Kinetics of an autocatalytic zymogen reaction in the presence of an inhibitor coupled to a monitoring reaction

Bdkrm

o~Ma&~~nticnl

Riologv,

Vol.

58, No.

1. pp. 19-41,

Elsevier Q 1996 Society

for Mathematical 0092.8240/96

1996

Science $15.00

Inc.

Biology + 0.00

009%8240(95hM304-9

KINETICS OF AN AUTOCATALYTIC ZYMOGEN REACTION IN THE PRESENCE OF AN INHIBITOR COUPLED TO A MONITORING REACTION l M. C. MANJABACAS, E. VALERO, M. GARCIA-MORENO, C. GARRIDO and R. VARGN* Departamento de Quimica-Fisica, Escuela Universitaria Polit&nica de Albacete, Universidad de Castilla-La Mancha, Avda. de Espaiia, s.n. Campus Universitario, 02006 Albacete, Spain

A global kinetic analysis of a model consisting of an autocatalytic zymogen-activation process, in which an irreversible inhibitor competes with the zymogen for the active site of the proteinase, and a monitoring coupled reaction, in which the enzyme acts upon one of its substrates, is presented. This analysis is based on the progress curves of any of the two products released in the monitoring reaction. The general solution is applied to an important particular case in which rapid equilibrium conditions prevail. Finally, we suggest a procedure to predict whether the inhibition or activation route dominates in the steady state of the system. These results generalize our previous analysis of simpler mechanisms.

1. Introduction. A number of proteolytic enzymes are synthesized as inactive precursors, termed proenzymes or zymogens, to protect the cells which produce them. These zymogens must undergo an activation process, usually a limited proteolysis, to change into the active form (Cohen, 1976). We must be aware of the phenomenon of zymogen activation to understand the fundamental physiological and biochemical processes. Physiological examples include the pancreatic zymogens such as proelastase, chymotrypsin A, B and C and procarboxipeptidase A and B, producing the active enzymes which take part in the digestion of proteins in the intestine (Hadorn, 1974; Baici, 1990; Vendrell et al., 19921, the activation of prothrombin, plasminogen and several enzyme factors in blood clotting and in *Author

to whom correspondence

should

be addressed. 19

20

M. C. MANJABACAS et

al.

fibrinolysis (Neurath and Walsh, 1976; Mann et al., 1988; Lijffler and Petrides, 1988; Fredenburgh and Nesheim, 1992; Gerads et al., 1992; Bergum and Gardell, 1992; Lorand and Radek, 1992; Ruf et al., 1992; Trimarchi et al., 1992, Ellis and Dano, 1992; Geppert and Binder, 1992) and the activation of protyrosinase initiating the melanization process (Galindo et al., 1983). Other examples of zymogen activation are the activation of the complement system, prohormone conversion such as the proinsulin-insulin conversion and the transformation of angiotensinogen into angiotensin, which are important steps in the response of the immune system, in the control of glucose concentration in blood and in blood pressure regulation, respectively (Mihalyi, 1972; Davie and Fujikawa, 1975; Holzer and Heinrich, 1980; Miiller-Eberhard, 1988; Rhodes et al., 1989; Havsteen and Vat-on, 1990). Different proteinase inhibitors presented in cells and body fluids can control these important physiological processes (Rappay, 1989). Specific inhibitors can be used as drugs in human therapy (Scharpe et al., 1991; Barrett and Salveson, 1986). Inhibitors such as E-aminocaproic acid, paminomethylbenzoic acid or aprotinin can be used to control pathological increases in fibrinolysis, e.g. in the treatment of leukemia or in operations involving organs with a high fibrinolysis activator content. In addition to inhibiting plasmin, they also inhibit trypsin, chymotrypsin and kallikrein, the last being the most important protein responsible for the release of bradykinin from kininogen (Liiffler and Petrides, 1988). When the activating enzyme and its activation product coincide, the process is an autocatalytic zymogen activation. Examples of these processes are the activation of prekallikrein (Tans et al., 1987), trypsinogen (Hadorn, 1974) and pepsinogen (Karlson, 1988) by kallikrein, trypsin and pepsin, respectively. Several experimental and theoretical models of autocatalysis have been extensively investigated. Some interesting simple autocatalytic enzyme systems have been studied in the absence of an inhibitor (Varon et al., 1991, 1992) as well as in the presence of an inhibitor (Manjabacas et al., 1992). In this paper we extend these studies to explore the kinetic behaviour of more complex systems. The mechanism proposed is that in Scheme I, which differs from the mechanism we recently studied (Manjabacas et al., 1992) (reactions (Ia) and (Ib)) by coupling of the reaction (Ic) corresponding to the action of E on one of its substrates. Reaction (Ic) is assumed to follow a uni-bi mechanism (Delaage et al., 1967; Hatfield et al., 1971; Dixon et al., 1979) and acts as a monitor, since it allows us to follow the activation process by measuring the accumulation of a chromophoric product, either X or Y.

