Kinetics of intra- and intermolecular zymogen activation with formation of an enzyme–zymogen complex ´ ´ ´ Matilde Esther Fuentes, Ramon Varon, Manuela Garcıa-Moreno and Edelmira Valero ´ ´ ´ ´ ´ Grupo de Modelizacion en Bioquımica, Departamento de Quımica-Fısica, Escuela Politecnica Superior de Albacete, Universidad de Castilla-La Mancha, Albacete, Spain Keywords autocatalysis; enzyme kinetics; pepsin; pepsinogen; zymogen Correspondence ´ E Valero, Grupo de Modelizacion en ´ ´ Bioquımica, Departamento de Quımica´ ´ Fısica, Escuela Politecnica Superior de Albacete, Universidad de Castilla-La Mancha, Avda Espana s ⁄ n, Campus ˜ Universitario, E-02071 Albacete, Spain Fax: +34 967 59 92 24 Tel: +34 967 59 92 00 E-mail: Edelmira.Valero@uclm.es Note The mathematical model described here has been submitted to the Online Cellular Systems Modelling Database and can be accessed free of charge at: http:// jjj.biochem.sun.ac.za/database/fuentes/ index.html A mathematical description was made of an autocatalytic zymogen activation mechanism involving both intra- and intermolecular routes The reversible formation of an active intermediary enzyme–zymogen complex was included in the intermolecular activation route, thus allowing a Michaelis–Menten constant to be defined for the activation of the zymogen towards the active enzyme Time–concentration equations describing the evolution of the species involved in the system were obtained In addition, we have derived the corresponding kinetic equations for particular cases of the general model studied Experimental design and kinetic data analysis procedures to evaluate the kinetic parameters, based on the derived kinetic equations, are suggested The validity of the results obtained were checked by using simulated progress curves of the species involved The model is generally good enough to be applied to the experimental kinetic study of the activation of different zymogens of physiological interest The system is illustrated by following the transformation kinetics of pepsinogen into pepsin (Received July 2004, revised September 2004, accepted September 2004) doi:10.1111/j.1432-1033.2004.04400.x Living organisms possess different systems of biological amplification that help them achieve a fast response to a given stimulus in substrate cycling [1–3], enzyme cascades [4,5] and limited proteolysis reactions [6–9] Limited proteolysis is an irreversible and exergonic reaction under normal physiological conditions, and there is no opposite reaction that regenerates the same hydrolyzed peptidic bond or that reinserts the corresponding released peptide Proenzyme activation therefore is a control mechanism that differs essentially from allosteric transitions and reversible covalent modifications Proenzyme activation by proteolytic cleavage of one or more peptide bonds requires the presence of an activating enzyme In those cases in which the activating FEBS Journal 272 (2005) 85–96 ª 2004 FEBS enzyme is the same as the activated one, the proenzyme activation process is termed autocatalytic Physiological examples include the activation of trypsinogen into trypsin [10,11], the conversion of pepsinogen into pepsin [12–14], and prekallikrein into kallikrein [15,16] Several reports describe the kinetic behaviour of enzyme systems involving autocatalytic zymogen activation – with or without steps in rapid equilibrium conditions – in the presence [17] and absence [18] of a substrate of the enzyme to monitor the reaction through the release of product, and also in the presence of an inhibitor of the enzyme [19,20] In all of these contributions, the zymogen was considered to be without enzyme activity Nevertheless, references to the enzyme activity of zymogens are increasingly more frequent [21–23] 85 Autocatalytic zymogen activation M E Fuentes et al Scheme Mechanism for the autoactivation of pepsinogen to pepsin [12] Pgn, pepsinogen; Pep, pepsin Al-Janabi et al (1972) [12] offered kinetic evidence for the existence of two activation pathways (intraand intermolecular) for the activation of pepsinogen to pepsin, as is indicated in Scheme They also obtained the concentration–time kinetic equation for the pepsinogen concentration, valid for the whole course of the reaction and which was still used in recent contributions [23] Subsequently, a number of different mechanisms for the activation process of pepsinogen were proposed by Koga and Hayashi (1976) [24] By comparing the simulated progress curves obtained for each of these mechanisms with the experimental results, these authors suggested a reaction mechanism including both intra- and intermolecular activation of the zymogen by the action of the active enzyme (Scheme 2) This mechanism takes into account the (irreversible) formation of a dimeric intermediate However, in the above contribution, no analytical approximate solutions of the suggested mechanism were obtained Taking into account the reaction in Schemes and concerning pepsinogen activation, we suggest a general mechanism (Scheme 3) applicable