) is even and positive (Ôodd cycleÕ) if the number of its negative steps is odd. A Ôminus combined autoinfluence path of order pÕ is defined as a set of minus ÔcyclesÕ suchthat(a)eachofasetof p species is involved precisely once in that set of cycles, a nd (b) t he various cycles in this combination have n o reactions or specie s in common (i.e. t he cycles in such a combination do not touch each other). The value (the sign) of a cycle is the product of the values (the signs) of its steps. The sign o f the magnitude of a Ôminus combined autoinfluence path of order pÕ equal those of the arithmetic product of its Ó FEBS 2004 Graphical tool to identify oscillophores (Eur. J. Biochem. 271) 3879 component Ôminus cyclesÕ. Accordingly, the influence (positive or negative) of a combined autoinfluence path depends only on the number of its Ôeven cyclesÕ,havingan even number of negative one-step influences (Eqn 9a) and any number of positive one-step influences (Eqn 8). In other words, if the number of Ôeven cyclesÕ in the combination is even, this combination contributes to the coefficient of characteristic equation a positive term. Thus, a single Ôeven cycleÕ gives rise to a negative in the characteristic equation (a positive autoinfluence). The absolute magnitude of a combined autoinfluence path is the product of all the rates divided by the product of all the concentrations of its species. Therewith the c oefficient of o rder p of the characteristic equation equals the sum of all simplest combined Ôminus autoinfluence paths of order pÕ in the network. That each coefficient of characteristic equation therewith corresponds to a sum of minus paths in the network, is the basis of the graphical analyses of the characteristic equation and of t he method we develop here. Inspection of Eqn 12 shows t hat in a ll coefficients a p the term consisting of negative half-steps (Eqn 9c) only, which corresponds to products of Jacobian elements b ii only, is always positive: of the term of o rder p the sign is ( )1) p multiplied by ()1) p . I ndeed, a ll these terms always constitute negative combined autoinfluence paths. This graphical procedure allows us to determine all coefficients of the characteristic polynomial for systems o f simple reactions. The graphical determination of character- istic polynomial co efficients for complex stoichiometries has been elaborated by Ivanova [23]. The concentrations are restricted by balance constraints (conserved sum concentrations, such as NADH + NAD) and by the requirement that they be positive. These restrictions define upper and lower limits for the values the concentrations can assume (i.e. borders of the so-called phase space). Any negative a i coefficient implies that the system is unstable [22]. Such instability could lead t o infinite growth (explosion) of some concentrations, unless the highest-order coefficient a m is or becomes (the reference state m ay shift) positive. If the system does not have steady states on its border, all phase trajectories lead inward. Steady states on the border are readily identified ([21] a nd an example below). That a m be positive and a i (i<m)benegativeina (unstable) steady state [25] (together with the border conditions mentioned above [21]) i s what w e shall here call the ÔoscillophoreticÕ condition, i.e. the condition for a stable limit cycle around the unstable s teady-state point. All phase trajectories (i.e. a ll time evolution o f the system through t he space of the concentrations) should then approach a cyclic trajectory m ore a nd more closely as time proceeds, approaching that stable limit cycle either from the outside or from the inside. In this paper we focus on this aspect of instability. We shall ask when the above instability condition, i.e. at least one a i being negative, is met. We shall not consider the condition that a m be positive. The formalism described in the preceding paragraphs will help us find the g raphical structures, i.e. ÔsubgraphsÕ (see below), in t he kinetic s cheme that contribute t erms to the coefficients of the characteristic polynomial of a predictable sign and that hence help determine th e stability properties of t he system. The aim of this paper is to identify ÔnegativeÕ subgraphs, because they can induce instability; their positive combined autoinfluence bestows them with oscillophoretic potential. The instability condition that a p be negative for s ome p < m translates to the c ondition that the positive combined autoinfluence paths of order p should outweigh the negative combined autoinfluence paths of that same order. From this, an Ôinstability ruleÕ follows. This is stated as ÔInstability is promoted (counteracted) by positive autoinfluence paths.