KINETICS OF AN AUTOCATALYTIC

k

k +I

E+Zz

k-1

_

2E

ki

EI -EI*

k;, C

k’

(Ib)

k’+3

EY -E

ES + k’,

21

(14

k-4

E+S

REACTION

w

k +4 E+I

k f3 EE -

EZ;

ZYMOGEN

+Y

(Ic)

X

Scheme I

In Scheme I, E is both the activating proteinase and the activated enzyme, Z is the inactive precursor of E, W is one or more peptides released from Z during the formation of E, I is an inhibitor of E, S is a chromogenic substrate of E, and X and Y are the products of the monitoring reaction of E on S (X, Y or both are the chromophoric products, the concentrations of which can be experimentally followed). The model I is a general model of an inhibited autocatalytic enzyme acting on a substrate S to form a product Y and an activation peptide X. In Scheme I, the autocatalytic reaction (Ia) describes the process of activation of Z, which is simultaneous with an irreversible inhibition reaction in two steps (Ib). The first step is the reversible formation of the inhibitor-proteinase complex, EI, which is followed by the irreversible formation of another complex, EI”. The kinetic behaviour of the autocatalytic activation of a zymogen in the absence of any inhibitor or substrate of the activating enzyme (reaction (Ia)) was studied theoretically (Varon et al., 1990) as well as experimentally (Garcia-Moreno et al., 1991). The kinetics of autoactivation in the presence of a substrate of the enzyme yielding a chromophoric product (subschemes consisting of both reactions (Ia) and (1~)) (Varon et al., 1991, 1992) as well as in the presence of an irreversible inhibitor of the enzyme (subschemes consisting of both reactions (Ia) and (Ib)) (Manjabacas et al., 1992) have been analyzed. Chemical reactions governed by the above subsystems can be observed easily in vitro and, therefore, the kinetic parameters can be evaluated experimentally. Nevertheless, the simultaneous presence of the zymogen, Z, the enzyme, E, the inhibitor, I and the substrate, S, is to be expected in a physiological medium such as the gastrointestinal tract, blood,

22

M. C. MANJABACAS

et al.

acinar cells, etc. Hence, adding reaction UC) makes the model a better description of in vivo situations. Lack of inclusion of reaction (Ic) in the model presents an experimental difficulty. Generally, the concentration of neither W nor E can be easily followed experimentally. To measure these constituents, we would have to use an experimental discontinuous method, e.g. a generally laborious, time-consuming procedure which consists of measuring the concentration of the activated enzyme E by removing periodic aliquots of the reaction mixture followed by an assay of the activity of E in the aliquots (Coulomb and Figarella, 1979; Hatfield et al., 1971). The catalytic activity of E is determined by measuring the initial rate of a reaction with one of the substrates of E under conditions of saturating substrate concentration. In contrast, the coupling of the inhibited autocatalytic reaction with reaction (Ic) leads to an advantageous experimental situation if the time course of a product, X or Y, can be experimental progress curves of X or Y, obtained using a continuous procedure that is easily measured. Then an experimental design and a method of kinetic data analysis can be made to discriminate between different mechanisms, analogously to the results for simpler zymogen activation models (Varon et al., 1986, 1987; Vazquez et al., 1993). The aim of this paper is to develop the kinetic model of enzyme reactions evolving according to Scheme I under linear conditions of an excess of inactive precursor Z, inhibitor I and substrate S, and from these equations to obtain approximate analytical solutions under conditions of rapid equilibrium. The linearization of enzyme systems, including the autocatalytic zymogen activation reactions is usual in kinetic analysis (Dixon et al., 1979; Segel, 1988). Comparison of our results with the simulated progress curves will indicate that the linearization is indeed appropriate.

2. Methods. The simulated progress curves have been obtained by numerical integration of the non-linear set of differential equations derived from Scheme I, using different sets of rate constants and initial concentrations. This numerical integration was carried out by using the Adams-Moulton method in combination with a Runge-Kutta fourth-order method (Gerald and Wheatley, 1989). Simulated experimental errors proportional to the value of [P] (Gray and Duggleby, 1989), with the mean 0 and a specified S.D. (l%>, were added to the value of the simulated concentrations using a random normal distribution. The random numbers were generated using a routine based on the algorithm of Box, Muller and Marsaglia (Rugg and Feldman, 1987). The algorithms for the numerical solution and the simulated errors were implemented in Turbo Pascal 6.0 (Garrido et al., 1992).