to any zymogen activation, for which we have carried out a kinetic analysis The above mechanism exhibits simultaneously two catalytic routes, an intramolecular activation process, Scheme General mechanism proposed for the autoactivation of zymogens involving both the intra (route a) and intermolecular (route b) steps Z is the zymogen, E is both the activating protease and the activated enzyme, EZ is the complex enzyme–substrate intermediate of the reaction, and W is one or more peptides released from Z during the formation of E route a, and an autocatalytic zymogen activation process catalyzed by the same enzyme it produces, route b This mechanism includes the reversible formation of an intermediary active enzyme–zymogen complex in the intermolecular activation step Both routes interact because route a diminishes zymogen concentration, increasing the active enzyme concentration, and therefore influences route b In turn, route b also decreases zymogen concentration, having an effect on route a Nevertheless, as we will see below, there are some experimental conditions in which it can be assumed that route b does not influence route a (but not vice versa), so that the latter can be analysed independently This mechanism is general enough to be applied to different zymogens exhibiting both intra- and intermolecular reactions including, as particular cases, those which reach rapid equilibrium (Scheme 4) and the simplest reaction showing the two mentioned routes in the absence of an EZ complex (Scheme 5) Scheme Mechanism shown in Scheme under rapid equilibrium conditions between E, Z and EZ Scheme Mechanism suggested by Koga and Hayashi [24] involving two pH-dependent steps and a nonlinear reaction containing a looped reaction with a dimeric intermediate, in which the peptide fragments are released and pepsinogen is converted to pepsin X1 and X2 are the unprotonated and protonated pepsinogen, respectively, while X3* and X4* are structural isomers of the active pepsin which are in an equilibrium involving proton binding X5 is the dimeric intermediate 86 Scheme Simplified general mechanism for the autoactivation of zymogens FEBS Journal 272 (2005) 85–96 ª 2004 FEBS M E Fuentes et al Autocatalytic zymogen activation Note that in Scheme 3, (Z) includes both X1 and X2 from Scheme and (E) includes both X3* and X4*, so that [Z] ¼ [X1] + [X2] and [E] ¼ [X3*] + [X4*] Also, note that Scheme corresponds to Scheme (previously reported by Al-Janabi et al [12]), when Z and E denote Pgn and Pep, respectively The aims of the present paper are: (a) to analyse the complete kinetics for Scheme 3, obtaining approximate analytical solutions and to confirm their goodness by numerical simulation; (b) from the above results, to derive other approximate solutions for Scheme in simplified conditions that arise from certain relations between the values of the first or pseudo first-order rate constants; (c) to derive the kinetic equations corresponding to Schemes and – which can be considered particular cases of Scheme when certain relations between the values of the first or pseudo first-order rate constants are observed – and (d) from the equations derived in (b), to suggest an experimental design and a kinetic data analysis to evaluate the kinetic parameters involved in Scheme 3, which is immediately applicable to Schemes and All of these results are illustrated by the kinetics of the autoactivation of pepsinogen to pepsin The mathematical model described here has been submitted to the Online Cellular Systems Modelling Database and can be accessed at: http://jjj.biochem sun.ac.za/database/fuentes/index.html free of charge Theory Notation and definitions [E], [Z], [EZ], [W]: instantaneous concentrations of the species E, Z, EZ and W, respectively [E]0, [Z]0, [EZ]0, [W]0: initial concentrations of the species E, Z, EZ and W, respectively The dissociation constant of the EZ complex will be: K2 ẳ k2 ỵ k3 k2 FEBS Journal 272 (2005) 8596 ê 2004 FEBS 2ị dẵEZ ẳ k2 ẵZẵE k2 ỵ k3 ịẵEZ dt 3ị dẵW ẳ k1 ẵZ ỵ k3 ẵEZ dt 4ị This set of differential equations is nonlinear and, in order to obtain analytical solutions, we shall assume that the concentration of Z remains approximately constant during the course of the reaction (Eqn 5), i.e ẵZ % ẵZ0 5ị Taking into account this assumption, the differential equation system that describes the mechanism shown in Scheme is given by Eqns (68): dẵE ẳ k1 ẵZ0 k2 ẵZ0 ẵE ỵ k2 ỵ 2k3 ịẵEZ dt 6ị dẵEZ ẳ k2 ẵZ0 ẵE k2 ỵ k3 ịẵEZ dt 7ị dẵW ẳ k1 ẵZ0 ỵ k3 ẵEZ dt 8ị The differential Eqns (6) and (7) constitute a nonhomogeneous linear system that may become homogeneous by further derivation and by performing the changes in the variables d[E] ⁄ dt ¼ X, and d [EZ] ⁄ dt ¼ Y, giving Eqns (9) and (10): dX ¼ Àk2 ẵZ0 X ỵ k2 ỵ 2k3 ịY dt 9ị dY ẳ k2 ẵZ0 X k2 ỵ k3 ịY dt 10ị the initial conditions of which are at t ẳ 0, X ¼ k1[Z]0, and Y ¼ 0, taking into account that [E]0 ¼ and [EZ]0 ¼ The solution to this system is given by Eqns (11) and (12): X¼À Time course differential equations and