Õ This c onnotes with instability be ing generated by positive feedback loops. Subgraphs favoring instability As mentioned above we d eal here with the formulation t hat decomposes biochemical networks into truly elementary reactions. A network then c onsists of a great many such reactions (each represented as a black node in our reaction equations), each of which connects a number of species (represented as white nodes). The entire network may become unstable when part of it w ould by itself be unstable. Consequently it can be useful to identify parts of the larger network that are unstable. The g raphical representation o f a subnetwork with e qual numbers of species a nd reactions is here called a subgraph. It is useful to consider subgraphs because all combined autoinfluence paths that visit a ll reactions and species within such a subgraph have equal absolute magnitudes (i.e. the product o f the rates divided by the products of all the species concentrations) but may differ in sign. By considering all such combined autoinfluence p aths of a subgraph together , one can therefore decide whether t he subnetwork as a whole promotes or counteracts instability: one simply determines whether more positive than negative combined a utoinflu- ence paths occur in that s ubnetwork. We shall speak of a negative s ubgraph in this case. We define the value of a subgraph that contains p reactions and p concentrations, as minus the sum of all its combined autoinfluence paths of order p. Note therefore that negativity of a subgraph and positivity of autoinfluence connote with instability. We shall now determine t he signs and hence the stability properties of a number of s ubgraphs. We shall do this first for subnetworks that consist of a single reaction, then for subnetworks of two reactions, then f or subnetworks of three reactions. F inally we shall consider subnetworks of arbi- trary size. One-reaction subgraphs. For the elements b ii,r,1 that correspond to the reactions r in which x i drives its o wn production (i.e. x i fi x i ), there is both a negative half step (because, as usual, x i stimulates its own removal; Eqn 9c) and a positive l oop [becau se x i now also stimulates its own production; compare Eqn (8) with i ¼ j ]. Adding these two, Eqn (7) shows t hat they cancel each other: Àa 1 3 b ii;r;1 ¼ðÀa ir a ir þ b ir a ir ÞÁ v r x i ¼ 0; ð13Þ where the symbol ’ means ÔcontainsÕ. The value of zero is obtained because a ll stoichiometric coefficients equal one (i.e. the reaction x i fi x i cannot lead to a net increase in x i because of the restrictions we here impose on the stoichio- metries; we can only have such a reaction produce a single 3880 B. N. Goldstein et al.(Eur. J. Biochem. 271) Ó FEBS 2004 molecule of x i ). These reactions are thus without any i nflu- ence, as the negative i nfluence is balanced by the positive one. In terms of autoinfluence p aths, the former negative half- step is one cycle with one negative o ne-step influence, hence a negative autoinfluence and stabilizing, whilst t he latter positive loop is one cycle with no negative one-step influence, hence a positive autoinfluence and destabilizing but of equal magnitude (because it belongs to the same subgraph): the two cancel. Autocatalytic processes such as x i fi 2x i are not described as single reactions in our formalism, as stoichiometry b would exceed one. What remains for a 1 is all the reactions that have x i as the substrate and not as the product. Therefore, a 1 is constructed only from the corresponding negative half-steps with the values [compare Eqns (7) and (9c)]: Àa 1 3 b ii;2 ¼Àa ir a ir Á v r x i ¼À v r x i ð14Þ This corresponds to a single cycle (from x i back onto itself) with a single n egative i nfluence step, i.e. it is negative in terms of autoinfluence and promotes stability. The sum total f or one-reaction subgraphs is thereby always positive (their total autoinfluence is negative). Indeed, it follows from Eqns (12) and (14) that the coefficient a 1 is always positive, favoring stability. We conclude that one-reaction subgraphs cannot give rise to instability. These need not be analyzed therefore for deciding on the potential instability o f large networks. Two-reactions subgraphs. We here consider examples of subgraphs with two species and two reactions, such as the branched cycle: According to Eqns (12) and (7) this subgraph contributes to minus the a 2 coefficient of the characteristic polynomial the two terms o n the right-hand side of the following equation: Àa 2 ¼Àb ii b jj þ b ij b ji ð15Þ where Àb ii b jj ¼ÀðÀa ir a ir ÞÁ ðÀa js a js Þþa js b js ÀÁ Á v r v s x i x j ¼Àða