KINETICS OF AN AUTOCATALYTIC

ZYMOGEN REACTION

23

The above programs were run on a Tandom 386/33 MHz (IBM AT-compatible) computer with an Intel 80387 arithmetic co-processor. 3. Kinetic Analysis. 3.1. Initial conditions. We assume that the concentrations of Z, S and I -[Z], [S], and [II-remain approximately constant during the whole reaction to reduce the corresponding set of differential equations to linear algebraic equations. This will be possible if the above-mentioned concentrations are much higher than the concentration of E at any time t, i.e. we suppose

(1)

[Zl=-[El

(2)

[Sl, VI x=-[El. To have condition (1) fulfilled it is necessary (but not sufficient) that

[Zlo x=-mo,

(3)

where [Z], and [E], are the initial concentrations of Z and E. As we will see below, [E] increases exponentially with time from [El, in some cases and, therefore, the mathematical condition (1) may not remain valid after a certain reaction time. Hence, in those circumstances in which the constancy of [Z] cannot be assumed, our analysis will not be valid. To have condition (2) fulfilled it is necessary and sufficient that

[Slo, [Ilo B [Zlo + [El0

(4)

because the maximal concentration that E can reach will be ([El, + [Z],), where [I], and [S], are the initial concentrations of I and S. If we insert condition (3) into condition (4), the latter becomes (5)

[~lo, [Ilo x=-[Zlo.

Hence, to obtain valid results, we will assume conditions (3) and (5) in the following text. 3.2. Set of differential equations. Taking into account the initial conditions, the kinetic behaviour of the enzyme species and products involved in Scheme I are described by the following set of differential equations: dE1 ~ dt

= -~~+,~~lo+k+,~~lo+k:,~~lo~~~l +k_,[EI]

+k’,[ES]

+k!+,[EY]

d[EZl ~ =k+,[Zlo [El -(k_, +k+,)[EZl dt

+k_,[EZl+2k+,LW

M. C. MANJABACAS et al.

24

dEEI

-

=k+2[EZl

dt

-k+,[EE]

dIEI - dt = k+4[110[El - (k4 + kJEI1 d[ EI”] = ki[EI] dt

&=I

~

=k’+,[Slo [El -WI

dt

+k’+,)[ES]

d[:EYl =k’+,[ES] -k:,[EY] dt

dX1 =k’+,[ES] dt

d[Yl = I?‘+~[ EY].

(6)

dt

3.3. Time course equations. Taking the above initial conditions into account, the dynamical system (6) is a homogeneous linear set of differential equations with constant coefficients. The solution of this system leads to the following expressions for the E, X and Y concentrations, where the symmetry of the system in X and Y permits us to express their concentrations using a single variable, P, that represents either X or Y,

[El = i

Y~,~exp(A,t)

(7)

h=l

[PI = P + hIiI Y~,~

e&V)

{P=X

n,

(8)

where the coefficients are

(k+3 + A&Y+3 + A&_1

+ k+z + Ah)

X(k_,+ki+Ah)(k’_l+k~,+A,)[EI~

3/E,h

=

fI p=l p+h

($74J

(h = 1,2,...,6)

(9)

KINETICS OF AN AUTOCATALYTIC ZYMOGEN REACHON

kL&+~(k_, + k+,)(k_, + ki)[ElONo p = (k’, + k:z){kik+,(k_, + k+,)Vlo - k+$+*(k_4 k;,k:,(k+, YX.h

=

+k,)Lzlo}

25

(lo)

+ hh)(k& + A&k4 +ki + Ah) x (k-1 + k,, + ‘+,)[~h[Sh

Ahpfil(hp-Ah) p+h

(h = 1,2,...,6) AY,h =

k:,k;,k:,(k+,

+ Ah)&

+ k,, + Ah)&

(11)

+ ki + AJE]o[S]o

‘hpfi,(‘,-‘,i p+h (h = 1,2,...,6) and the eigenvalues

(12)

Ah(h = 1,2,. . . , 6) are the roots of the equation

5 Fj;):h6-’= 0

(13)

i=O

where F, = 1 and the expressions of the other coefficients I;;:(i = 1,2,. . . ,6) are given in Appendix A. 4. Results and Discussion. 4.1. Some considerations. Previously, kinetic analyses of reactions where an inhibitor competes irreversibly with a substrate of a proteinase have been carried out and have been used for experimental determinations of kinetic parameters (Leytus et al., 1984; Liu and Tsou, 1986). In these cases, at infinite time, [E] must be zero since inactivation occurs. A complete analysis of the case in which the substrate of the proteinase is the zymogen from which it derives, and where the proteinase is irreversibly inhibited, without a monitoring reaction (reactions (Ia) and (Ib)) has recently been studied (Manjabacas et al., 1992). In this case, there are three possibilities for the concentration of E at infinite reaction time: it can become zero, a constant value or increase exponentially with time. The experimental study of the subscheme consisting of reactions (Ia) and (Ib) requires a discontinuous method which generally becomes laborious. Nevertheless, the analysis we introduce here, in which subscheme (1~) is coupled to reactions (Ia) and (Ib) could let us easily monitor the reaction. The subscheme consisting of reactions (Ia) and (Ib) could easily be followed experimentally by adding the substrate to the reaction medium at t = 0; i.e.