mass balances The kinetic behaviour of the species E, Z, EZ and W involved in Scheme is described by the following set of differential equations (Eqns 14): 1ị dẵE ẳ k1 ẵZ k2 ẵZẵE ỵ k2 ỵ 2k3 ịẵEZ dt k2 k2 The presence of EZ complex allows the definition of a Michaelis–Menten constant for the activation of zymogen towards its active enzyme as follows: Km ẳ dẵZ ẳ k1 ẵZ k2 ẵZẵE ỵ k2 ẵEZ dt k1 ẵZ0 k2 ẵZ0 ỵ k2 ị k1 t k1 ẵZ0 k2 ẵZ0 þ k1 Þ k2 t e þ e k1 À k2 k1 k2 11ị Yẳ k1 k2 ẵZ2 k1 t ðe À ek2 t Þ k1 À k2 12ị where: 87 Autocatalytic zymogen activation k1 ẳ k2 ẵZ0 þkÀ2 þk3 Þþ M E Fuentes et al qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk2 ẵZ0 ỵk2 ỵk3 ị2 ỵ4k2 k3 ẵZ0 13ị k2 ẳ k2 ẵZ0 ỵk2 ỵk3 ị q k2 ẵZ0 ỵk2 ỵk3 ị2 ỵ4k2 k3 ẵZ0 14ị Note that both k1 and k2 are real quantities, k1 always being positive and k2 negative, and that the relations between k1 and k2 are as follow (Eqns 1517): k1 ỵ k2 ẳ k2 ẵZ0 ỵ k2 ỵ k3 ị 15ị k1 k2 ẳ k2 k3 ẵZ0 16ị k1 < jk2 j 17ị To return to our original symbolism, Eqns (11) and (12) are integrated and, taking into account the initial conditions mentioned above, gives: ẵE ẳ A1;0 ỵ A1;1 ek1 t ỵ A1;2 ek2 t 18ị ẵEZ ẳ A2;0 ỵ A2;1 ek1 t ỵ A2;2 ek2 t 19ị The expressions corresponding to Ai,j (i ¼ 1, 2, 3, 4; j ¼ 0, 1, 2) are given in the Appendix A (Eqns A1–A12) If the progress of the reaction is followed by measuring the instantaneous zymogen concentration, the following mass balance must be taken into account: ẵZ ẳ ẵZ0 ẵE 2ẵEZ ð20Þ Inserting Eqns (18) and (19) into Eqn (20), the following time-concentration equation (Eqn 21) is obtained: ẵZ ẳ A3;0 þ A3;1 ek1 t þ A3;2 ek2 t ð21Þ This equation could also be obtained by integration of Eqn (1) after inserting into it condition (Eqn 5) and Eqns (18) and (19) To obtain the equation describing the accumulation of the peptide product of catalysis, Eqn (19) is inserted into Eqn (8) and, by integrating again, and taking into account the initial condition [W]0 ¼ 0, we obtain Eqn (22): ẵW ẳ A4;0 ỵ A4;1 ek1 t ỵ A4;2 ek2 t ð22Þ This equation could also be obtained from Eqns (19) and (21), taking into account the following mass balance: 88 ẵW ẳ ẵZ0 ẵZ ẵEZ 23ị Equation (21) for zymogen consumption is different from the equation reported previously in the literature for the simplified reaction mechanism shown in Scheme [12] To obtain this latter equation, the reaction mechanism was simplified, disregarding the intermediary zymogen-active enzyme complex, as this is the only way to obtain a concentration–time relation for the whole course of the reaction, but which clearly corresponds to a reaction mechanism which does not take into account reality The equations derived here have the advantage that they respond to a mechanism close to that which occurs in reality, including the formation of an EZ complex in the intermolecular activation step However, they have the disadvantage of being only valid for a relatively short time, with the corresponding experimental difficulties The measurement of zymogen concentrations not far from the initial value in a short-time reaction leads to unavoidable experimental errors Nevertheless, taking into account that the values of the kinetic parameters are independent of the reaction time registered, this will allow the evaluation of kinetic parameters involved in the system whenever the reaction can experimentally be followed Once the value of the kinetic parameters are obtained, the behaviour of the reaction can be predicted until the zymogen is exhausted Results and Discussion We obtained the time course equations for the species involved in the reaction corresponding to the autocatalytic activation of a zymogen, including the formation of an active enzyme–zymogen complex (Scheme 3) The reaction scheme suggested is the most simple one that covers the main features described in the literature, i.e a route of intramolecular activation of the zymogen into the active enzyme, E, and one or more peptides represented by W [route (a), Scheme 3] [12,22,25–27] and a route of autocatalytic activation of zymogen by the active enzyme formed (route (b), Scheme 3, [12,26,28]) Route (a) of Scheme condenses, in a single step, the whole process corresponding to a conformational change of Z molecules brought about by low pH and the subsequent cleavage of the N-terminal peptide [14] Thus, k1 is actually an apparent rate constant corresponding to the whole process leading from Z to E and W by intramolecular activation Route (b) of Scheme has been assumed to follow a single Michaelis–Menten mechanism instead of the more general Uni–Bi mechanism This approach is the usual one used to describe FEBS Journal 272 (2005) 85–96 ª 2004 FEBS M E Fuentes et al mechanisms of autocatalytic zymogen activation and has been sufficiently justified [11,29–31] Previously, kinetic analyses of the reactions, whereby a zymogen is activated both intra- and intermolecularly by the action of the active enzyme, have been made and used for the experimental determination of the kinetic parameters involved in pepsinogen