ir a ir a js a js À a ir a ir a js b js ÞÁ v r v s x i x j ¼ 0 ð16Þ b ij b ji ¼ða js b is ÞÁða ir b jr ÞÁ v r v s x i x j ¼ v r v s x i x j ð17Þ In the first facto r of Eqn (16), which corresponds to the direct self-influence t erm b ii , one recognizes the influence that x i has on itself through its own d egradation v r (the path of influence –a ir a ir ). In the s econd factor of Eqn (16), which corresponds to the direct influence of j on itself (b jj ), one recognizes the two direct influences x j has on itself, i.e. one through its degradation (–a js a js ) and a second influence through its autocatalytic feedback (+a js b js ) through the same reaction, v s . The latter influences cancel each other, again as all stoichiometries are equal to one. The second phrasing of Eqn (16) corresponds to the sum of two autoinfluence paths of order two. Both of these contain t he negative half step of x i back onto itself which is negative. O ne of th em multiplies w ith the negative half-step of x j back on to itself and constitutes n egative a utoinfluence (even number of cycles and even number o f negative influences). The other multiplies w ith the positive loop of x j back onto itself through v s : two cycles with one negative influence constituting a positive (destabilizing) autoinflu- ence. These two autoinfluence paths of order two cancel each other. In Eqn (17) one re cognizes the influence that x j has o n x i (+a js b is ) b ecause t he former is the substrate of the reaction v s that produces the latter, as well as the a nalogous influence x i has on x j (+a ir b jr ). Together they constit ute the positiv e influence that x i has on itself through the negative path constituted by the sequel of reactions v r and v s . Equation (17) corresponds to a single even cycle with two positive influence steps, i.e. promoting positive autoinfluence a nd hence instability. The net sign of the three autoinfluence paths is )1+(+1)+(+1)¼ +1, i.e. Graph (3) contri- butes the negative (instability) term [Eqn (17)] to a 2 .Graph (3) is ne gative. As Graph (3) will be part of a larger network, whether it actually will be able to cause instability ( oscillations) depends on whether the precise kinetic parameter values make its p ositive a utoinfluence dominate t he negative autoinfluence in the other s ubgraphs of order two of the network. It m ay b e noted that the corresponding subgr aph that lacks the loop, i.e. in which reaction s does not reproduce its substrate x j , cannot be negative, and hence cannot cause instability. Then only the negative autoinflu- ence path of Eqn (16) remains, which then cancels the positive autoinfluence of Eqn (17). Revolving around the cycle then does not lead to an increase in the number of molecules. Elementary reactions producing more types of product than types of substrate are essential for the occurrence of instability, due to the restriction s on stoichiometries that d erive f rom ou r descent t o t he molecular level. Another branched cycle, Graph (4), having two bran- ches, contributes to –a 2 the same positive, destabilizing term: Graph 3. Graph 4. Ó FEBS 2004 Graphical tool to identify oscillophores (Eur. J. Biochem. 271) 3881 Negative Graphs (3) and (4) involve positive paths with branching that can be interpreted as positive feedback interactions (autocatalysis). They can also be interpreted as product activation in some enzyme reactions, because a reaction product here stimulates the same re action. Another example of such positive paths of influence occurs in the case of the antiport of two ligands by a protein molecule through the membrane [24]. Three-reaction subgraphs. Inthesamewayweidentify the negative ( instability generating) subgraphs with three species and three reactions, by pointing out that they have positive p aths of influence. We divide these g raphs into two classes, i.e. those with positive influence steps only, and those including an even number (two) of negative s teps in the cycle. The former class is s hown in Graph (5): The latter c lass of graphs, involving two negative s teps in the cycle with three s pecies and three reactions, i s presented in Graph (6). All of the subgraphs in G raphs (5) and (6) contribute the following 3! terms to the coefficient a 3 : a 3 ¼ðÀ1Þ 3 ðb 11 b 22 b 33 þ b 21 b 13 b 32 þ b 31 b 12 b 23 À b 11 b 23 b 32 À b 22 b 13 b 31 À b 33 b 12 b 21 Þ The first term here corresponds to all h alf-steps multiplied, the second and the third terms correspond to the circular paths running through all three reactions and all three species. The other three terms correspond each to a single half-step, multiplied by a circular p ath running through