26

M. C. MANJABACAS et al.

by coupling it to another

enzyme reaction (reaction (Ic)), in which, from substrate which releases at least product that can be sensitively monitored.

t = 0, the enzyme E acts on a chromogenic

one chromophoric

4.2. Particular cases of Scheme I. In the following, we identify different autocatalytic mechanisms which are particular cases of the mechanism indicated in Scheme I with one or more of the rate constants much higher than the others (and mutually of the same scale) and/or some’ rate constants are zero. We denote by Ai (i = 1,2,. . . ,20) the subscheme of Scheme I consisting of reactions (Ia) and (Ib) with parameters as indicated in Table 1. The action of E on S, i.e. reaction (Ic) with the parameter conditions indicated in Table 2, are denoted by Mj (j = 1,. . . ,4). The complete mechanism in which (Ia)satisfy scheme Ai and (Ic) satisfies Mj is denoted by AiMj (i = 1,. . . ,2O;j = 1,. . . ,4; i and j not being 1 simultaneously because A,M, coincides with the mechanism indicated as Scheme I). Note that in schemes A,M,, Cm = 3, F-10, 12-16, l&20; II = 2, 4) one or more of the reversible steps is in rapid equilibrium conditions. As examples, in Table 3 the standard reaction schemes corresponding to four particular cases are indicated.

Table 1. Rate constants which must be much higher than the others or null so that subscheme of Scheme I consisting of reactions (Ia) and (Ib) becomes the corresponding subscheme A,(i = 1,2,. . . ,20) Subscheme A, A, A3 A4 A5 A, A7

A8

iy A:(: A,, ;I3

A:: A,,

First or pseudo-first rate constants much higher than the others

Null rate constants

-

-

k+,El,,k-, k+J:j3

k-

k+,M,,E’,,~,, k+1[Zl,,k-,,k+,[Zl,,k-, k+,Ul,, k-4, k+, k+,[Zl~,k-,,k+~[Zl~,k-~,k+3 k+,lZl,,k-, k_,:Ij3

k-

k+,lZl,,~-,,~+3 k+,[Zl,,k-,,k+,[Zl,,k-, k+,lZl,,,k-,,k+3 k+,[Zl,,k-,,k+,[Zl,,k-,,k+,

A,,

A 20

ki ki

ki ki -

-

A17 A,,

ki k, k, ki

k+,L&,,k, k+3vk

k+,lZl,,k-,,ki>k+,

-

-

KINETICS OF AN AUTOCATALYTIC ZYMOGEN REACTION

27

Table 2. Rate constants which must be much higher than the others or null so that subscheme (Ic) becomes the corresponding subscheme Mi (j = 1,2,. . . ,4) First or pseudo-first rate constants much higher than the others

Subscheme

Null rate constants

The kinetic equations for each of the 80 A,Mj (i = 1,2,. . . ,20; j = 1,2,..., 4) reactions could be obtained by solving the corresponding system of differential equations. However, these kinetic equations can be obtained from equations (7) and (8) by replacing in them the constants indicated in Tables 1 and 2 for the particular corresponding case. 4.2.1. Example: Derivation of the kinetic equations of the particular case The scheme of this mechanism is indicated in Table 3. Taking into account Tables 1 and 2, the kinetic equations of this mechanism can be obtained from those of Scheme I by inserting in them:

A,,M,.

k+,[~l,,k_,,k+,,k+,~~l,,k-,,k:,~~l,~k’,~k:,~xl k+,[Zl,,k_,,k+,,k+,~~l,,k_,,k:,~~l,,k~,~k~,

mutually

ofthe

same scale.

(14)

In Appendix B the procedure to obtain the time course equations of the enzyme species E and the products X or Y is indicated. The result is

[El = P exp(At) [PI

=

a[1 - exp(ht)]

(15)

P=X,Yl,

(16)

where the expressions for p, u and A are given approximately by

Pz a=

wqqao 6

(17)

k’+,~,&[~loLSlo

(18)

G6 A z--

K,K;k,(l - r)[llo Km-a0

+ K$qIlo

+ K,K,[S]o + K,K,zq *

(19)

28

M. C. MANJABACAS et

al.

