Báo cáo khoa học: Kinetics of intra- and intermolecular zymogen activation with formation of an enzyme–zymogen complex

Kinetics of Intra- and Intermolecular Zymogen Activation with Formation of an Enzyme-Zymogen Complex

Tác giả: Matilde Esther Fuentes, Ramón Varón, Manuela García-Moreno and Edelmira Valero

Lĩnh vực: Modelización en Bioquímica, Departamento de Química-Física, Escuela Politécnica Superior de Albacete, Universidad de Castilla-La Mancha, Albacete, Spain

Nội dung tài liệu: Tài liệu này trình bày một mô tả toán học về cơ chế hoạt hóa zymogen tự xúc tác, bao gồm cả các con đường nội phân tử và liên phân tử. Một phức hợp trung gian enzyme-zymogen có thể đảo ngược được đưa vào con đường hoạt hóa liên phân tử, cho phép định nghĩa hằng số Michaelis-Menten cho quá trình chuyển đổi zymogen thành enzyme hoạt động. Các phương trình theo thời gian-nồng độ mô tả sự biến đổi của các loài tham gia vào hệ thống đã được thu được. Ngoài ra, các phương trình động học tương ứng cho các trường hợp cụ thể của mô hình tổng quát đã được suy ra. Các quy trình thiết kế thực nghiệm và phân tích dữ liệu động học để đánh giá các tham số động học, dựa trên các phương trình động học đã suy ra, được đề xuất. Tính hợp lệ của các kết quả thu được đã được kiểm tra bằng cách sử dụng các đường cong tiến triển mô phỏng của các loài tham gia. Mô hình này nhìn chung đủ tốt để áp dụng cho nghiên cứu động học thực nghiệm về sự hoạt hóa của các zymogen khác nhau có ý nghĩa sinh lý. Hệ thống được minh họa bằng việc theo dõi động học chuyển đổi pepsinogen thành pepsin.

Mục lục chi tiết:

• Kinetics of intra- and intermolecular zymogen activation with formation of an enzyme-zymogen complex
• Keywords
• Correspondence
• Note
• A mathematical description was made of an autocatalytic zymogen activation mechanism involving both intra- and intermolecular routes.
• 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 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].
• Autocatalytic zymogen activation
• Scheme 1. 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 (intra- and intermolecular) for the activation of pepsinogen to pepsin, as is indicated in Scheme 1.
• 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 1 and 2 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, 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 2. 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.
• Scheme 3. 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.
• Scheme 4. Mechanism shown in Scheme 3 under rapid equilibrium conditions between E, Z and EZ.
• Scheme 5. Simplified general mechanism for the autoactivation of zymogens.
• Note that in Scheme 3, (Z) includes both X1 and X2 from Scheme 2 and (E) includes both X3* and X4*, so that [Z] = [X1] + [X2] and [E] = [X3*] + [X4*].
• Also, note that Scheme 5 corresponds to Scheme 1 (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 3 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 4 and 5 – which can be considered particular cases of Scheme 3 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 4 and 5.
• 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]: initial concentrations of the species E, Z, EZ and W, respectively.
• The dissociation constant of the EZ complex will be:
• The presence of EZ complex allows the definition of a Michaelis–Menten constant for the activation of zymogen towards its active enzyme as follows:
• Time course differential equations and mass balances
• The kinetic behaviour of the species E, Z, EZ and W involved in Scheme 3 is described by the following set of differential equations (Eqns 1–4):
• 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.
• Taking into account this assumption, the differential equation system that describes the mechanism shown in Scheme 3 is given by Eqns (6–8):
• 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 = X, and d[EZ]/dt = Y, giving Eqns (9) and (10):
• the initial conditions of which are at t = 0, X = k₁[Z]₀, and Y = 0, taking into account that [E]₀ = 0 and [EZ]₀ = 0.
• The solution to this system is given by Eqns (11) and (12):
• where:
• Note that both λ₁ and λ₂ are real quantities, λ₁ always being positive and λ₂ negative, and that the relations between λ₁ and λ₂ are as follow (Eqns 15–17):
• To return to our original symbolism, Eqns (11) and (12) are integrated and, taking into account the initial conditions mentioned above, gives:
• The expressions corresponding to Aij (i = 1, 2, 3, 4; j = 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:
• Inserting Eqns (18) and (19) into Eqn (20), the following time-concentration equation (Eqn 21) is obtained:
• This equation could also be obtained by integration of Eqn (1) after inserting into it condition 5 (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 = 0, we obtain Eqn (22):
• This equation could also be obtained from Eqns (19) and (21), taking into account the following mass balance:
• Equation (21) for zymogen consumption is different from the equation reported previously in the literature for the simplified reaction mechanism shown in Scheme 1 [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 3 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 3 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 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 5 (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 3 were derived by solving the nonhomogeneous set of ordinary, linear (with constant coefficients), differential Eqns (6–8).
• These kinetic equations are valid whenever condition 5 (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 3 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.
• Experimentally, it is possible to determine whether the assumption 5 (Eqn 5), which is always true at the onset of the reaction, is still fulfilled at a certain reaction time.
• The fraction, p, of the remaining zymogen is introduced as:
• and we may arbitrarily set the p-value (e.g. p = 0.7) above which the approximate solutions remains applicable.
• Thus, the [Z]-values for which the equations obtained are applicable are:
• For example, if [Z]₀ = 10⁻³ M and p = 0.7, then, according to Eqn (25), the analytical equations derived here will be valid only when [Z] ≥ 7 × 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 3 (Eqns 1–4), for an