two reactions and two species. Using the graphical rules mentioned above, we now consider the left hand subgraph in Graph (6) as an example. This subgraph contains the following three simplest com- bined autoinfluence paths (simplest subgraphs): 1. One combined autoinfluence p ath is the positive c ycle going through all three species and all three reactions, from species one to species two to species three (corres- ponding to b 12 b 23 b 31 ). This one cycle i nvolves two negative steps such as shown in Eqn (9a) and t hen a positive step [as in Eqn (8)]. The rule then makes for odd (number of cycles), even (number of steps), hence positive combined autoinfluence: + 1. The second such auto- influence path, i.e. the reverse of this cycle i s absent here. 2. A second combined autoinfluence path consists of one cycle going through the two species and two reactions on the l eft hand side of t he subgraph, and one n egative half step on the right hand side (as i ndicated in bold; b 12 b 32 b 33 ). The arithmetic product of these two positive autoinfluence paths is positive (once cycle, no negative influences): +1. The mirror image combined autoinflu- ence path (b 12 b 12 b 33 ) would run through the two reactions on the right hand side (negative), a nd one negative half-step on the left (positive). However, the latter would touch the cycle, hence this one does not count. The third term, i.e. b 11 b 23 b 32 is also ÔemptyÕ for this diagram. 3. The third type of combined autoinfluence is negative: a combined influence of three separate anti half-steps: )1. The sum of these t wo positive and one negative combined autoinfluences contributes a negative term into the coeffi- cient a 3 , and therewith promotes instability. The subgraph on the left of Graph (6) is therewith negative. In fact all of t he subgraphs in G raphs (5) a nd (6) contribute the positive term v 1 v 2 v 3 x 1 x 2 x 3 to minus the a 3 coefficient of the characteristic polynomial (the negative term to the coefficient a 3 ), where subscripts 1, 2 and 3 refer to the different sp ecies an d r eactions in the subgraph. C onsequently they promote i nstability. All of these subgraphs are negative. All subgraphs represented in Graphs (5) and (6) have a single branched reaction, constituting two so-called Ôeven cyclesÕ.AnÔeven c ycleÕ involves an even number (here 0 or 2 ) of equally directed influence steps and any number of positive influence steps. Such cycles cause graphs to become negative. In full reaction schemes some reactions start f rom species in a n egative subgraph. These efflux reactions can eliminate the negative subgraph, wh en the system r eaches its steady state. In the steady sta te the equality of rates for internal and external opposite reactions eliminates the negative term, induced by the negative subgraph. Then only damped oscillations can be observed (see below for examples). To obtain sustained oscillations, the efflux reaction should be reversible, leading, for example, to an inhibitory enzyme complex (see below for an example). In the latter case the reversible steady-state efflux equals zero and the negative subgraph is upheld. n-Reaction subgraphs. We can now formulate the proper- ties of any negative subgraph t hat contains an arbitrary equal number of species and reactions. Such a negative graph should be constructed at least of two even cycles, formed by a branched reaction. Moreover, species of the negative subgraphs should not be connected with other parts of the full scheme by outgoing irreversible reactions. The outgoing i rreversible reactions cause t he correspond- ing opposite stationary fluxes to be equal, canceling the negative s ubgraph with a positive graph of th e same absolute value (see b elow for a n example). Therefore, only damped oscillations can be obtained in such a case. Additional reversible reactions, leading through the species of the n egative graph to dead-end species , do not eliminate the negative graph. Graph 5. Graph 6. 3882 B. N. Goldstein et al.(Eur. J. Biochem. 