multicyclic

This paper presents a kinetic analysis of the whole reaction course, i.e. of both the transient phase and the steady state, of open multicyclic enzyme cascade systems. Equations for fractional modifications are obtained which are valid for the whole reaction course. The steady state expressions for the fractional modifications were derived from the latter equations since they are not restricted to the condition of rapid equilibrium. Finally, the validity of our results is discussed and tested by numerical integration. Apart from the intrinsic value of knowing the kinetic behaviour of any of the species involved in any open multicyclic enzyme cascade, the kinetic analysis presented here can be the basis of future contributions concerning open multicyclic enzyme cascades which require the knowledge of their time course equations (e.g. evaluation of the time needed to reach the steady state, suggestion of kinetic data analysis, etc.), analogous to those already carried out for open bicyclic cascades.

on-line at: www.actabp.plEnzyme cascades are ubiquitous in biological systems.They play an important role in the regulation of many physiological processes, e.g.regulation of metabolism, repair of lesions, protection against infectants, regulation of neurotransmi�er receptor function and of the efficiency of synaptic transmission, or determination of the balance between cell activation and cell death.The special significance of enzyme cascades is their ability to impose upper and lower boundaries on the rates of a biological process.Besides, the abundance of the design features in enzyme cascades provides many possibilities of response and adaptation to environmental cues and challenges.Such cascades are therefore essential to the success of evolutionary systems.The complicated structure of many enzyme cascades renders the kinetic analysis difficult.However, it is a prerequisite for the understanding of biological regulation at a high level.
Enzyme cascades may be classified into noncyclic and cyclic ones.The non-cyclic cascades are irreversible and unidirectional and involve activation of zymogens.A kinetic analysis of a general model of non-cyclic cascades has already been carried out (Havsteen et al., 1993).Cyclic cascades are a common and important type of enzyme cascades which operate by allosterically regulated chemical modification/demodification of the active site of key metabolic enzymes.Some examples of cyclic cascades are the cascade involved in the modulation of glycogen synthase and glycogen phosphorylase activity (Krebs, 1972;Chock et al., 1980;Edstrom et al., 1990;Cárdenas & Goldbeter, 1996;Schulz, 1998;Gall et al., 2000;Hanashiro & Roach, 2002;de Paula et al., 2002;Rozi & Jia, 2003), the one involved in the regulation of Escherichia coli glutamine synthetase (Stadtman et al., 1976;Chock et al., 1990;Stadtman, 1990;2001;Jiang et al., 1998;Mutalik et al., 2003), the G-protein cascade mediating phototransduction (Lamb, 1996), the mitogen-activated protein kinase cascades in Saccharomyces cerevisae (Gustin et al., 1998;Pomerance et al., 2000) and Xenopus lae-2005 R. Varón and others vis (Ferrell & Machleder, 1998), the genetic cascades such as the E. coli flagellar regulatory cascade (Kalir et al., 2001;Tha�ai & van Oudenaarden, 2002) or the cascade involved in the regulatory mechanism of inflammation and autoimmunity (Van den Steen et al., 1998).Another type of cyclic cascades involved in metabolic regulation are substrate cycles, where a target metabolite is reversibly interconverted into another chemical species by two different enzymes coupled in the opposite directions (Newsholme et al., 1984).The kinetics of substrate cycling are well known, both in the steady state (Passonneau & Lowry, 1978) and in the transient phase (Valero & García-Carmona, 1996), allowing these systems to be applied to the quantitative determination of low levels of a metabolite or an enzyme activity (Passonneau & Lowry, 1993;Valero et al., 1997;2000).
Concerning enzyme cascades, the steady state kinetics of monocyclic (Stadtman & Chock, 1977;Goldbeter & Koshland, 1987;1990;Cárdenas & Cornish-Bowden, 1989;1990;Szedlacsek et al., 1992) and multicyclic (Chock & Stadtman, 1977;Stadtman & Chock, 1979) systems has been analysed under some simplifying conditions that facilitate the derivation of the steady state kinetic equations.Besides, a complete kinetic analysis for both the transient phase and the steady state of monocyclic (Varón & Havsteen, 1990) as well as open and closed bicyclic cascades (Varón et al., 1994a;1994b) has been performed.However, a complete analysis of multicyclic enzyme systems, valid from the begin of the process, i.e. for both the steady state and the transient phase, remains to be carried out yet.Only simulated progress curves of uni-, bi-and tricyclic cascades obtained under severe assumptions were obtained (Stadtman & Chock, 1979), but no analytical time course equations for any of the species involved have been derived yet.
From the above it is clear that there are gaps in the kinetic analysis of multicyclic cascades.The steady state kinetic behaviour under different simplifying conditions is known (Chock & Stadtman, 1980), but there is no information about the way the system reaches this steady state, i.e. about the transient phase of the reaction.Analysis of the transient phase of multicyclic enzyme cascades is biologically important for the following reasons: 1) The transient phase kinetic analysis is the first step to establishing the time course of the different regulatory properties of a system and not only in the steady state of the reaction.Moreover, emerging from the transient phase equations of the cascades, studies could be carried out introducing new regulatory parameters allowing one to estimate the time needed by the system to reach the steady state or the time elapsed to generate a biochemical response to any primary stimuli, as already propose by us for mono-and bicyclic enzyme cascades (Varón & Havsteen, 1990;Varón et al., 1994a;1994b).This parameter is important, because a cascade of a high steady state sensitivity, but a long transient time may, in real time, behave like a low sensitivity system and, conversely, a cascade of a moderate steady state sensitivity could, in real time, displays a considerable sensitivity, if it has a short transient phase.
2) From the transient phase kinetic analysis an experimental design and a kinetic data analysis can be suggested which allow the evaluation of more kinetic parameters than those based on the steady state only.This has already been carried out for monocyclic and bicyclic cascades (Varón & Havsteen, 1990;Varón et al., 1994a;1994b).
3) From the transient phase equations those corresponding to the steady state can be obtained immediately.
4) Although most of the cyclic enzyme cascades known are either bicyclic (Stadtman & Chock, 1979;Chock & Stadtman, 1980;Cárdenas & Cornish-Bowden, 1989;Varón et al., 1994a;1994b) or tricyclic (Mutalik et al., 2004), we think that in the future multicyclic cascades involving more cycles will be reported and the analysis presented here could be a useful tool for the workers in this field and, of course, it can be applied, as a particular case, to any of the open multicyclic enzyme cascades actually known which fits the model.5) Finally, from a general open multicyclic cascade containing N-cycles a be�er appreciation of the relationship between the number of cycles in an open enzyme cascade and its regulatory properties can be obtained (Stadtman & Chock, 1979).
Therefore, the biological objectives of this paper are: a) To present a general kinetic analysis of open multicyclic cascade enzyme systems covering the whole course of the reaction, assuming certain conditions that render possible the derivation of explicit analytical equations that provide the transient phase equations for any form of the interconvertible enzymes involved in the model.From this kinetic analysis any of the tasks mentioned in points 1) and 2) above can be undertaken.b) To obtain the kinetic equations for the steady state as a particular case of the corresponding solution for the transient phase when time assumes sufficiently high values.
c) To apply the results to some particular cases of open cyclic cascades.d) To discuss the validity of our kinetic analysis and to check the quality of our results by comparison with those obtained by numerical integration using a specific so�ware for enzyme reactions previously developed by us (García-Sevilla et al., 2000) which allows the simulation of the kinetic behaviour of enzyme systems (e.g.any open multi-

