By com-paring the simulated progress curves obtained for each of these mechanisms with the experimental results, these authors suggested a reaction mechanism inclu-ding both intra- and i
Trang 1with formation of an enzyme–zymogen complex
Matilde Esther Fuentes, Ramo´n Varo´n, Manuela Garcı´a-Moreno and Edelmira Valero
Grupo de Modelizacio´n en Bioquı´mica, Departamento de Quı´mica-Fı´sica, Escuela Polite´cnica Superior de Albacete, Universidad de Castilla-La Mancha, Albacete, Spain
Living organisms possess different systems of
biologi-cal 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
corres-ponding released peptide Proenzyme activation
there-fore is a control mechanism that differs essentially
from allosteric transitions and reversible covalent
modi-fications
Proenzyme activation by proteolytic cleavage of one
or more peptide bonds requires the presence of an
acti-vating enzyme In those cases in which the actiacti-vating
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 pep-sin [12–14], and prekallikrein into kallikrein [15,16] Several reports describe the kinetic behaviour of enzyme systems involving autocatalytic zymogen activa-tion – with or without steps in rapid equilibrium condi-tions – 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 contribu-tions, the zymogen was considered to be without enzyme activity Nevertheless, references to the enzyme activity
of zymogens are increasingly more frequent [21–23]
Keywords
autocatalysis; enzyme kinetics; pepsin;
pepsinogen; zymogen
Correspondence
E Valero, Grupo de Modelizacio´n en
Bioquı´mica, Departamento de
Quı´mica-Fı´sica, Escuela Polite´cnica Superior de
Albacete, Universidad de Castilla-La
Mancha, Avda Espan˜a 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
(Received 6 July 2004, revised 6 September
2004, accepted 9 September 2004)
doi:10.1111/j.1432-1033.2004.04400.x
A mathematical description was made of an autocatalytic zymogen activa-tion 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 Micha-elis–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
Trang 2Al-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
pepsi-nogen concentration, valid for the whole course of the
reaction and which was still used in recent
contribu-tions [23] Subsequently, a number of different
mecha-nisms for the activation process of pepsinogen were
proposed by Koga and Hayashi (1976) [24] By
com-paring the simulated progress curves obtained for each
of these mechanisms with the experimental results,
these authors suggested a reaction mechanism
inclu-ding 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
acti-vation, for which we have carried out a kinetic
ana-lysis The above mechanism exhibits simultaneously two
catalytic routes, an intramolecular activation process,
route a, and an autocatalytic zymogen activation pro-cess 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 there-fore 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 independ-ently This mechanism is general enough to be applied
to different zymogens exhibiting both intra- and inter-molecular 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 1 Mechanism for the autoactivation of pepsinogen to
pepsin [12] Pgn, pepsinogen; Pep, pepsin.
Scheme 2 Mechanism suggested by Koga and Hayashi [24]
invol-ving 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 X 1
and X2are the unprotonated and protonated pepsinogen,
respect-ively, while X3* and X4* are structural isomers of the active pepsin
which are in an equilibrium involving proton binding X 5 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.