271) Ó FEBS 2004 Examples of biochemical oscillations Two interacting enzymes. Here we discuss one of the simplest biochemical osc illators. Its kinetic scheme (Fig. 1) contains a negative graph [Graph (4)] of second order (two species and two reactions). This sug gests that the system of Fig. 1 may oscillate. A more detailed analysis should then be undertaken to determine whether it will actu ally oscillate. We shall now do this. In Fig. 1, two enzymes, E 1 and E 2 , modify each other by releasing group P in the reactions E 2 -P fi P+E 2 and E 1 -P fi P+E 1 . The former reaction is catalyzed by E 1 , the latter by E 2 . T he arrows indicate the p referential reaction orientation. Various biochemical systems can be recognized in Fig. 1, for example, mutual dephosphory- lation (or phosphorylation) if P represents phosphate. P is not considered to be a variable; it should be present in excess. The scheme in Fig. 1 is open for the fluxes through E 1 (e.g. synthesis, degradation) but the total concentration of E 2 is conserved: ½E 2 -Pþ½E 2 ¼E 2 or x 4 þ x 2 ¼ 1 ð18Þ The reaction participants and their normalized concentra- tions x i (i.e. their concentrations divided by the total concentrations of E 2 ) are shown i n Fig. 1. The following kinetic equations correspond to Fig. 1: dx 1 dt ¼ k 2 x 2 x 3 À k 4 x 1 dx 2 dt ¼ k 1 x 1 x 4 À k 3 x 2 dx 3 dt ¼ k 5 À k 2 x 2 x 3 ð19Þ Taking into account the constraint in Eqn (18), Eqns (19) involve three independent variables, i.e. x 1 , x 2 , x 3 . We shall now analyze the characteristic polynomial o f the t hird order f or Eqns (19). We know that the coefficient a 1 is always positive, and it is readily seen that the coefficient a 3 is also positive h ere. The coefficient a 2 contains a negative t erm, which corresponds to the negative graph highlighted in Fig. 1 by the heavy lines. This negative term equals –v 1 v 2 /x 1 x 2 . At steady state v 2 ¼ v 4 and v 1 ¼ v 3 . Then the positive term +v 4 v 3 /x 1 x 2 ,which is also present in the coefficient a 2 for Fig. 1, cancels the term –v 1 v 2 /x 1 x 2 of the negative graph. Therefore, in the steady state a 2 is positive. Consequently, only damped oscillations can be observed in Fig. 1. Considering the structure of the reaction scheme in Fig. 1, one recognizes that irreversible effluxes of species of the negative graph must be present in order for the steady state condition to be satisfied. For the same reason, all the biochemical schemes involving a negative graph of second order can only induce damped oscilla- tions. Damped oscillations calculated for F ig. 1 are shown i n F ig. 2. Substrate inhibited bifunctional enzyme. Many kinetic graphs that generate oscillations with only positive auto- influence p aths are known from the literature. Some of them have been classified [25]. A lthough the graphs with positive paths implemented here e.g. Gra phs (3–5) are simpler t han the kinetic graphs in [25], t hey can represent biochemical reality. For example, t he second of the subgraphs in Graph (5) has been used to analyze the network topological basis for oscillatory antiport of t wo different ions across the cell membrane [24]. Less studied are the graphs that include negative paths, such as those in G raph (6). Two negative p aths in the c ycle of subgraphs here reflect the competition of two r eactions for a single species. For example, the competition of protein X and the enzyme E 2 for the acetyl group in pyruvate dehydrogenase c omplex has been shown to be important for the prediction of o scillatory behavior [28]. The phenomenon of substrate inhibition is often associ- ated with the potential for oscillations [33]. An earlier graph- Fig. 2. Calculated time dependence of the normalized E 1 concentration (X 1 ) for Fig. 1. Time scale in relative units. The following parameter values were used for these calculations: k 1 ¼ 0.1, k 2 ¼ 1, k 3 ¼ 2.2, k 4 ¼ 5, k 5 ¼ 1. The dimensions of these parameters values are not specified because they depend on the time scale and can be d ifferent for different actual systems. Species concentrations were normalized. The initial values of the normalized concentratio ns were: x 1 (0) ¼ 1, x 2 (0) ¼ 0.4, x 3 (0) ¼ 0.2, x 4 (0) ¼ 0.6. Calculations used the computer program DBSOLVE (created by I. I. Goryanin, Institute of Theoretical and Experimental Bi ophysics, RAS, M oscow Region, Russia). Fig. 1. Reaction scheme of two enzymes d emodify ing e ach other. Filled circles represent reactions, open circles represent substances. T he rate of the reaction t hat combines E 2 with P to yield E 2 -P is given as v 3 .IfP represents a p hosphate group, reaction nu mbe r 3 could be a p rotein kinase, and v 1 should t hen represent the dephosphorylation of E 2 -P, as catalyzed by E 1 . This reaction re leases P a nd E 2 . In this reaction E 1 is used but im mediately r eleased a s it i s a catalyst. The rate of reaction 2 i s v 2 , which is catalyzed by E 2 (which then functions as a protein p hos- phatase) and dephosphorylates E 1 -P. Reaction 4 d egrades E 1 . Reac- tion 5 synthesizes E 1 -P de n ovo (i.e. not from E 1 ). Ó FEBS 2004 Graphical tool to identify oscillophores (Eur. J. Biochem. 