MATERIALS AND METHODS
Simulated progress curves were obtained by numerical solution of the non-linear set of differential equations (B2) in Appendix B 1 , using arbitrary sets of rate constants and initial concentration values.This numerical solution was found by the Runge-Ku�a-Fehlberg algorithm (Fehlberg, 1970;Burden & Faires, 1985) using a computer program implemented in Visual C++ 6.0 (García-Sevilla et al., 2000).The above program was run on a PC-compatible computer based on a Pentium III/450 MHz processor with 128 Mbytes of RAM.Data thus obtained and the corresponding analytical solutions were plotted using the SigmaPlot Scientific Graphing System for Windows version 4.00 which was also used to obtain the corresponding values of the statistical parameters g(j) (j=3,6,9) given by Eqn.(84).

A MODEL OF AN OPEN MULTICYCLIC CAS-CADE SYSTEM
We studied a N-cycle cascade system that coincides, except in part of the notation, with the one proposed for the steady state by Chock and Stadtman (1977;1980).This system is shown in the following scheme: where e 1 ,e 2 ,...,e N+1 are the allosteric effectors, E i ,R 1,i ,...,R N,i the inactive converting enzymes, E a ,R 1,a ,...,R N,a the active converting enzymes, I 1,i ,I 2,i ,...,I N,i the inactive interconvertible enzymes and I 1,a ,I 2,a ,...,I N,a the active interconvertible enzymes.
The set of reactions of this system is given in the more detailed Scheme 2.