Trang 3Note that in Scheme 3, (Z) includes both X1 and X2
from Scheme 2 and (E) includes both X3* and X4*, so
that [Z]¼ [X1] + [X2] and [E]¼ [X3*] + [X4*] Also,
note that Scheme 5 corresponds to Scheme 1
(previ-ously 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
sim-plified conditions that arise from certain relations
between the values of the first or pseudo first-order rate
constants; (c) to derive the kinetic equations
correspond-ing to Schemes 4 and 5 – which can be considered
par-ticular 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]0: initial concentrations of the species E, Z, EZ and
W, respectively
The dissociation constant of the EZ complex will be:
K2¼k2
k2
The presence of EZ complex allows the definition of a
Michaelis–Menten constant for the activation of
zymo-gen towards its active enzyme as follows:
Km¼k2þ k3
k2
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):
d½Z
dt ¼ k1½Z k2½Z½E þ k2½EZ ð1Þ d½E
dt ¼ k1½Z k2½Z½E þ ðk2þ 2k3Þ½EZ ð2Þ d½EZ
dt ¼ k2½Z½E ðk2þ k3Þ½EZ ð3Þ d½W
dt ¼ k1½Z þ k3½EZ ð4Þ 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):
d½E
dt ¼ k1½Z0 k2½Z0½E þ ðk2þ 2k3Þ½EZ ð6Þ d½EZ
dt ¼ k2½Z0½E ðk2þ k3Þ½EZ ð7Þ d½W
dt ¼ k1½Z0þ k3½EZ ð8Þ The differential Eqns (6) and (7) constitute a nonho-mogeneous linear system that may become homogen-eous 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
dt ¼ k2½Z0Xþ ðk2þ 2k3ÞY ð9Þ dY
dt ¼ k2½Z0X ðk2þ k3ÞY ð10Þ the initial conditions of which are at t¼ 0, X ¼
k1[Z]0, and Y¼ 0, taking into account that [E]0¼ 0 and [EZ]0¼ 0 The solution to this system is given by Eqns (11) and (12):
X¼ k1½Z0ðk2½Z0þ k2Þ
k1 k2
ek 1 tþk1½Z0ðk2½Z0þ k1Þ
k1 k2
ek 2 t
ð11Þ
Y¼k1k2½Z
2 0
k1 k2
ðek1t ek2tÞ ð12Þ where:
Trang 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðk2½Z0þk2þk3Þ2þ4k2k3½Z0 q
2
ð13Þ
k2¼ðk2½Z0þk2þk3Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðk2½Z0þk2þk3Þ2þ4k2k3½Z0 q
2
ð14Þ Note that both k1and k2are real quantities, k1 always
being positive and k2 negative, and that the relations
between k1and k2are as follow (Eqns 15–17):
k1þ k2¼ ðk2½Z0þ k2þ k3Þ ð15Þ
k1k2¼ k2k3½Z0 ð16Þ
To return to our original symbolism, Eqns (11) and
(12) are integrated and, taking into account the initial
conditions mentioned above, gives:
½E ¼ A1;0þ A1;1ek1 tþ A1;2ek2 t ð18Þ
½EZ ¼ A2;0þ A2;1ek 1 tþ A2;2ek 2 t ð19Þ
The expressions corresponding to Ai,j (i¼ 1, 2, 3, 4;
j¼ 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:
½Z ¼ ½Z0 ½E 2½EZ ð20Þ Inserting Eqns (18) and (19) into Eqn (20), the
follow-ing time-concentration equation (Eqn 21) is obtained:
½Z ¼ A3;0þ A3;1ek1 t
þ A3;2ek2 t
ð21Þ 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]0¼ 0, we obtain Eqn
(22):
½W ¼ A4;0þ A4;1ek 1 tþ A4;2ek 2 t ð22Þ
This equation could also be obtained from Eqns (19)
and (21), taking into account the following mass
bal-ance:
½W ¼ ½Z0 ½Z ½EZ ð23Þ 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 reac-tion mechanism was simplified, disregarding the inter-mediary 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 cor-responds 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 corres-ponding experimental difficulties The measurement of zymogen concentrations not far from the initial value
in a short-time reaction leads to unavoidable experi-mental errors Nevertheless, taking into account that the values of the kinetic parameters are independent of the reaction time registered, this will allow the evalua-tion 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 autocata-lytic 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 litera-ture, 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 corres-ponding to the whole process leading from Z to E and
