[R] = b + at +
3 h = 1
r
h
e
l
h
t
4 where l
h
h = 1 – 3 are the eigenvalues of the matrix of coefficients of the Eq. A.8, i.e. the
roots of the equation: l + j l
2
+ F
1
l + F
2
= 0 5
in which: F
1
= k
1
[S] +
k
3
[I] +
k
2
+ k
4
+ k
5
6 F
2
= k
1
k
4
[S] +
k
3
[I] +
k
4
k
2
+ k
5
7 a =
k
1
k
4
k
5
[S] [E]
l
1
l
2
8 [P]
= a
j 9
b = k
1
k
5
[S] [E]
{ j[k
4
l
1
+ l
2
+ l
1
l
2
] − k
4
l
1
l
2
} jl
1
l
2 2
10 The parameters g
i
i = 1 – 3 and r
i
i = 1 – 3 are given in Appendix A.
One of the roots of Eq. 5 is equal to − j. The other two, which we shall name l
1
and l
2,
are the roots of the equation l
2
+ F
1
l + F
2
= 0.
If we admit that k
3
[I] and k
4
are much smaller than k
1
[S] and k
2
, which is true in the case of a slow-binding inhibition Szedlacsek and Dug-
gleby, 1995; Sculley et al., 1996, and when we assume that a pseudo steady state exists in the
catalytic route from the onset of the reaction Waley, 1980; Tatsunami et al., 1981; Topham,
1990; Wang, 1990, i.e. when k
1
[S] , k
2
, k
5
k
3
[I] , k
4
, j, then, using the Eq. A.12 of Ap- pendix A, it is easy to show that:
k
4
B l
1
Bk
3
[I] +
k
4
11 l
2
: −
k
1
[S] +
k
2
+ k
5
12 l
2
l
1
13 Now, Eq. 3 can be simplified to:
[P] = [P] + g
e
− jt
− {[P]
+ g }e
l t
14 where:
l = − k
4
+ K
m
k
3
[I] K
m
+ [S]
16 g =
k
5
j − k
4
[S ][E]
j{K
m
k
3
[I] −
j − k
4
K
m
+ [S]
} 17
If we denote a as d[P]dt at t = 0, then we
obtain from the Eq. 14: a
= k
5
[S ][E]
K
m
+ [S]
18 From Eqs. 15, 16 and 18 we then obtain:
k
4
j = −
[P] l
a 19
3. Materials and methods
The simulated curves were obtained by the numerical solution of the differential equations
Eqs. A1 – A7 using a set of arbitrary but realistic rate-constants and initial concentrations accord-
ing to the conditions defined by Eq. 1. The numerical solution was obtained by the Adams –
Moulton
method in
combination with
the Runge – Kutta fourth-order formulas Gerald and
Wheatley, 1989; Garrido-del Solo et al., 1992. The analytical solutions were obtained using
Laplace transformation Darvey, 1977; Jacquez, 1985.
4. Results and discussion
In this paper, we have, under the conditions described by Eq. 1, derived the kinetic Eqs. 3
and 4, which describe the evolution of the spe- cies P and R in the enzyme reaction that is shown
in Scheme 1.
Fig. 1a shows the progress curve of the product according to the Eq. 3 and is obtained by nu-
merical solution. It can be noted that the two curves overlap. The Fig. 1a shows that the pro-
gress curve of the product reaches a maximal value, [P]
max
, when t = t
max
and that it goes through an inflexion point, [P]
inflex
, at t = t
inflex
. If
Fig. 1. a Plot of Eq. 3 and the simulated progress curve of the product obtained by numerical solution of the system of
differential equations Eqs. A1 – A7. The two curves overlap. b Plot of d[P]dt against time. In both figures, the singular
points [P]
max
and [P]
inflex
are shown. The values used for the initial concentrations and for the rate constants were, [E]
= 1
nM, [I] =
10 mM, [S] =
1 mM, k
1
= 1 × 10
6
M
− 1
s
− 1
, k
2
= 500 s
− 1
, k
3
= 5000 s
− 1
, k
4
= 0.01 s
− 1
, k
5
= 100 s
− 1
and j = 0.1 s
− 1
.
the reaction, i.e. the [P] value can be obtained
from t
max
and t
inflex
. The latter is equal to the concentration of the product, when t = t
= 2t
max
− t
inflex
see Fig. 1a. The evaluation of the co-ordinates of the singu-
lar points of the progress curve of P can be carried out, for example, from a plot of the
d[P]dt against time, since the intersection point of the curve obtained with the abscises axis was
located at t = t
max
. The value of t corresponding to the minimum value of the curve d[P]dt was
equal to t
inflex
see Fig. 1b.
4
.
1
. Particular cases The particular cases of Scheme 1 can be ob-
tained by insertion of the corresponding values for the rate constants, i.e. the Schemes 2 – 4 of
Table 1, and their associated equations, will emerge with j = 0 and k
4
= 0, k
3
= k
4
= 0, respec-
tively see Table 1. In the case in which the disappearance of the
product is due to its irreversible reaction with a reactant Y [Y]
[P] + [R], in which [Y] :
[Y] along the entire course of the reaction, then the
constant j may be substituted for k
j
[Y] in the
equations, where k
j
is the rate constant corre- sponding to the process of transformation of the
product.
5. Kinetic data analysis