autoactivation [12,21,23,32] However these contributions used the simplified reaction mechanism shown in Scheme (which coincides with Scheme 1), i.e the equilibrium between the species E, Z and EZ in the intermolecular activation step was not taken into account It is this step that we include in the present paper, with the additional advantage that the results obtained using this novel approach are nearer reality [24,26] For greater clarity and to better imitate the physiological conditions, we assumed in our analysis that no active enzyme is present at the onset of the reaction, but only the zymogen Validity of the time course equations derived Kinetic equations for all the species involved in Scheme were derived by solving the nonhomogeneous set of ordinary, linear (with constant coefficients), differential Eqns (6–8) These kinetic equations are valid whenever condition (Eqn 5) holds, and for this reason they are approximate analytical solutions They can be further simplified in such a way that a kinetic analysis of the experimental kinetic data make it possible to completely characterize the system Obviously, the approximate analytical time course equations derived here are also applicable to any zymogen activation mechanism described by Scheme in the same initial and experimental conditions As [Z] continuously decreases from the beginning of the reaction, the longer the reaction time, the less accurate the analytical solutions This is usual in enzyme kinetics, where to derive approximate analytical solutions corresponding either to the transient phase or the steady-state of an enzymatic reaction, substrate concentration (the zymogen in this case) is usually assumed to remain approximately constant [33–35] and therefore the results obtained are only valid under this condition It is obvious that if the reaction is allowed to progress, the final concentration of zymogen will be zero Thus, as is common practice in assays on enzyme kinetics, the reaction can only be allowed to evolve to a small extent during the assays compared with the total reaction time taken for the substrate to vanish [36] Obviously, the more the zymogen concentration diminishes, the less accurate the equations obtained become FEBS Journal 272 (2005) 85–96 ª 2004 FEBS Autocatalytic zymogen activation Experimentally, it is possible to determine whether the assumption (Eqn 5), which is always true at the onset of the reaction, is still fulfilled at a certain reaction time The fraction, q, of the remaining zymogen is introduced as: q¼ ½ZŠ ½ZŠ0 ð24Þ and we may arbitrarily set the q-value (e.g q ¼ 0.7) above which the approximate solutions remains applicable Thus, the [Z]-values for which the equations obtained are applicable are: ẵZ ! qẵZ0 25ị For example, if [Z]0 ¼ 10)3 m and q ¼ 0.7, then, according to Eqn (25), the analytical equations derived here will be valid only when [Z] ‡ · 10)4 m To illustrate the degree of validity of our approach, in Fig 1A we show the time progress curves obtained by numerical integration of the entire differential equation system obtained directly from the mechanism shown in Scheme (Eqns 1–4), for an arbitrary set of rate constants values and [Z]0-value A comparison of the results obtained above for [Z] with those obtained from the equation derived here Eqn (21) and from the equation previously reported in the literature for Scheme (Eqn B1, Appendix B) [12] is shown in Fig 1B, using the same values for the rate constants and initial conditions Table shows a numerical comparison of these data for different q-values, including the relative errors of the [Z] values predicted by the two integrated equations with respect to those obtained from the numerical solution at the same times As can be seen, as long as q remains higher than 0.7, the relative error committed using the equations derived here remain below 10%, nevertheless it is greater when the EZ complex is not taken into account Uni-exponential kinetic behaviour The time course equations here obtained are of the bi-exponential type Nevertheless, because k1 is positive and k2 negative, and due to the relationship in Eqn (17), the negative exponential term in Eqn (21) can be neglected from a relative short time after the onset of the reaction, so that the kinetic behaviour of all of the species becomes uni-exponential from this time The higher the value of |k2| compared with k1, the shorter the time from which the kinetic behaviour can be considered uni-exponential In this way, the kinetic equations for Z (Eqn 26) and W (Eqn 27) become: 89 Autocatalytic zymogen activation M E Fuentes et al Table Values of [Z] obtained from the simulated curves ([Z]sim) compared with those obtained from Eqn (21) ([Z]Eqn 21) and from Eqn (B1) in Appendix B [12] ([Z]Eqn B1) The values of the rate constants used were those indicated in Fig and the q-values correspond to [Z]sim-values In the third column we have indicated the corresponding t-value at which the [Z]-values are reached The fifth and seventh columns correspond to the