Table 3. Examples of mechanisms which can be considered to be particular cases of mechanism indicated in Scheme I taking into account the notation and the rate constant values indicated in Tables 1 and 2. K,, K, and K; are the dissociation constants of the complexes EZ, EI and ES; i.e. K,=k_,/k,, and Ki =kL,/ki, K, =k-,/k+,, Scheme

Mechanism Kl

E+Ze +

A7M2

EZ-+EEzZE

k

iv

I K4 II

EI

k;z

G E+S

-ES

7

k;J

EY -E+Y X

Kl

E+Ze

-47M4

+

k EZ$+EE~ZE W

I K4 II

EI E+S

_

E+Z---L +

A16M2

G K

k’+z ES E+X+Y k EZf2_2E+W

I K4 II

EI -

E+S

ki

=

G Kl

E+Ze +

A16M4

EI*

k;z ES\,

k;3 EY -E+Y

X k EZ+2_2E+W

I K4 II

EI -

E+S

ki

_

EI* K

62 ES-

E+X+Y

KINETICS OF AN AUTOCATALYTIC

The parameter r which appears dimensionless quantity

ZYMOGEN

in equations

(18) and (19) is the

k+,Kplo

r= k,K,[l]o

29

REACTION

(20)

-

In equations (17) and (181, G, and G, are given by equations (B8) and (B9) in Appendix B. From equation (16), the rate of the product formation, V, is given by v = - ah exp( At >

cm

Other cases can be solved by proceeding analogously. The behaviour of this system in the steady state (t + a) depends on whether Y is equal to, less than or greater than unity. These cases are analyzed as follows: Case (a). r < 1.

In this case from equation (191, A<0

(22)

and therefore the exponential term in equation (15) as t -+ ~0 (i.e. in the steady state) may be neglected and the steady-state concentration of E is

[El s,=O.

(23)

Similarly, the steady-state concentration of the product P {P = X, Y}, taking into account equations (16) and (22) as t -+ ~0 (i.e. in the steady state), becomes

[PISS=

k;&[Elo islo qql - r)[Ilo

{P=X,Y}.

(24)

It follows, from equations (21) and (22) that the rate of the product formation, V, is approximately null: v=: 0.

(25)

Figure la shows variations of [El and [PI (P = X, Y} versus time in this case (r < 1) obtained by plotting equations (15) and (16) for two chosen sets of rate constants and initial concentrations. Note that the irreversible inhibition of E prevails over the autocatalytic activation of Z. Figure 2a

30

M. C. MANJABACAS et al.

b)

[P]

(b)

0

10

20

~~ (W Figure 1. Plots of [E] and [P] (P = X,Y} versus time obtained from equations (15) and (161, respectively, corresponding to (a) r = 0.2, (b) r = 2 and (c) r = 1. The rate constant values, the initial molar concentrations and the equilibrium constants (M) are: K, = 10m5; K’, = lo-‘; ki2 = 0.5 set-‘; I?‘+~= 0.5 set-‘; K, = 10V3; ki = 0.05 set-‘; [S], = 5.10m4; [El, = 10e9 and for cases (a)(i) [Z], = 1O-6 and [I], = 5.10w4 and (ii) [Z], = 2.10w6 and [I], = 10e3; for case (b)(i) [Z], = 5.10-’ and [IIs = 5.10m4 and (ii) [Z], = 10e6 and [I], = lo-“; for case (c)(i) [Z], = lo-’ and [I], = 5.10e4 and (ii) [Z], = 2.10-’ and [ 1],]10-3.

KINETICS OF AN AUTOCATALYTIC ZYMOGEN REACTION

0.03

0.02 z ,o

A

3

z

0.01 3

0

0

5

10

tim8 o(r) 0.02

0.0

0

10

20

0

~-64

Figure 2. The simulated progress curves obtained as indicated in the Methods section corresponding to Cases (a) r = 0.2, (b) r = 2 and (c) r = 1 using the same values of the rate constants and initial concentrations as in Fig. 1.

31

32

M. C. MANJABACAS et al.

shows the corresponding simulated progress curves obtained from the differential equations. Note that in those cases in which [I] is in great excess, r + 0 and equation (23) will be valid, i.e. the actual final value of E will be zero in all cases. Case (b). r > 1. In this case, and taking equation (19) into account, the eigenvalue, A, is positive, A>0

(26)

and [El,,, [ PI,, {P = X or Y} and v increase exponentially with time. Therefore, equations (15), (16) and (21) are used without further simplification. Note that the rate of product formation in the steady state, according to equation (211, is not constant. Therefore, a finite steady state does not exist. Figure lb shows variations of [E] and [P] versus time in this case obtained by plotting equations (15) and (16). Thus, the autocatalytic activation of Z prevails over the irreversible inhibition of E. In Fig. 2b the corresponding simulated progress curves are shown. Case (c). r = 1. Taking equation (19) into account, we obtain h+O.