Assumptions
We assume that: 1) The reactions of the converter enzymes with their allosteric effectors, i.e. the steps E i + e 1 E a and R j,i + e j+1 R j,a , are in a state of rapid equilibrium.This assumption is already implied in Schemes 1 and 2. 2 From the relations (1) and (2) as well as from Scheme 2, we deduce: 3) [e j ] (j=1,2,...,N+1) are maintained at constant levels.This implies that the allosteric effectors e j either are present in excess or continuously produced and fed into the system at a rate commensurate with their conversion.Assumptions 1-3 predict that the E a and R j,a (j=1,2,...,N) concentrations remain constant from the onset of the reaction and that their values are given by the equations: (5) (j=1,2,...,N) (6) where K' 1 and K' j+1 are the dissociation constants of E a and R j,a (j=1,2,...,N), respectively.These assumptions, among others, were made by other authors to obtain steady state equations of these cascades (Chock & Stadtman, 1977;1980).Hence, the kinetics of Scheme 2 are equivalent to that of the following Scheme 3: 1 The appendices are published only as pdf files on the www (www.actabp.pl)line version not in the printed form.where [E a ] and [R j,a ] (j=1,2,...,N) are given by Eqns.
(5) and ( 6). 4) The concentrations of the unmodified enzymes I 2,i ,I 3,i ,...,I N,i remain approximately constant during the whole course of the reaction time assayed.This implies that, at any time, the species I j,i (j=2,3,...,N) either are present in excess with regard to I j-1,a (j=2,3,...,N) or continuously produced and fed into the system at a rate commensurate with their conversion.This assumption is reasonable and necessary to linearise the differential equations of the reactions in Scheme 3.

Notation and definitions
The kinetic analysis presented here requires the introduction of appropriate notations and definitions to make it easier and comprehensible.In this section we summarise those notations and definitions which will be used in the following sections.

Species and mechanism
Systematic treatment of the complex Scheme 3 requires a simple set of notation and definitions.We suggest the following one: Using the above notation, Scheme 3 takes the form: Note that with the assumptions 1-4 and the notation for the species and mechanism above the interconversions of the species X i (i=1,2,...,3N+1) are either of first order or of pseudo-first order, being k -(4n-3) (n=1,2,...,N) and k 4n-2 (n=1,2,...,N) the first order rate constants and k 4n-3 [Y n ] 0 (n=1,2,...,N) and k 4n-1 [Z n ] 0 (n=1,2,...,N) the pseudo-first rate constants.It is easy to represent these first order interconversions through a directed graph as that shown in Scheme 5.In this scheme we have indicated the steps corresponding to the connection of two cycles by means of dashed lines.The circles delimit the classes C 1 ,C 2 ,...,C N in the graph.Each class contains those species X i so that any of them has influence on each of the others belonging to the same class (Gálvez & Varón, 1981).One must not confound a class with a cycle.Note that to class C 1 belong the species involved in the first cycle (i.e.X 1 , X 2 , X 3 and X 4 ) and also the species X 5 (involved in the second cycle).Likewise, to the n-th class (n=2,3,...,N-1) belong the species X 3n and X 3n+1 , which are involved in the n-th cycle (the species X 3n-1 , also involved in the n-th cycle, belongs to class C n-1 ) and the species X 3n+2 involved in the next cycle.Finally, to the last class (C N ) belong the two species X 3N and X 3N+1 .The concept of class in a directed graph is also called a strong component of the graph (Jacquez, 1996).i (i=1,2,...,3N+1): Index which can take any of the values 1,2,...,3N+1 corresponding to the subscripts in the notation X i of any of the 3N+1 enzyme species so denoted.u(i): The subindex corresponding to the class to which the species X i (i=1,2,...,3N+1) belongs.Therefore u(i) can take the values 1,2,...,N.For example,