271) 3883 theoretic analysis, however, [21] showed that some kinetic schemes with substrate inhibition cannot induce sustained oscillations. W e s hall here discuss a n e xample of subgraphs with substrate inhibition, in the context of oscillations observed f or phosphofructo-2-kinase:fructose-2,6-biphos- phatase [21,27]. Nonlinear oscillations or bistable switches in this bifunctional e nzyme could be highly important for a switching mechanism between the opposing fluxes in glycolysis/gluconeoge nesis. This case has been analyzed before (e.g. in [21,27]), but this paper will now present the detailed an alysis of conditions necessary for the negative graph t o induce sustained oscillations. We shall analyze the steady states, because the presence of steady states on the phase-space border and the irreversible steady-state efflux from the species of the negative g raph can eliminate oscillations. The following kinetic graph (compare [21,27]) is drawn to illustrate the analysis of the steady states. Graph (7) shows the substrate cycle, S 1 fi S 2 fi S 1 ,as catalyzed by the bifunctional enzyme (E 1 /E 2 ;E 1 and E 2 are two states of a single protein). H ere t he arrows between symbols correspond to the preferential reaction orientation. In reaction 1, E 1 catalyzes the f orward reaction S 1 fi S 2 and E 2 catalyzes reaction 2 , which runs in the opposite direction, S 2 fi S 1 . Reaction 2 may be coupled to a s ource of external free energy. Alternations between two enzyme activities are caused by conformational transitions, i nduced by the modifying enzymes, E 3 (catalyzing reaction 3) and E 4 (E 4 is not shown in the scheme). The reaction S 1 fi S 2 is merely catalyzed by E 1 alone, but during the reaction S 2 fi S 1 the e nzyme undergoes cyclic conformational transitions E 2 fi E 1 fi E 2 , where t he latte r transition is catalyzed by E 3 . G raph (7) does not contain irreversible effluxes from the s pecies (E 2 ,E 3 and S 2 ) of the negative graph that i s shown b y h eavy ar rows, a nd contai ns only the influx to S 2 , i.e. 1 fi S 2 (from reaction 1 to S 2 ). Therefore it retains its oscillophoretic potential. Inhibitory reversible reactions, added to the negative graph, do not interfere with that potential. The subgraph highlighted by the heavy arrows in the full Graph (7), is one of the negative g raphs identified in this paper [i.e. the second left of the subgraphs in Graph (6)]. This negative graph is the b ranched cycle with one positive loop, represen ting the E 3 catalyzed reaction tha t ma kes E 2 out of E 1 , and one longer cycle, involving two negative i nfluence steps, E 3 fi 4 ‹ S 2 and S 2 fi 2 ‹ E 2 . T hese two negative influence steps corres- pond to the competitive interactions of S 2 with E 3 and with E 2 . The reaction E 3 fi 4 ‹ S 2 is the forward reaction for E 3 inhibition by S 2 . For simplicity, the reverse reaction (not participating in the negative graph) is not shown. Oscillations can be expected if we add reversible inhibi- tion of E 3 by substrate S 1 to Graph ( 7). The reversible inhibitions of E 3 by both S 2 and S 1 do not eliminate the negativity of the negative subgraph, because these reversible steps d o not contribute additional terms to the terms of the negative graph. Their contributed effluxes are eq ual to influxes. However, t he number of s pecies of reactions becomes larger with this new inhibition. Accordingly, a positive graph with four species and four reactions, as well as a negative graph with three species and three reactions, are obtained in Graph (7). This is a sufficient condition for oscillations to arise. We shall now show how a necessary condition for oscillations to occur follows from the absence of steady states on the border of the phase space. The full s ystem contains seven species variables: x 1 ¼½E 1 ; x 2 ¼½S 1 ; x 3 ¼½E 2 ; x 4 ¼½S 2 ; x 5 ¼½E 3 ; x 6 ¼½E 3 S 2 ; x 7 ¼½E 3 S 1  These species are interdependent through the following three balance constraints: x 2 þ x 4 þ x 6 þ x 7 ¼ S ¼ constant x 1 þ x 3 ¼ E ¼ constant ð20Þ x 5 þ x 6 þ x 7 ¼ E 0 ¼ constant These constraints reflect the conserved total concentra- tions of the substrates (we shall use S ¼ 3.3 relative units) of the b ifunctional e nzyme (E ¼ 0.2 relative units), and of the modifying enzyme ( E¢ ¼ 0.31 relative units). The following four equalities for the steady s tate reaction rates a re deduced from the structure of Graph (7): v 1 ¼ v 2 ¼ v 3 ; v 4 ¼ v 5 ; v 6 ¼ v 7 ð21Þ where indices 1,2,3,… relate to various reactions, v 4 and v 5 relate to the reversible reaction E 3 +S 2 b « E 3 S 2 , v 6 and v 7 relate to the reversible reaction E 3 +S 1 b « E 3 S 1 .The equalities [Eqn (21)] together with the constraints [Eqn (20)] allow u s t o obtain all seven concentration values for one of the steady states in the phase space [E 1 ] ¼ 0, [S 2 ] ¼ 0, [E 3 S 2 ] ¼ 0, [E 2 ] ¼ E, and [S 1 ], [E 3 ], [E 3 S 1 ] inside of the phase space. No steady states exist on the borders of the phase space. This is the necessary condition for sustained oscillations to be observed in this s ystem. The stability of the steady states was analyzed by using four differential equations for four independent species variables: Graph 7. 