Wby intramolecular activation Route (b) of Scheme 3 has been assumed to follow a single Michaelis–Menten mechanism instead of the more general Uni–Bi mech-anism This approach is the usual one used to describe
Trang 5mechanisms 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
intermole-cularly by the action of the active enzyme, have been
made and used for the experimental determination of
the kinetic parameters involved in pepsinogen
autoacti-vation [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
nonhomogene-ous 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
pos-sible to completely characterize the system Obviously,
the approximate analytical time course equations
derived here are also applicable to any zymogen
acti-vation 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
accu-rate the analytical solutions This is usual in enzyme
kinetics, where to derive approximate analytical
solu-tions corresponding either to the transient phase or the
steady-state of an enzymatic reaction, substrate
con-centration (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 reac-tion time The fracreac-tion, q, of the remaining zymogen is introduced as:
q¼ ½Z
and we may arbitrarily set the q-value (e.g q ¼ 0.7) above which the approximate solutions remains applic-able Thus, the [Z]-values for which the equations obtained are applicable are:
For example, if [Z]0¼ 10)3m and q¼ 0.7, then, according to Eqn (25), the analytical equations derived here will be valid only when [Z]‡ 7 · 10)4m
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 equa-tion system obtained directly from the mechanism shown in Scheme 3 (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 1 (Eqn B1, Appendix B) [12] is shown in Fig 1B, using the same values for the rate constants and initial conditions Table 1 shows a numerical com-parison 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 k1is 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 con-sidered uni-exponential In this way, the kinetic equa-tions for Z (Eqn 26) and W (Eqn 27) become:
Trang 6½Z ¼ A3;0þ A3;1ek1 t ð26Þ
½W ¼ A4;0þ A4;1ek1 t ð27Þ
The case in which one exponential term can be
neglected after approximately t¼ 0
In such a case, the following relations (Eqns 28–30)
must be fulfilled:
i.e
Under these conditions, the uni-exponential behaviour
of the species can be assumed from t¼ 0 Thus, if the relationships 29 and 30 [Eqns (29) and (30)] are inser-ted into Eqns (26) and (27), we obtain:
½Z ½Z0k1½Z0ðk2 k2½Z0Þ
k1k2 ðek 1 t 1Þ ðfrom t 0Þ
ð31Þ
½W k1½Z0
k1
ðek1 t
1Þðfrom t 0Þ ð32Þ Bearing in mind the relation 29 (Eqn 29), Eqn (15) becomes:
k2 ðk2½Z0þ k2 ¼ k3Þ ð33Þ and from Eqns (16) and (33) we obtain:
k1 k3½Z0
The kinetic behaviour from the onset of the reaction is
a consequence of assumption 28 (Eqn 28) This condi-tion is only fulfilled if certain relacondi-tions between the
Fig 1 (A) Simulated progress curves corresponding to the species
involved in the mechanism shown in Scheme 3 The values of the
rate constants used were: k1¼ 4.0 · 10)3Æs)1, k2¼ 1.0 · 10 3
M )1Æs)1, k
)2 ¼ 2.1 · 10)4Æs)1 and k 3 ¼ 5.4 · 10)4Æs)1 The initial
zymogen concentration used was [Z] 0 ¼ 2.4 · 10)5M (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.