Vol. 52 769
Kinetics of open multicyclic enzyme cascades X 6 belongs to class C 2 (see Scheme 5) so that u(6)=2.Note that u(1)=u(2)=...=u(5)=1 and if i > 2, then u(i) coincides with the integer part of the quotient i/3.f: takes the value of 0 if u(i) < N and 1 if u(i)=N v(i): Integer, i-dependent number defined as the minimum value of the index i of the species X i belonging to the class C u(i) .For example, for i=8 (u(i)=2), v(i)=6.w(i): Integer, i-dependent number defined as the total number of species X i belonging to the classes C 1 , C 2 ,...,C u(i) .. For example, if i=6 (and, therefore u(i)=2) then w( 6) is the sum of the number of species X i belonging to C 1 (i.e. 5) and to C 2 (i.e.3), i.e. w(6)= 5+3=8.Note that: From Eqns. ( 8) and ( 9) and the meaning of the number f defined above we can state that the sum v(i) + w(i) + f is always even.
The polynomial theory and the notation used for the roots yield the equations: The matrices G 1 (0), G 2 (0),..., G u(i) (0) are irreducible with dominant main diagonal (with respect to rows).Hearon (1963) has shown that the non-null eigenvalues of this type of matrices are negative or complex with a negative real part.Therefore the roots λ 2 ,λ 3 ,...,λ u(i) have these characteristics.

Kinetic equations
From Scheme 5, from the meaning of v(i), u(i) and w(i) and taking into account that we consider the concentrations [Y 1 ], [Z 1 ], [Z 2 ],..., [Z N ] constant (and therefore approximately equal to their initial values [Y 1 ] 0 , [Z 1 ] 0 , [Z 2 ] 0 ,..., [Z N ] 0 ), the set of 4N differential equations describing the kinetic behaviour of species X i (i=1,2,...,3N+1) and Y j (j=2,3,...,N) involved in the cascade is given by the system of (non linear) differential equations ( B1 In the following, we derive the approached analytical equations giving the time course equation of any of the species X i (i=1,2,...,3N+1).In the derivation of the kinetic equations the notations and definitions above must be used.
The linear system of differential equations (B3) in Appendix B admits analytical integration using any of the available mathematical method for solving a linear system of differential equations, e.g. the Laplace transform method.The analytical solutions obtained will be obviously approximate.In the main text we discuss the validity of these analytical equations.
To derive the approached analytical time course equation of any of the species X i (i=1,2,...3N+1) we only need to solve the system formed by w(i) first linear differential equations in (B3), because the species X w(i)+1 ,...,X w(i)+2 ,...X 3N+1 have no influence on X i .For example, if we want to derive the time course equation of the species X 6 in a cascade we only need to solve the following set of differential equations: if N ≥ 3 (w(6)=8) or the following set if N = 2 (w(6)=7): Generally, the set of w(i) first equations in (B3) needed to derive the time course equation of X i (i=1,2,...,w(i)) can be expressed in a matricial form as: If the Laplace transform is applied to both sides of Eqn. ( 44) and assuming that the only enzyme species X i (i = 1, 2, ..., w(i)) present at the onset of the reaction is X 1 with a concentration [X 1 ] 0 , we have: where λ is the operator of the Laplace transform.From eqn. ( 45) we have: The form of matrix D u(i) (λ) i permits its determinant, i.e. d u(i) (λ) i , to be expressed as the product: where r 1 (λ), r 2 (λ),...., r u(i)-1 (λ) are given by Eqn.(33) with j=1,2,...u(i)-1 and r u(i) (λ) i is given either by Eqn.
Eqn. ( 53) is inserted into Eqn.( 52) to give: Note that in order to obtain analytical expressions of the roots λ 1 , λ 2 ,..., λ w(i) involved in the kinetic equations of the model [see Eqns.( 38)-( 40)] it is always necessary to solve one quartic equation, as well as N-2 cubic equations (if N ≥ 3) and one quadratic equation (if N ≥ 2).In Appendix C we summarise an algebraic procedure for deriving analytical expressions of the roots of both cubic (Cardano's method) and quartic equations (Ferrari's method).