3884 B. N. Goldstein et al.(Eur. J. Biochem. 271) Ó FEBS 2004 dx 1 dt ¼ k 2 x 3 x 4 À k 3 x 1 x 5 dx 2 dt ¼Àk 1 x 1 x 2 þ k 2 x 3 x 4 À k 6 x 2 x 5 þ k 7 x 7 dx 3 dt ¼ k 3 x 1 x 5 À k 2 x 3 x 4 dx 4 dt ¼ k 1 x 1 x 2 À k 2 x 3 x 4 À k 4 x 4 x 5 þ k 5 x 6 ð22Þ In addition to referring to the a bsence of steady states on the border of t he phase space, the procedure b y Clarke [22] enables us to identify qualitatively phase trajectories that lead to a stable limit c ycle. The c haracteristic polynomial o f the system in Eqn (22) reads: k 4 þ a 1 k 3 þ a 2 k 2 þ a 3 k þ a 4 ¼ 0 ð23Þ If in this polynomial a 4 > 0 for all concentration values and a 3 < 0 in the unstable steady state, oscillations can be obtained. Analysis of negative and positive subgraphs and comparison of their values gives rise to the estimation of the kinetic parameters t hat enable such oscillations. The main result of such an analysis is that oscillations arise if t he parameter k 3 is the l argest and the parameters k 6 and k 7 are the smallest in the system. Oscillations in this system can indeed be observed [27]. Discussion Oscillatory phenomena in biochemical systems are be ing studied more and more intensively. All known kinetic models for calcium oscillations have been reviewed recently [4]. Models for other oscillatory phenomena continue to appear [33–39] and many more w ill appear in the future with the increasing possibilities for inspecting the d ynamics inside single c ells [40,41]. Our classification of kinetic schemes (or ÔmotifsÕ [40]) into ones that may and ones that cannot exhibit oscillations may be useful for the analysis of th e existing models that are responsible for the oscillations. Such an analysis may help to understand the mechanisms underlying the oscillations, and perhaps even suggest ways to influence the dynamics of such systems. Our method is based on the molecular mechanisms without any preliminary simplifications and without using phenomenological equations. It may t here- fore be suitable, especially now that functional genomics is unraveling more and more of the molecular specifics that underlie cell fun ction. Our method to classify potential biochemical oscillators is based on t he graphical analysis of t he kinetic schemes. Our approach is similar in s ome aspects to the procedure described previously [25,26]. However, the representation o f the k inetic schemes i n terms of dual grap hs [ 21] is different, and has enabled us t o simplify the identification a nd the classification of oscillophoretic networks. Because above we were most concerned with demon- strating the basis of our method, we here summarize how the approach may be implemented in the context of a known reaction network. First the network kinetics should b e drawn o ut i n a detailed molecular s cheme making all molecular interactions, such as the binding of a ligand to a n enzyme, exp licit. Then one should try to recognize subgraphs of known s ign in that s cheme. Here one may resort to the subgr aphs identified i n this p aper, or to subgraphs that may appear i n future w ork analyzing networks more extensively. Alternatively, one may u se the method of making an inventory o f t he autoinfluences within each subgraph and determine whether there are more positive ones than negative ones, in which case the subgraph is negative (unstable). Having identified the (negative) subgraphs with oscillop horetic potential, one may then analyze their effect quantitatively and compare the results to those obtained for through analysis of all other subgraphs of the same order in the same network, as was illustrated for the two examples in this paper. The network outside the former s ubgraph may do away with the oscillophoretic potential of the subgraph or maintain it by contributing subgraphs of equal order but of different or equal sign and magnitude: the dynamic development of a system is ultimately dictated by the i nfluences the concentrations of its substances have on each others (and their o wn) develop- ment in time [7,35,42]. When systems are analyzed on a more coarse-grained level than we do here, the influences are not defined in t erms of rates, concentrations and reaction stoichiometries only. In such analyses, other properties such as elasticity coeffi- cients [31], Michaelis