(27)

We now use the Maclaurin series expansion:

exp(At)=l+At+-

( Atj2 + (AtI 2! 3!

+ ... .

(28)

Inserting equation (28) into equations (15) and (161, one obtains (AtI [E]=P+PAt+P2,+P3,+*** a(Atj2 [PI=

-aAt-,,--

( Atj3

a(Atj3 31

+ ***

(29)

(P=X,Y}.

(30)

Neglecting the higher-order ( At) terms in equations (29) and (301, one obtains a first-order approximation:

[El = P [PI = Pt

(31) {p=

x,yj,

(32)

KINETICS OF AN AUTOCATALYTIC

ZYMOGEN

RELKTION

33

where p is

(33)

G, being given by equation (B8) in Appendix B. Note that in this case the progress curve of the enzyme species E and the products X or Y increase in a linear way from t = 0, as indicated by equation (32) and the rate of product formation, taking equation (21) into account, reaches a constant value given by equation (33). Hence, both processes, the autocatalytic activation and the irreversible inhibition are balanced. This case and its corresponding simulation are illustrated in Figs. lc and 2c, respectively. For each of the r values used in Figs. 1 and 2, we have shown two of the different progress curves. Note that in all cases, the shapes of the simulated progress curves in Fig. 2 agree with those in Fig. 1 plotted using our approximate solution equations (15) and (16), which shows the consistency of our analysis. The above results obtained for Cases (a)-(c) show that according to whether Y is equal to, lower than or higher than unity, neither route, inhibition or activation, respectively, prevails in the steady-state behaviour of the particular solution. The limit values of r are 0 and ~0.The value 0 corresponds to a pure inhibition process, whereas the value ~0corresponds either to a pure activation process or to an activation process with reversible inhibition. Analogously, we could define parameters similar to r for the other A,Mi cases, which let us study the kinetic behaviour of these systems. Note that according to equation (201, there are many sets of values of the rate constants and [Z], and [I], compatible with a same Y value. Therefore, there are several possible progress curves for E and the product P (X or Y> for each r value, but all of them show the same kinetic behaviour with regard to the dominant route. Under the experimental conditions of a particular reaction which evolves according to scheme A,,M, in Table 3, the rate constants involved in equation (20) have fixed values and, therefore, the Y values are exclusively dependent on the ratio [Z],/[I],, so that by varying this ratio one can modify the catalytic behaviour of the system. 4.2.2. Application of the model to specific biologicalsystems. The above results concerning the competence of both the activation and the inhibition routes in autocatalytic enzyme systems in the presence of an irreversible inhibitor can be applied to some specific real systems. For example, they

34

M. C. MANJABACAS et

al.

allow one to predict the effect of different irreversible inhibitors on autocatalytic zymogen systems. Moreover, the introduction of the parameter r in the analysis of autocatalytic zymogen systems may possibly allow one to explain, from a kinetic viewpoint, the causes for the beginning of some illnesses, such as acute pancreatitis. Bovine trypsinogen activation by trypsin according to the scheme E+Z2EZk”-2E+W where E, Z and W denote trypsin, trypsinogen and a hexapeptide, has recently been studied at 31°C and pH 8.1 (Garcia-Moreno et al., 1991) and the dissociation constant, K,, and the rate constant, k+2, were evaluated. The K, and k,, values yield k+,/K, = (22.2 f 9.4) M-’ set-‘. From this value of the quotient k+,/K, and our results, the effect of the presence of any complexing, irreversible, competitive inhibitor of the trypsin on the autocatalytic system bovine trypsinogen-trypsin can be established. According to equation (20) and using the central value of k+JK,, we have K4[210 ’ = 22’2 ki[ I],,

i?

in iVsec)