RESULTS AND DISCUSSION
We performed a complete kinetic analysis of the general model of multicyclic cascades shown in Scheme 1, under a minimal set of assumptions, to obtain the explicit analytical solution of the corresponding system of differential equations which is required for the derivation of the kinetic equations, Eqn. ( 50) and subsequent expressions.These equations are valid for the whole course of the reaction, i.e. for both the transient phase and the kinetic steady state.Previously, only a kinetic analysis of the steady state of Scheme 1 has been carried out (Chock & Stadtman, 1977;1980) under the same assumptions 1-3 used here and other ones that make the steady state results more restrictive than the corresponding ones obtained here.Equation ( 50), which describes the time dependence of the variation of the concentrations of the species X i (i=1,2,...,3N+1) contains a number of exponential terms, w(i)-1, which depends upon the i-value.According to this equation the enzyme species X w(i)+1 , X w(i)+2 ,...,X 3N+1 do not influence the behaviour of species X i , as expected.
Assumption 1 of a rapid equilibrium in the reversible steps in which the allosteric effectors are involved during the whole course of the reaction means that these steps reach the equilibrium practically at the onset of the reaction, i.e. from t ≈ 0. According to Varón et al. (2000) that requires that both the pseudofirst rate constant k' j [e j ] (j=1,2,... N+1) (because from assumption 3 [e j ] is constant) and the first rate constant k' -j (j=1,2,...N+1) involved in each of the reversible steps (note that K' j = k' -j / k j ) in which an allosteric effector binds the modifier enzyme are much higher than all of the other ones involved in the cascade.Thus, by deriving our equations we have implicitly presumed (through assumptions 1 and 3) that all of the modification steps (reversible or not) are much slower than those of activation and deactivation of the modifier enzymes.Obviously, in a cascade there must be a step slower than the other ones, i.e. the rate-limiting step.Nevertheless, our analysis is not based on any rate-limiting step, but on the analytical integration of the set of differential equations (B1) a�er using assumptions 1-4 which linearises it.A good general discussion about the time scales of enzyme regulatory mechanisms is provided by O�away (1988).

The steady state equation
Because the roots λ 2 , λ 3 ..., λ w(i) are negative or complex with a negative real part, the exponential term in Eqn.(50) can be neglected beginning from a reaction time high enough (t  ∞) that the steady state can be assumed reached.If in Eqn.(50) we make t infinite, we have for the concentration of species X i at the steady state, [X i ] ss : From Eqns. ( 71) and ( 58), [X i ] ss is given by:

Validity of our kinetic analysis
We have obtained, for the first time, time course equations giving the concentration of all of the species X i (i=1,2,...,3N+1) involved in an open multicyclic cascade system from t=0 to the steady state (t  ∞).
To obtain these equations we used assumptions 1-3 (which coincide with some of the assumptions used in previous contributions of other authors regarding the steady state analysis only) which allowed us to set [Y 1 ] ≈ [Y 1 ] 0 and [Z j ] ≈ [Z j ] 0 (j=1,2,...,N).Moreover, we used assumption 4 that the concentration of the inactive enzyme species Y 2 , Y 3 ,...,Y N (i.e.unmodified enzymes I 2,i , I 3,i ,...,I N,i ) remain approximately constant during the whole course of the reaction, i.e. we assume that [Y j ] ≈ [Y j ] 0 (j=2,3,...,N).These four assumptions allowed us to linearise the corresponding system of differential equations describing the kinetic behaviour of the system indicated in Scheme 5. Note that all four assumptions are of the same, operational nature, and were required simply to reach the goal -linearisation of the system.
Assumptions 1-3 are widely used in contributions concerning cascade systems.Assumption 4 requires some discussion.The fulfilment of assumption 4 implies that, at any time, the concentration of species Y j (j=2, 3,...,N) must be approximately equal to its initial concentration [Y j ] 0 (j=2,3,...,N).From the system of differential equations in (B1) it is easy to see that all of the species X i (i=1,2,...,3N+1) and Y j (j=2,3,....,N) will reach a steady state in which their conversion rates are null and their concentrations [Y j ] ss remain constant.This result is shown in Figs.1-4 in which we plot simulated time course curves for a tricyclic cascade obtained from numerical integration of Eqns.(B2).
Obviously, the assumed constancy of [Y j ] (j=2,3,...,N) in an assay requires that: Thus, the more the condition (80) is fulfilled, the more accurate are the analytically approached equations.The two coupled reactions in which Y j is involved are (see Scheme 4): X 3(j-1) + Y j X 3j-1  X 3j + X 3(j-1) Z j + X 3j X 3j+1  Y j + Z j Because X 3 can reach, at a maximum, the value [X 1 ] 0 and X 3(j-1) (j=3,4,...,N) could reach, at a maximum, the value [Y j-1 ] 0 (j=3,2,...,N), an experimental The general kinetic equations obtained here are valid for any N-cyclic cascade system, in which N ≥ 2, i.e. for all multicyclic cascades which fit our model.An analogous analysis has already been done (Varón & Havsteen, 1990) for monocyclic cascade systems (N = 1).If the general equations presented here are applied to the case in which N = 1, then 0/0 indeterminations arise, the solution of which gives the same results as those obtained from the individualised study of monocyclic cascades systems.