constants, and kinetic powers [29,30] also determine the dynamics and s tability of the system [12]. To t he extent that these analyses deal with the origin of dynamics in terms of n etwork topology, then that topology is the topology of influences. This type of more coarse- grained analysis is useful when the systems are not yet understood to molecular kinetic detail, or when the systems are so large that a detailed m olecular analysis is beyond reach and modularization is required. Because we here analyze at the level of complete molecular detail (i.e. only reactions with zero and first order kinetics), the topology of the influences coincides with the topology of the network stoichiometries. Our method has this as an important advantage, which comes w ith i ts ot her advantage of b eing completely molecular. This very advant- age can of course become a d isadvantage in cases where molecular detail is not known or required. The d ynamics of cellular systems are determined at many different levels of the cellular control hierarchy. For different levels of the hierarchy, different methods for the analysis of th e d ynam- ics are needed. We demonstrated how reaction networks that are formulated down to the detail of simple unimolecular and bimolecular reactions can be organized into topologies. The latter can then be examined for their potential to induce oscillations. Oscillophoretic topologies involve branched directed cycles, c onstructed of a n even number of negative paths a nd any number of positive paths. Our approach has the advantage th at it considers positive and negative interactions in a unified manner. The implication o f the identification o f an oscillophoretic subgraph is that if such a subgraph is found in a large network, then that network may be unstable and give rise to oscillations; the presenc e of an oscillophoretic subgraph is a necessary condition for the network t o engage in t he oscillations. However, i t is not a sufficient condition. Ó FEBS 2004 Graphical tool to identify oscillophores (Eur. J. Biochem. 271) 3885 Whether the overall network actually engages in an oscillation when an oscillophoretic subgraph is present depends on the precise parameter values. To estimate the parameter domain where oscillatory phenomena can be observed, the numerical value of the negative graph should be compared with the values of other graphs of the same order in the system. In practice this means that to produce oscillations, reactions involved in the negative graph should be rapid enough a s compared with their surrou nding reactions. We here p erformed such an analysis for two examples, one with positive and one with negative inter- actions. We classified g raphs of different t opologies with two species and two reactions as well as with three species and three reactions, which can i nduce oscillations, if they are connected with other parts of the system. Sustained oscillations can be induced if these connections are irreversible influxes or reversible dead-end reactions. All considered topologies involved a single branched reaction. More complicated topologies with additional reaction branching do not eliminate o scillations. Graphs of similar topologies but with different numbers of species and reactions (the number o f species and r eactions in the analyzed graphs is the same) retain the oscillophoretic property. On the basis of their network topologies, our approach can predict a number of new biochemical oscillators that fit the classification developed here, but were not included in fo rmer classifications [25]. It turned out that not only well-known substrate inhibition and product activation induce oscillations. Any competition of a single, channeled intermediate for multiple active sites in multienzyme com- plexes can also induce osc illatory kinetics [28]. Our approach to Ôoscillophore topologiesÕ can b e com- bined with other known theoretical approaches [29–32] to simplify the study of complex b iochemical systems. 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Westerhoff, H.V (2004) Coordinated behavior of mitochondria in both space and time: a reactive oxygen species-activated wave of mitochondrial depolarization Biophys J in press Westerhoff, H.V & Van Dam, K (1987) Thermodynamics and Control of Biological Free Energy Transduction Elsevier, Amsterdam . alysis of biochemical networks. It enables us to develop a graph-theoretical identification of networks that may, and of networks that cannot, serve as oscillophores (i.e oscillations. The main result of such an analysis is that oscillations arise if t he parameter k 3 is the l argest and the parameters k 6 and k 7 are the smallest