(34)

where K,, ki, [Z& and [El, have been defined previously. Therefore, the r value depends on the inhibitor used (through the (K,/ki)) and the quotient [Z],/[ I],. For a given inhibitor, the r value only depends on the quotient El,/[~l,. For those ([Zl,/[QJ va 1ues compatible with condition (5) and such that Y is higher or less than unity, the activation route dominates the inhibition route or vice versa, respectively, at the steady state. For that ([Z],/[ 1]& value for which Y= 1, the activation and inhibition routes are balanced and the trypsin concentration remains constant at the steady state. We refer to ([Z],/[Il,> value corresponding to r = 1 as the critical concentration ratio. In Fig. 3 equation (34) is plotted for different (K,/ki) values. Note the increase of the critical concentration ratio when the quotient K,/k, decreases. Trypsin is synthesized as trypsinogen by the acinar cells of the pancreas and stored there, together with trypsin irreversible inhibitors in vesicles of the cells. The synthesis of the trypsin from its inactive precursor trypsinogen as well as the action of the inhibitors avoid the self-digestion of the pancreas by the trypsin and other proteases activated by trypsin. One theory about the beginning of acute pancreatitis is the self-digestion of the pancreas, which suggests that the trypsinogen is seldom activated into the acinar cells due to factors such as endotoxins, exotoxins, virose infections,

KINETICS OF AN AUTOCATALYTIC ZYMOGEN REACTION

35

0.5

for the autoactivation of bovine Figure 3. Dependence of r on [Z],/[I], trypsinogen at 30°C and pH 8.1 in the presence of the complexing, competitive, irreversible trypsin inhibitors according to equation (34). The abscissa of the intercept of the dashed line with the plotted solid straight lines is the corresponding value of the critical concentration ratio. For lines 3 and 4 these values are 0.045 and 0.023, respectively.

ischemias, anoxias and direct traumatisms (Greenberger et al., 1989) without the normally efficient safety system. This theory is based on the presence of the irreversible trypsin inhibitors to block this activation. According to our theoretical analysis, a possible kinetic explanation for the beginning of the illness is that the factors that initiate the acute pancreatitis affect one or more of the (k+JK1), (K,/ki) and ([Z],/[I],) values, so that the r value becomes higher than unity for at least one of the present irreversible inhibitors. This work was partially supported by grant number PB92-098%C02-CO1 of the Direction General de Investigation Cientifica y TCcnica (DGICYT) (Spain). APPENDIX A Expressions

of the Coefficients

Fi (i = 1,2,. . . ,6) in Equation (13).

F,=k+,~Z~o+k,~~~o+k’,,~~~o+~_, +k,+k’,

+k;,+k;,

+k+,+k+,+k-, (Al)

36

M. C. MANJABACAS et al.

F*=kl[Zlo(k+~+k+,+k_,+k~+k’_,+k:,+k:,) +k,[1lotk_,

+ki,+v+,)

+k+,+k+3+ki+k’1

+k:1[S]Otk_,+k+2+k+3+k_4+ki+k;,+k;3)

+(k_j+k+*+k+~)(k_4+ki+r_l+k;*+ki3) +(k_,+ki)tk’_,+k:,+k:,)+(k-,+k,)k, +(k’-, +k’+2W+3

(A3

F3=k+l[Z]o((k+2+k+3)(km4+ki+k’-l +(k_,+ki)(k’, +k+,[llo((k_,

+k;,+k;,-k+,k+,)

+k:,+k:,)+(k’_,

+k+,+k+,)(k,+k’_,

+k;,)k;,)

+k;,+k:J

+ki(k’,+k:2+k;3)+(k_,+k+2)k+3+(k’l+k:2)k:3) +k’+,[Slo((k_,+k+,+k+,)(k_,+ki+k’+,+k:,) +(k_,

+ k+,)k+,

+ (k-, + k&V+, + k’+,))

+(k_,+k+,+k+,)((k_,+ki)(kL, +(k_,

+k\,+k!+,)+k’+,(k’,

+k+2)(k+3(k-4+ki+kl-l

+k’+,(k_,+ki)(k’_,

+kL,))

+k:,+k’+,))

+k;,)

(A31

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +k’+,((k_,+ki)(kL, -k+,k+,(k_,

+ ki + k’, + ‘:2 + k:,))

+k+,[l]o(k\,(kL, +k+,ki(k_, +(k_,

+k’+,))

+k;,)(k+,+ki+k-1

+k+,+k’_,

+k+z)(k+j

+k’+,[S]o((k_,

+k;,+k’+,)

+ki)(kL,

+k;,

+k:,))

+k+,)((k’+,+k’+,)(k+,+k-,+ki))

+k+,(k_, +k_,((k_,+ki)(kL,

+ k$k_,

+k+*k+3((k_4+ki)(k’,

+ k,, +k:,

+k:,))

+k$,)(k+3+k’+,)

+k+3k’+,(k_,+ki+k’_,

+k’+,(kY,

+k+,)

+k:,))

+k;,+k;,)+(k’_,

+k’+,)(k_,+ki)(k+2+k+3)

+k;,)k;,’ (A41

KINETICS OF AN AUTOCATALYTIC

F,=k+,[ZIo((k+,+k+,)(k_,+k+i)k:3(k11

+k+,[Zlo((k_,

+k:,[Slo((k_,

+k:,)))