relative error of [Z]-values obtained with Eqn (21) and Eqn (B1), respectively, compared with [Z]sim-values q (%) [Z]sim (lM) t (s) [Z]Eqn (lM) 100 99 98 95 90 85 80 75 70 65 60 50 24.00 23.76 23.52 22.80 21.60 20.40 19.20 18.00 16.80 15.60 14.40 12.00 0.00 2.55 5.02 11.43 22.56 31.82 42.09 51.64 62.36 72.06 83.12 108.68 24.00 23.75 23.49 22.77 21.34 20.03 18.46 16.92 15.12 13.43 11.46 6.73 Relative error (%) [Z]Eqn (lM) 0.00 0.05 0.13 0.15 1.20 1.83 3.85 5.99 10.00 13.89 20.43 43.88 21 24.00 23.75 23.49 22.77 21.32 19.91 18.16 16.40 14.32 12.43 10.34 6.23 B1 Relative error (%) 0.00 0.05 0.11 0.12 1.31 2.40 5.40 8.91 14.76 20.35 28.21 48.09 jk2 j ) k1 28ị k1 ỵ k2 % k2 29ị k1 k2 % Àk2 ð30Þ i.e Fig (A) Simulated progress curves corresponding to the species involved in the mechanism shown in Scheme The values of the rate constants used were: k1 ẳ 4.0 à 10)3ặs)1, k2 ẳ 1.0 à 103 )1 )1 )4 )1 M ặs , k)2 ẳ 2.1 à 10 ặs and k3 ẳ 5.4 à 10)4ặs)1 The initial zymogen concentration used was [Z]0 ¼ 2.4 · 10)5 M (B) Progress curves corresponding to Z consumption obtained from numerical integration (curve i), from Eqn (21) (curve ii) and from the equation corresponding to the mechanism proposed by Al-Janabi et al [12] (Eqn B1), Appendix B (curve iii) Conditions as indicated in Fig 1A ẵZ ẳ A3;0 ỵ A3;1 ek1 t 26ị k1 t 27ị ẵW ẳ A4;0 þ A4;1 e ½ZŠ % ½ZŠ0 À k1 ½ZŠ0 ðk2 k2 ẵZ0 ị k1 t e 1ị from t % 0ị k1 k2 31ị ẵW % k1 ẵZ0 k1 t ðe À 1Þðfrom t % 0Þ k1 In such a case, the following relations (Eqns 28–30) must be fulfilled: ð32Þ Bearing in mind the relation 29 (Eqn 29), Eqn (15) becomes: k2 % k2 ẵZ0 ỵ k2 ẳ k3 Þ The case in which one exponential term can be neglected after approximately t ¼ 90 Under these conditions, the uni-exponential behaviour of the species can be assumed from t ¼ Thus, if the relationships 29 and 30 [Eqns (29) and (30)] are inserted into Eqns (26) and (27), we obtain: ð33Þ and from Eqns (16) and (33) we obtain: k1 % k3 ẵZ0 Km ỵ ẵZ0 ð34Þ The kinetic behaviour from the onset of the reaction is a consequence of assumption 28 (Eqn 28) This condition is only fulfilled if certain relations between the FEBS Journal 272 (2005) 85–96 ª 2004 FEBS M E Fuentes et al Autocatalytic zymogen activation first- and pseudo first-order rate constants apply We have demonstrated that condition 28 (Eqn 28) leads to Eqn (33) and therefore taking into account Eqn (14), the following relationship is deduced: k2 ẵZ0 ỵ k2 ỵ k3 ị ) 4k2 k3 ẵZ0 35ị i.e condition 33 (Eqn 33) is a sufficient condition for relationship 35 (Eqn 35) to exist In turn, condition 28 (Eqn 28) is a sufficient condition for relationship 29 (Eqn 29) Indeed, if we insert condition 29 into Eqn (15), k2 is given by Eqn (33) and therefore according to Eqn (14), relation 35 (Eqn 35) is observed Thus, conditions 28 (Eqn 28) and 35 (Eqn 35) are equivalent This is expressed mathematically as: jk2 j ) k1 , ðk2 ½ZŠ0 ỵ k2 ỵ k3 ị ) 4k2 k3 ẵZ0 36ị That condition 35 (Eqn 35) is fulfilled, which justifies the uni-exponential kinetic behaviour, is reasonable to expect because k3 is a rate constant corresponding to the cleavage of a peptidic bond, i.e to a covalent modification, whereas k-2 and k2[Z]0 are rate constants corresponding to the dissociation and formation of the EZ complex It is therefore reasonable to think that: k3 ( k2 37ị and or k% k3 ẵZ0 K2 ỵ ẵZ0 The case in which the activation can be represented by Scheme From a comparison of Schemes and 5, it can be seen that the latter formally arises from Scheme if: k3 ! ð38Þ In both of the above cases condition 36 (Eqn 36) is fulfilled In the following we will denote, for greater clarity, k1 as k Thus, we rewrite Eqns (31) and (34) as: ½ZŠ % ½ZŠ0 À k1 ½ZŠ0 ðk2 À k2 ẵZ0 ị kt e 1ị kk2 k% k3 ẵ Z Km ỵ ẵ Z ð39Þ ð40Þ ð42Þ If we take into account condition 42 (Eqn 42), we see that Eqn (36) is fulfilled and therefore uni-exponential Eqns (39) and (40) are applicable, but now: k % k2 ½ZŠ0 which is obtained as Eqn (40) ð43Þ lim k; where k is given by k3 !1 The case in which the intramolecular activation of pepsinogen is predominant In this case, the amount of zymogen activated intermolecularly by the active enzyme (route b) in Scheme may be considered negligible and so it can be assumed that: ð44Þ k2 % Therefore Eqn (33) is rewritten as: k2 ẳ k2 ỵ k3 ị k3 ( k2 ẵZ0 ð41Þ ð45Þ Under these conditions, Eqn (39) can be rewritten as: ½ZŠ % ½ZŠ0 À k1 ½ZŠ0 kt ðe À 1Þ k ð46Þ which may be transformed into the uni-exponential equation reported by Al-Janabi et al [12] by substituting the exponential term by a series development, only considering the two first terms for short reactions times, and then returning to the exponential notation This gives: ẵZ ẳ ẵZ0 ek1 t ð47Þ Rapid equilibrium assumptions: Scheme A particular case of uni-exponential behaviour is that corresponding to rapid equilibrium conditions, i.e the assumption that the reversible reaction step in Scheme is in equilibrium from the onset of the reaction For that, relations 37 and 38 (Eqns 37 and 38) must be observed simultaneously All equations