Table 1 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 con-stants used were those indicated in Fig 1 and the q-values corres-pond 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
(l M ) t (s)
[Z] Eqn 21
(l M )
Relative error (%)
[Z] Eqn B1
(l M )
Relative error (%)
Trang 7first- 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:
ðk2½Z0þ k2þ k3Þ2 4k2k3½Z0 ð35Þ
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:
jk2j k1, ðk2½Z0þ k2þ k3Þ2 4k2k3½Z0 ð36Þ
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
con-stants corresponding to the dissociation and
forma-tion of the EZ complex It is therefore reasonable to
think that:
and⁄ or
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:
½Z ½Z0k1½Z0ðk2 k2½Z0Þ
kk2 ðekt 1Þ ð39Þ
k k3½ Z0
Rapid equilibrium assumptions: Scheme 4
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 3 is in equilibrium from the onset of the
reac-tion 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:
k k3½Z0
The case in which the activation can be represented by Scheme 5
From a comparison of Schemes 3 and 5, it can be seen that the latter formally arises from Scheme 3 if:
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:
which is obtained as lim
k 3 !1k; where k is given by Eqn (40)
The case in which the intramolecular activation
of pepsinogen is predominant
In this case, the amount of zymogen activated inter-molecularly by the active enzyme (route b) in Scheme
3 may be considered negligible and so it can be assumed that:
Therefore Eqn (33) is rewritten as:
k2¼ ðk2þ k3Þ ð45Þ Under these conditions, Eqn (39) can be rewritten as:
½Z ½Z0k1½Z0
which may be transformed into the uni-exponential equation reported by Al-Janabi et al [12] by substitu-ting 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:
½Z ¼ ½Z0ek1 t ð47Þ
Kinetic data analysis The uni-exponential kinetic behaviour of the reaction evolving according to Scheme 3 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-exponen-tial behaviour given by Eqns (39–40) In this kinetic analysis, it is assumed that the remaining zymogen,
Trang 8[Z], can be experimentally monitored by a
discontinu-ous 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
experimen-tal [Z]-values obtained by any discontinuous method
at different reaction times, t, and at different [Z]0
-val-ues, and fitting them to Eqn (26), gives the
correspond-ing 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
depend-ence on initial zymogen concentration, [Z]0 Therefore,
the kinetic parameters k3 and Kmcan be evaluated by
a nonlinear least-squares fit of the experimental
k-val-ues obtained in step (a) to this equation Furthermore,
these parameters can also be obtained by linear
regres-sion 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 intercept¼Km
k3
ð48Þ
slope¼ 1
k3
ð49Þ abscissa intercept¼ Km ð50Þ
Therefore, the kinetic parameters k3 and Km can be
evaluated
Particular cases of Scheme 3
Schemes 4 and 5 can be considered formally as
partic-ular cases of the reaction mechanism shown in
Scheme 3 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, 4 and 5
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 4 (Eqns 39 and 41) and Scheme 5
(Eqns 39 and 43), which are particular cases of
Scheme 3 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
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 accord-ing to Scheme 5, takaccord-ing into account Eqns (39) and (43), the intercept and the slope of the straight line ari-sing from step (b) will become:
ordinate intercept¼ lim
k 3 !1ðKm
k3
Þ ¼ 1
k2
ð51Þ
slope¼ lim
k3!1ð1
k3
In this way, the suggested procedure for evaluating the kinetic parameters allows us to discriminate between Scheme 5 and Schemes 3 and 4 If the straight line ari-sing from step (b) has a slope of zero or nearly zero, then a compatible mechanism reaction is that des-cribed by Scheme 5 If this is not the case, the mechan-ism reaction is compatible with both Schemes 3 and 4, between which it is impossible to discriminate Never-theless, because Scheme 4 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 men-tioned 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 4 and that Eqn (41) is ful-filled To illustrate this, Eqn (40) is plotted in linear form in Fig 2 for fixed values of k2and k-2at different
k3values leading to Schemes 3, 4 and 5
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 experimen-tal progress curves corresponding to the remaining pepsinogen in the reaction medium The inset shows the same results as percentage of remaining pepsino-gen Taking into account assumption 5 from the The-ory 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, A3,1and k at the different initial pepsi-nogen 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: k3¼ [6.13 ± 0.14] · 10)4Æs)1, Km¼ [1.50 ± 1.29],· 10)7m This value of k3 cannot be compared with the value of the second order rate constant k2
Trang 9reported 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 Kmvalues
for the pepsinogen–pepsin system have not been
repor-ted either Taking into account the discrimination between Schemes 3, 4 and 5 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 3 and 4
It can be seen that the curves fitting the experimen-tal data in Fig 3A are approximately straight lines This fact can be explained by the following: the expo-nential 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 consid-ered significant, this equation is transformed into the straight line equation:
½Z ½Z0k1½Z0ð2k2½Z0þ k2þ k3Þ
k2½Z0þ k2þ k3
whose ordinate intercept and slope are:
ordinate intercept¼ ½Z0 ð54Þ
slope¼ k1½Z0ð2½Z0þ KmÞ
½Z0þ Km
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, k1¼ [5.14 ± 0.56] ·
10)3Æs)1 This value, which was obtained at 5C and
pH¼ 2, together with the value obtained at 28 C and the same pH by Al-Janabi et al [12] (k1
Fig 3 (A) Time course of pepsinogen consumption at different initial concentrations Experimental conditions were as indicated in Experi-mental 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)6and (j) 8.17 · 10)6M 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).