Limits of applicability of the postulated model
The limits of applicability of the postulated model are precisely those for which assumptions 1-4 are no more fulfilled.Extensive discussions about the fulfilment of assumptions 1-3 have already been made by those authors who carried out the kinetic analysis of multicyclic enzyme cascades (Chock & Stadtman, 1977).
In enzyme kinetics it is frequent to assume an approximately constant concentration of a ligand species (i.e.substrate, activator or inhibitor) during the assayed reaction time in order to be able to de-rive approached analytical kinetic equations, which will remain valid while the assumption of the constancy of the ligand concentration prevails (Segel, 1975;Cornish-Bowden, 1995).Quantitatively, variation of the ligand concentration of 5% in comparison with its initial value is normally accepted as the limit of the validity of the approached equations (Segel, 1975).Obviously, approached kinetic equations could also be considered valid for a drop in the ligand concentration higher than 5%, but the accuracy of the analytical solutions results are the worse the more variation is allowed.A high decrease of the ligand concentration makes advisable to use the Michaelis Integrated Method (Segel, 1975).
Let us fix an upper limit G (expressed per unity of concentration) in the variation of the concentration of any of the ligand species initially present in an enzyme system under which we accept the validity of the approached analytical equations (e.g.G = 0.05).In our analysis, in each cycle the specie Y j  50) for X 3 , X 6 and X 9 and N=3.(---) Simulated time progress curves of Y 2 , Y 3 , X 3 , X 6 and X 9 obtained by numerical integration of the set of differential equations (B1).(B).The same curves for X 3 , X 6 and X 9 as in (A) but using a more suitable scale.50) for X 3 , X 6 and X 9 and N=3.(---) Simulated time progress curves of X 3 , X 6 and X 9 obtained by numerical integration of the set of differential equations (B1).50) for X 3 , X 6 and X 9 and N=3.(---) Simulated time progress curves of X 3 , X 6 and X 9 obtained by numerical integration of the set of differential equations (B1).which surely is much higher than the true value g(j) for Y j , if the true values of the differences were used.But these hypothetical differences, G[Y j ] 0 , between [Y j ] 0 and [Y j ] would be acceptable.Thus, g(j) = G for the species Y j is acceptable as a border parameter for the accuracy of the theoretical progress curve of [Y j ] in comparison to its experimental (simulated) progress curve.We suggest taking this same arbitrary g(j) upper value, G, for all of the different species X j [j=1,2,3,...w(j)] involved in order to know, in each case, the goodness of our analysis.
In Table 2 we give examples for a possible use of this upper value of g(j), the limit parameter G, in order to know the goodness of the approached solutions in the case of a tricyclic cascade.Note that if we fix the limit parameter G as 0.05, the approach is good in the reaction time considered (100 s) for X 3 in the four cases, for X 6 in cases 2, 3 and 4 and for X 9 in cases 3 and 4. If we consider in case 1 a reaction time of 8 s then the approach is good in this time for X 3 , X 6 and X 9 .Graphically this degree of accuracy can be observed in Figs.1-4.

Analytic value of the presented work
Apart from the considerations made in the Introduction section about the advantages of having transient phase equations of multicyclic enzyme cascades and their biological importance, they have, in our opinion, the following additional advantages: (a) Metabolic control analysis (MCA) as proposed by Kacser and Burns (1973) and Heinrich and Rapoport (1974) and most other contributions in this field are limited to the steady state.The transient phase equations obtained here, giving the instantaneous concentration of the species involved, allow one to extend the MCA to study the effect of parameters on different variables of enzyme cascades.The basic definitions and relationships to analyse some control features of the instantaneous values of meta-Table 2. Values of g(j) (j=3,6,9) and their standard errors corresponding to the approached analytical equations for [X 3 ], [X 6 ] and [X 9 ] in an open tricyclic cascade.
Cases 1, 2, 3 and 4 correspond to those in Figs. 1, 2, 3 and 4, respectively, but using a reaction time T (equal to t Q ) of 100 s (instead of the 40 s in the figures) at which the steady state is largely reached in the four cases.In case 1 (inset) the reaction time, T, used was 8 s (as in inset of Fig. 1) at which the [Y 2 ] and [Y 3 ] values do not considerably differ from their initial values [Y 2 ] 0 and [Y 3 ] 0 so that our analysis could be used.The g(j) values were obtained from the values given by the analytical approach at any time in comparison with that from the corresponding simulated curve at the same time, according to Eqn. (84).The values of time points, Q, in each simulation are also indicated in the last column.Note that if we fix the limit parameter G as 0.05, then the approach is good in the reaction time considered (100 s) for X 3 in the four cases, for X 6 in cases 2, 3 and 4 and for X 9 in cases 3 and 4. If in case 1 we consider a reaction time of 8 s (inset) then the approach is good for X 3 , X 6 and X 9 .