+k+,)(k+,k,(k’,

+k;,+k\,)

+k>,)k\,)

+k+,)k+Jk_,

+k+,k,k:,(k’,

+k:,))

-tki)(k>, +k:,))

+(k_, +k+2)k+3(k_4+ki)k’+3(k’1 +k:,) F6=k+,[Zlok+,k,k’+,(k’_,

37

+kT+,)

+k;)(k’,

+(k+, +k,)(k’,

REACTION

+k!+,)

-k+*k+,(k;,(k_,+ki+k’, +(k_,

ZYMOGEN

+k’+,)(k_,

bw

+k+,)

-k+Ik+2k+3[zlo(k:3(k_4 -k,wc’_, +K+2)).

(A61

APPENDIX B

Derivation of the Time Course of the Enzyme Species E and the Products P {P = X or Y) of the Particular Case A,,M,. If we insert condition (14) into equations (Al)-(A6), we obtain: F, =k+,[Z]o+k+,[~lo+k’+,[~lo+k-1

+k+,+k-,+k’,

+k;,

(Bl)

F,=k+IIZlo(k+,+k_,+k’,+k’+~) +k+,[llo(k_,

+k+, +k’,

+k:,)

+k’+,[Slo(k_,

+k+3 +k-,

+k:,)

+(k_,

+k+,Kk_,+k’,

+&,I

+k_,(k’,+k;3)+k_lk+3+k)-lk;3

(B2)

F3=k+l[Z]o(k+3(k_4+kk)l+k:3)+k-,(k’1+k:3)+k’Ik:3) +k+,[Zlo((k_,

+k+,W_,

+k;,[Slo((k_,

+k+3K_4 +k’+,) +k-,k+j +k-,k;d

+(k_,

+k+3)(k_,W,

+k_l(k+3(k-4+k),

+P+,) +k-,k+,

+k’-,k;,)

+k:,)+k;,k’,) +k;,))

+k:,k-,k’,

(B3)

M. C. MANJABACAS et al.

38

+k:,)k’-,

F,=k+,[Zlo(k+3(tk-4

+ k’+,k_,) +k’+,k_,k’_,)

+k+,[Zlo(k:3k’,(k+3+k_,)+k_lk+3(kl-1+k:3)) + k;,[Slo(k_,k’+,(k+,

+ k_,) + k+,k&_,

+ k’+,))

+k_l(k_,k’_l(k+3+k;3)+k+3k;3(k_4+kl1))+k;sk’_lk-4k+3 F, = k+,[Zlok+,k_,k:,k’_,

034)

+ k+,[Zlok_,k+,k’,k’+,

+k;,[Slok_Ik+3k-4k;3+k_,k+3k_4k:3kl-,

(B5)

F~~k+~k~~kl~(k+~kik_~[ZIo-k+~k+~k-~[Zlo)~

(B6)

According to equations (Bl)-(B61, the following simplifications Fi + 00

(i= 1,...,6)

Fi +O

(j=

E

are possible:

1,...,4)

F6 G z=cG5

(B7)

where G,=K,K;[ZIO+K,K;[IIO+K,K,[S]~+K,K,K;

(~8)

G, = K;K,kJZlo(l

(B9)

-r)

and r is given by equation (20) in the text. We divide both the left and the right sides of equation (13) by F, to obtain

(B10) Since l/F, and FJF, (i = 1,2,. . . , 4) are small, the first five terms in the left side may be ignored and (BlO) converts into

,+;=o.

(Bll)

5

Therefore, taking equation (Bll) into account, one finite root is obtained. The roots of equation (BlO) are the same as those of equation (13). Hence, if condition (14) prevails, lAJ-)m h, - hi = A\h

(h =2,...,6)

(B12)

(h=2,...,6;i=l).

U313)

KINETICS OF AN AUTOCATALYTIC ZYMOGEN REACTION

39

Equation (B12) allows us to ignore the exponential terms in equations (7) and (8) because they are smaller in comparison with the others; therefore, the enzyme concentration, [El, and the product concentration, [P] {P = X or Y}, is given by equations (15) and (16) in the text. To obtain the parameters involved in these equations, we must take into account the following: If in equations (9), GO), (11) and (12) we consider condition (14) and divide both the numerator and denominator by k, , k’+ 1k, 4k, 3kT+3 taking into account equations (B9), (B121 and (B13) and G, =

hlA2h3h4h5h6

k+,k’+rk+&+++3 we obtain [El, and the parameters respectively, in the text.

(B141 ’

(T and h which appear in equations

(17) and (181,

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Received 29 June 1994 Revised version accepted 24 May 1995