for the uni-exponential behaviour are applicable but, in this case the Michaelis constant Km should be replaced in Eqn (40) by the equilibrium constant K2: FEBS Journal 272 (2005) 85–96 ª 2004 FEBS Kinetic data analysis The uni-exponential kinetic behaviour of the reaction evolving according to Scheme from the onset is the most realistic because of condition [28] will probably be fulfilled for the reasons given above Thus, we will confine ourselves to the general case of a uni-exponential behaviour given by Eqns (39–40) In this kinetic analysis, it is assumed that the remaining zymogen, 91 Autocatalytic zymogen activation M E Fuentes et al [Z], can be experimentally monitored by a discontinuous method [12,23] The procedure we suggest is valid whenever [E] + [EZ] remains much lower than [Z]0 and consists of the following two steps: (a) plotting the experimental [Z]-values obtained by any discontinuous method at different reaction times, t, and at different [Z]0-values, and fitting them to Eqn (26), gives the corresponding A3,0, A3,1, and k-values for the different initial zymogen concentrations used; (b) Eqn (40) indicates that the kinetic parameter k has a hyperbolic dependence on initial zymogen concentration, [Z]0 Therefore, the kinetic parameters k3 and Km can be evaluated by a nonlinear least-squares fit of the experimental k-values obtained in step (a) to this equation Furthermore, these parameters can also be obtained by linear regression by using any linearizing transformation of Eqn (40), such as a Hanes–Woolf type plot ([Z]0 ⁄ k vs [Z]0) In this case, a straight-line will be obtained, with the following properties: ordinate intercept ¼ slope ¼ Km k3 k3 abscissa intercept ¼ ÀKm ð48Þ ð49Þ ð50Þ Therefore, the kinetic parameters k3 and Km can be evaluated Particular cases of Scheme Schemes and can be considered formally as particular cases of the reaction mechanism shown in Scheme The kinetic equations for these mechanisms could be obtained from their corresponding system of differential equations However, they can also be obtained faster and more easily from the differential equations of the mechanism indicated in Scheme 3, by converting it into the mechanism under study [37–39], as has been done in the present paper Discrimination between Schemes 3, and The above described step (b) for evaluating the kinetic parameters k3 and Km involved in Eqns (39) and (40) is also valid for evaluating the kinetic parameters involved in Scheme (Eqns 39 and 41) and Scheme (Eqns 39 and 43), which are particular cases of Scheme It also serves to discriminate between them Thus, if the enzymatic system under study evolves according to Scheme 4, in which case relations 37 and 38 (Eqns 37 and 38) are fulfilled, the Km value 92 obtained in step (b) of the above described procedure will approximately coincide with K2, according to Eqn (41) In turn, if the enzyme system evolves according to Scheme 5, taking into account Eqns (39) and (43), the intercept and the slope of the straight line arising from step (b) will become: Km ordinate intercept ¼ lim ð Þ ¼ k3 !1 k3 k2 ð51Þ slope ¼ lim ị ẳ k3 !1 k3 52ị In this way, the suggested procedure for evaluating the kinetic parameters allows us to discriminate between Scheme and Schemes and If the straight line arising from step (b) has a slope of zero or nearly zero, then a compatible mechanism reaction is that described by Scheme If this is not the case, the mechanism reaction is compatible with both Schemes and 4, between which it is impossible to discriminate Nevertheless, because Scheme corresponds to a situation in which relations 37 and 38 (Eqns 37 and 38) are observed, it is reasonable to think that the lower the k3 value, the more probable it is that the above mentioned relations will be observed Thus, the higher the ordinate intercept of the straight line arising from step (b), the more probable the reaction scheme will be the one described by Scheme and that Eqn (41) is fulfilled To illustrate this, Eqn (40) is plotted in linear form in Fig for fixed values of k2 and k-2 at different k3 values leading to Schemes 3, and Pepsinogen autoactivation kinetics The theoretical results obtained in the present paper are illustrated by the kinetics of the activation of pepsinogen to pepsin Figure 3A shows the experimental progress curves corresponding to the remaining pepsinogen in the reaction medium The inset shows the same results as percentage of remaining pepsinogen Taking into account assumption from the Theory section and the results shown in Table 1, the time course of the reaction was followed in all cases until a q-value of 0.7 was reached These data were fitted by nonlinear regression to Eqn (26), thus providing the values of A3,0, A3,1 and k at the different initial pepsinogen concentrations used Figure 3B shows these data plotted according to the kinetic analysis here proposed Taking into account Eqns (48–50), the following values for the kinetic parameters involved in the system were obtained: k3 ẳ [6.13 0.14] à 10)4ặs)1, Km ¼ [1.50 ± 1.29], · 10)7 m This value of k3 cannot be compared with the value of the second order rate constant k2 FEBS Journal 272 (2005) 85–96 ª 2004 FEBS M E Fuentes et al Autocatalytic zymogen activation ted either Taking into account the discrimination between Schemes 3, and proposed here and the experimental results plotted in Fig 3B, the reaction mechanism is compatible with the formation of an EZ complex, although it is not possible to discriminate between Schemes and It can be seen that the curves fitting the experimental data in Fig 3A are approximately straight lines This fact can be explained by the following: the exponential term in Eqn (39) can be substituted by a series development, and taking into account that, for short reaction times, only the two first terms may be considered significant, this equation is transformed into the straight line equation: ½ZŠ % ½ZŠ0 À k1 ½ZŠ0 2k2 ẵZ0 ỵ k2 ỵ k3 ị t k2 ẵZ0 þ kÀ2 þ k3 ð53Þ whose ordinate intercept and slope are: Fig Plot of [Z]0 ⁄ k vs [Z]0 according to Eqn (40) for three different k3 values Values of the rate constants k2 and k)2 were as indicated in Fig The values used for k3 were the following: curve i, · 10)2 s)1; curve ii, · 10)3 s)1 and curve iii, · 10)4 s)1, which correspond to Schemes 5, and 4, respectively The inset shows an expansion of this graph near the coordinate origin reported in the literature, as their corresponding units are not the same [12] In addition, because kinetic data taking into consideration the formation of the EZ complex have not been obtained before, the Km values for the pepsinogen–pepsin system have not been repor- ordinate intercept ẳ ẵZ0 slope ẳ k1 ẵZ0 2ẵZ0 ỵ Km ị ẵZ0 ỵ Km ð54Þ ð55Þ From this equation it can be seen that the value of k1 can be obtained from the slopes of plots of [Z] vs time at relatively short reaction times once Km is known, giving the following value, k1 ¼ [5.14 ± 0.56] · 10)3Ỉs)1 This value, which was obtained at °C and pH ¼ 2, together with the value obtained at 28 °C and the same pH by Al-Janabi et al [12] (k1 % Fig (A) Time course of pepsinogen consumption at different initial concentrations Experimental conditions were as indicated in Experimental procedures The inset shows the same results as remaining pepsinogen (%) Lines have been shifted at five unit intervals for greater clarity The following initial concentrations of pepsinogen were used: (d) 1.52 · 10)6 (s) 3.18 · 10)6 (m) 4.84 · 10)6 (n) 6.49 · 10)6 and (j) 8.17 · 10)6 M The points represent experimental data (they are the mean of three assays), the error bars represent SD, and the lines correspond to data obtained by nonlinear regression analysis to Eqn (26) (B) Secondary plot of the above data as [Z]0 ⁄ k vs [Z]0 k Values were obtained by fitting experimental progress curves from Fig 3A by nonlinear regression to Eqn (26), according to the kinetic analysis here proposed The points represent experimental data and the line corresponds to data obtained by linear regression analysis according to a Hanes–Woolf rearrangement of Eqn (40) FEBS Journal 272 (2005) 85–96 ª 2004 FEBS 93 Autocatalytic zymogen activation 4.33 · 10)2Ỉs)1) make it possible to estimate the values of the preexponential factor, A, and the activation energy, Ea, involved in the Arrhenius equation [k1 ¼ A exp(–Ea ⁄ RT)], which provides the variation of the rate constant, k1, corresponding to step (a) in Scheme The estimated values are A ¼ 6.75 · 109 s)1, Ea ẳ 64.53 kJặmol)1 Furthermore, it can be observed from Fig 3A that the slopes of the plots obtained at different initial zymogen concentrations tend to infinite when [Z]0 fi 1, and to zero when [Z]0 fi 0, in agreement with Eqn 55 Concluding remarks In conclusion, we have obtained new approximate solutions for the kinetics of zymogen activation in conditions where both intra- and intermolecular processes take place The proposed reaction scheme (Scheme 3) is a modification of previous mechanisms for this kind of processes [24], which were only treated by numerical integration The main innovation of the present paper is that the kinetic behaviour of the system has been analysed in both analytical and numerical ways, thus showing the goodness of the analysis The above suggested mathematical analysis has been applied to the pepsinogen–pepsin activation, which is an interesting physiological enzymatic system M E Fuentes et al of pepsinogen were precooled at °C Then, 100 lL of 0.1 m sodium citrate ⁄ HCl buffer, pH 2.0, also at °C, were added to this solution, stirred, and after the appropiate time intervals, 300 lL of 0.5 m Tris ⁄ HCl buffer, pH 8.5, were added These additions, by syringe, were made as quickly as possible The test tubes were introduced in a water bath at 37 °C for 20 min, after which the solutions were assayed for remaining pepsinogen activity Solution (100 lL) was now added to mL of 0.2 m sodium citrate ⁄ HCl buffer, pH 2.0, and allowed to activate for 20 Then, mL hemoglobin solution was added to each tube and, after exactly 10 min, mL of 5% trichloroacetic