CONCLUDING REMARKS
All of the objectives presented in the Introduction section have been accomplished.Apart from the intrinsic value of knowing the time kinetic behaviour of any of the species involved in an open multicyclic enzyme cascade, the presented kinetic analysis can be the basis of future contributions regarding open multicyclic enzyme cascades which require the knowledge of their time course equations (e.g.evaluation of the time needed to reach the steady state, suggestion of kinetic data analysis, etc.), analogously as that already carried out for open bicyclic cascades (Varon et al., 1994a).Likewise, the steady state kinetics of these cascades could be compared with those steady state results obtained by other authors under other assumptions.
The role of multicyclic interconvertible enzyme cascades (e.g.mitogen-or messenger-activated, or extracellular signal-regulated protein kinase cascades) in the regulation of basic and critical biological functions such as cell growth and division (Ballif & Blenis, 2001;Santen et al., 2002;Bardwell et al., 2003;Bardwell, 2004), changes of volume (Hamish & Christine, 2003;Yakar et al., 2003) andcompartmentization (Labrecque et al., 2003) is the topic of a number of current contributions.The paper presented here may contribute to a be�er knowledge of the kinetic aspects of these features.
40) Insertion of Eqns.(38)-(40) into Eqn.(37) yields the expression: ) in Appendix B. Assuming also that [Y 2 ], [Y 3 ],...,[Y N ] remain approximately constant during the whole course of the reaction (and therefore approximately equal to their initial values [Y 2 ] 0 , [Y 3 ] 0 ,...,[Y N ] 0), the 3N+1 first differential equations in the system of differential equations (B1) become a set of linear differential equations with constant coefficients from which the time course of any of the species X 1 , X 2 ,...X 3N+1 can be obtained by analytical integration.If we are interested in the time course of any of the species X i (i=1,2,...,3N+1) we only need to integrate the set of the first w(i) linear differential equations (B1) in Appendix B.
. all the species X i reach a constant concentration given by Eqn.(72) in the steady state, which allows Eqn.(50) to be rewri�en as: a time course equation for the whole course of the reaction, i.e. for the transient phase and the steady state, at which Eqn. (72) becomes:[X i ] = [X i ] ss(steady state; i=1,2,...,w(i)) (73)

Figure 2 .
Figure 2. Time progress curves in an open tricyclic cascade under the same set of values as in Fig. 1 except that the value used for [X 1 ] 0 was 2 × 10 -6 M. (A).(-) Curves obtained from plot of Eqn.(50) for X 3 , X 6 and X 9 and N=3.(---) Simulated time progress curves of Y 2 , Y 3 , X 3 , X 6 and X 9 obtained by numerical integration of the set of differential equations (B1).(B).The same curves for X 3 , X 6 and X 9 as in (A) but using a more suitable scale.

Figure 3 .
Figure 3.Time progress curves in an open tricyclic cascade under the same set of values as in Fig. 1 except that the value used for [X 1 ] 0 was 10 -7 M. (-) Curves obtained from plot of Eqn.(50) for X 3 , X 6 and X 9 and N=3.(---) Simulated time progress curves of X 3 , X 6 and X 9 obtained by numerical integration of the set of differential equations (B1).

Figure 4 .
Figure 4. Time progress curves in an open tricyclic cascade under the same set of values as in Fig. 1 except that the values used for [X 1 ] 0 and [Y 3 ] 0 were 2 × 10 -6 M and 10 -4 M, respectively.(-) Curves obtained from plot of Eqn.(50) for X 3 , X 6 and X 9 and N=3.(---) Simulated time progress curves of X 3 , X 6 and X 9 obtained by numerical integration of the set of differential equations (B1).