Fig 2 Plot of [Z] 0 ⁄ k vs [Z] 0 according to Eqn (40) for three
differ-ent k3values Values of the rate constants k2and k)2were as
indi-cated in Fig 1 The values used for k3were the following: curve i,
2 · 10)2s)1; curve ii, 2 · 10)3s)1 and curve iii, 2 · 10)4s)1,
which correspond to Schemes 5, 3 and 4, respectively The inset
shows an expansion of this graph near the coordinate origin.
Trang 104.33· 10)2Æs)1) make it possible to estimate the values
of the preexponential factor, A, and the activation
energy, Ea, involved in the Arrhenius equation [k1¼
A exp(–Ea⁄ RT)], which provides the variation of the
rate constant, k1, corresponding to step (a) in Scheme 3
The estimated values are A¼ 6.75 · 109s)1, Ea¼
64.53 kJÆmol)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]0fi 1, and to zero when [Z]0fi 0, in agreement
with Eqn 55
Concluding remarks
In conclusion, we have obtained new approximate
solutions for the kinetics of zymogen activation in
con-ditions 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
Experimental procedures
Materials
Pepsinogen from porcine stomach (3300 unitsÆmg protein)1),
hemoglobin from bovine blood, pepstatin A, sodium citrate
and trichloroacetic acid were purchased from Sigma
(Madrid, Spain) Stock solutions of pepsinogen were
pre-pared daily by dissolving 6.5 mg of the zymogen in 5 mL of
0.02 m Tris⁄ HCl buffer, pH 7.5 The hemoglobin solution
was also prepared daily by 4 : 1 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
deter-mined by active-site pepstatin A titration as a tight-binding
inhibitor [40] All other buffers and reagents were of
analyt-ical grade and used without further purification All
solu-tions 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
of pepsinogen were precooled at 5C Then, 100 lL of 0.1 m sodium citrate⁄ HCl buffer, pH 2.0, also at 5 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 37C for 20 min, after which the solutions were assayed for remaining pepsinogen activity Solution (100 lL) was now added to 1 mL of 0.2 m sodium cit-rate⁄ HCl buffer, pH 2.0, and allowed to activate for
20 min Then, 1 mL hemoglobin solution was added to each tube and, after exactly 10 min, 1 mL of 5% trichloro-acetic acid solution was added The mixture was filtered through a poly(vinylidene difluoride) filter paper (pore size¼ 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 5C 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 ± 0.1C A Precisterm Selecta water bath was used for the experiments at 37C Spectrophotometric readings were obtained on a Uvikon 940 spectrophotometer from Kontron Instruments, Zurich, Switzerland
The experimental progress curves thus obtained were fit-ted by nonlinear regression to Eqn 26 using the sigmaplot scientific graphing system, version 8.02 (2002, SPSS Inc)
Numerical integration Simulated progress curves were obtained by numerical integration of the nonlinear set of differential equations directly obtained from Scheme 3 (Eqns 1–4), using arbitrary sets of rate constants and initial concentration values This numerical solution was found by the Runge–Kutta–Fehl-berg algorithm [42,43] using a computer program imple-mented in Visual C++ 6.0 [44] The above program was run on a PC compatible computer based on a Pentium
IV⁄ 2 GHz processor with 512 Mb of RAM
Acknowledgements
This work was supported by grants from the Comisio´n 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 Formacio´n de Perso-nal Investigador (MCyT, Spain), associated to the above Project, cofinanced by the European Social Fund