acid solution was added The mixture was filtered through a poly(vinylidene difluoride) filter paper (pore size ¼ 0.45 lm, diameter ¼ 13 mm) and the absorbance of the filtrate was read at 280 nm against a blank containing no enzyme All the assays were performed in polypropylene tubes [41] Other activating pepsinogen concentrations were assayed by appropriate dilution of the stock solution in 0.02 m Tris ⁄ HCl buffer, pH 7.5 Assays at °C were performed using a Hetofrig Selecta bath with a heater ⁄ cooler using a commercial antifreeze and checked using a Cole-Parmer digital thermometer with a precision of ± 0.1 °C A Precisterm Selecta water bath was used for the experiments at 37 °C Spectrophotometric readings were obtained on a Uvikon 940 spectrophotometer from Kontron Instruments, Zurich, Switzerland The experimental progress curves thus obtained were fitted by nonlinear regression to Eqn 26 using the sigmaplot scientific graphing system, version 8.02 (2002, SPSS Inc) Experimental procedures Numerical integration Materials Pepsinogen from porcine stomach (3300 unitsỈmg protein)1), hemoglobin from bovine blood, pepstatin A, sodium citrate and trichloroacetic acid were purchased from Sigma (Madrid, Spain) Stock solutions of pepsinogen were prepared daily by dissolving 6.5 mg of the zymogen in mL of 0.02 m Tris ⁄ HCl buffer, pH 7.5 The hemoglobin solution was also prepared daily by : dilution in 0.3 m HCl of a stock solution of 2.5% (w ⁄ v) hemoglobin, filtered previously through glass wool The zymogen concentration was determined by active-site pepstatin A titration as a tight-binding inhibitor [40] All other buffers and reagents were of analytical grade and used without further purification All solutions were prepared in ultrapure deionized nonpyrogenic water (Milli Q, Millipore Iberica, SA, Barcelona, Spain) Methods Assay for pepsinogen activation The general scheme for these experiments was the same as used earlier [12] Aliquots of 100 lL of the stock solution 94 Simulated progress curves were obtained by numerical integration of the nonlinear set of differential equations directly obtained from Scheme (Eqns 1–4), using arbitrary sets of rate constants and initial concentration values This numerical solution was found by the Runge–Kutta–Fehlberg algorithm [42,43] using a computer program implemented in Visual C++ 6.0 [44] The above program was run on a PC compatible computer based on a Pentium IV ⁄ GHz processor with 512 Mb of RAM Acknowledgements ´ This work was supported by grants from the Comision Interministerial de Ciencia y Tecnologı´ a (MCyT, Spain), Project No BQU2002-01960 and from Junta de Comunidades de Castilla-La Mancha, Project No GC-02–032 M E F has a fellowship from the Prog´ rama de Becas Predoctorales de Formacion de Personal Investigador (MCyT, Spain), associated to the above Project, cofinanced by the European Social Fund FEBS Journal 272 (2005) 85–96 ª 2004 FEBS M E Fuentes et al Autocatalytic zymogen activation References Newsholme EA, Challiss RAJ & Crabtree B (1984) Substrate cycles: their role in improving sensitivity in metabolic control Trends in Biochem 9, 277–280 ´ Valero E, Varon R & Garcı´ a-Carmona F (1995) Kinetic study of an enzymic cycling system coupled to an enzymic step: determination of alkaline phosphatase activity Biochem J 309, 181–185 ´ Valero E, Varon R & Garcı´ a-Carmona F (1997) Mathematical model for 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Me´todos Nume´ricos Con MATLAB (Capella, I, ed.) 3rd edn, pp 505–509 Prentice Hall, Madrid, Spain 96 M E Fuentes et al 44 Garcı´ a-Sevilla F, Garrido del Solo C, Duggleby RG, ´ ´ ´ Garcı´ a-Canovas F, Peyro R & Varon R (2000) Use of a windows program for simulation of the progress curves of reactants and intermediates involved in enzyme-catalyzed reactions Biosystems 54, 151–164 Appendix A k1 ẵZ0 k2 ỵ k3 ị k1 k2 A1ị k1 ẵZ0 k2 ỵ k3 ỵ k1 ị k1 k1 k2 ị A2ị A1;0 ẳ A1;1 ẳ A1;2 ẳ k1 ẵZ0 k2 ỵ k3 ỵ k2 ị k2 k1 k2 ị k1 k2 ẵZ2 k1 k2 A4ị k1 k2 ẵZ2 k1 k1 k2 ị A5ị A2;0 ẳ A2;1 ẳ A2;2 ẳ A3;0 ẳ ẵZ0 ỵ A3;1 ẳ A3ị k1 k2 ẵZ2 k2 k1 k2 ị k1 ẵZ0 k1 ỵ k2 k2 ½ZŠ0 Þ k1 k2 k1 ½ZŠ0 ðk2 À k2 ½ZŠ0 Þ k1 ðk1 À k2 Þ ðA6Þ ðA7Þ ðA8Þ k1 ½ZŠ0 ðk1 À k2 ½ZŠ0 Þ k2 ðk1 À k2 ị A3;2 ẳ A9ị k1 ẵZ0 k1 ỵ k2 Þ k1 k2 ðA10Þ k1 ½ZŠ0 k2 k1 ðk1 À k2 ị A11ị A4;0 ẳ A4;1 ẳ k1 ½ZŠ0 k1 k2 ðk1 À k2 Þ ðA12Þ ½ZŠ0 keÀkt k1 ỵ k2 ẵZ0 ekt B1ị A4;2 ẳ Appendix B ẵZ ẳ k ẳ k1 ỵ k2 ẵZ0 B2ị FEBS Journal 272 (2005) 85–96 ª 2004 FEBS ... mechanisms of autocatalytic zymogen activation and has been sufficiently justified [11,29–31] Previously, kinetic analyses of the reactions, whereby a zymogen is activated both intra- and intermolecularly... for each of these mechanisms with the experimental results, these authors suggested a reaction mechanism including both intra- and intermolecular activation of the zymogen by the action of the... route of intramolecular activation of the zymogen into the active enzyme, E, and one or more peptides represented by W [route (a), Scheme 3] [12,22,25–27] and a route of autocatalytic activation of