#### ABSTRACT

The analysis of dependence of the length projection of Li vectors of biparametrical inhibited and activated (La) enzymatic reactions from the length projection of vectors of monoparametrical inhibited and activated enzymatic reactions on the basic 0 σ plane in three-dimensional coordinate system, allows to deduct the quadratic forms of equations for the calculation of the constants of inhibition (Ki) and activation (Ka) of enzymes. Examples of calculation of constants are given.

#### Introduction

In previous articles [1-9], devoted to construction of a vector
method representation of enzymatic reactions in the threedimensional coordinate system the properties of L vectors
of enzymatic reactions was analyzed, from which the parametriacal
classification of the types of enzymatic reactions and the equations
for calculation of initial activated (Va ) and inhibited (Vi ) reaction
rates was suggested. In these article the equations of traditional
form (t.f.) for calculation of the constants of activation (Ka) and
absent in practice nontrivial types of biparametrical constants of
inhibition (Ki) of enzymes (Table 1), was deduced.

This work is devoted to deduction of quadratic form (q.f.) of the
equations for calculation of biparametrical constants of inhibition
and activation of enzymes (Table 1t & 1f), opening additional ability
in the analysis of enzyme action what help of quadratic forms of
equation (Table 1q& 1f).

The examples of comparative using traditional and quadratic
form of equations for calculation of Ki and Ka constants of inhibition
and activation are given.

*The symbol of a graph in Figs. 1-15 corresponds to the type of reaction under study. For example: the line (0) characterizes the position of initial (nonactivated) enzymatic reaction, line I – the position of a graph representing the type of activated enzymatic reaction etc.

#### Deduction of Traditional form of Equations

From (Figures 1, 1a and 2) it easy to see, that ( Ii l ) length of (LIi) projection of LIi vector of biparametrically coordinated, i I type (or mixed type [10 -12]) of enzyme inhibition on Pi semiaxis will be determined by divide (i-0) parameters on ( Ii l ) length of (LIi) projection of LIi vector of – by summation of the quadratic (l2) lengths (orthogonal between them self) LIIIi and LIVi projections of monoparametrical LIIIi and LIVi vectors of i III and type of enzyme inhibition, (which also are the coordinate of these vectors) but in the same time they taking adjacent place relative to orthogonal LIi projection of LIi vector (Figure 2), determined by equation:

The Ii l length of LIi projection on 0 σ plane of Figures.; (1 - 2) may be determined as

Having expressed from Eq. (2) the lIIIi length of LIIIi projection of LIIIi vector on P0V semiaxis of coordinate (Figures 1, 1a):

from Eq. (3) – the lIVi length of the second adjacent of LIVi vector projection on semiaxis:

and substituted them in Eq. (4):

we shall obtain traditional form (t.f.) of equation for calculation of the Ii K constant of biparametrically coordinated, i I type, inhibition of enzymes, taking in to consideration the lIi length of orthogonal projection of LIi vector on basic 0 σ plane of Figure (1a):

where , as it follows from (Figures 1&2).

It is analogous for length of adjacent projections: for all other lIIa, lVa … of vectors projections of biparametrical reactions (Figures 1,1a & 2).

#### Deduction of Quadratic form of Equations

From analysis of Equations (1-4) one can easily see that substitution in Eq. (4) of the dimensionless coordinates of the lengths of LIIIi and LIVi vector projections is equal to substitution in this equation of the i /KIIIi and i /KIVi parameters then it is not difficult to become the alternative equations for calculation of i K and a K constants of biparametrical types of inhibition and activation of enzymes. Having substituted in Eq. (4) of the dimensionless coordinates of the lengths of LIIIi and LIVi vector projections is equal to substitution in this equation of the i /KIIIi and i /KIVi parameters.

we find that such as:

this substitution will lead to equation:

or, in quadratic form:

convenient for calculation of constant inhibition of enzymes (Eq. 1, q.f., in Table 1).

It is analogous for all the other equations of biparametrical types of inhibition (Eqs: 2, 5 – 7), and activation (Eqs: 9 – 11 and 14-15 of enzymes, Table 1q & 1f) such as orthogonal projections of correspond L vectors on the basic 0 σ plane, easy to determine by data of two-dimensional (scalar) coordinate system (Figure 2), taking into account orthogonal L projections of tree-dimensional L vectors on basic 0 σ plane of (Figures 1a).

Examples of constants calculation.

### Example 1: Calculation of Constant Inhibition

The inhibitory effect of Tungstic acid anions 2 4 4 WO − (0.5×10− M) on the initial rate of pNPP cleavage by calf alkaline phosphatase (Figure 3). shows that the presence 0.5×10−4M of these anions in the enzyme-substrate system makes the binding of the enzyme to the substrate cleaved difficult and leads to a decrease in the maximum reaction rate (V0= 2.56, V’= 1.74μmol/ (min per μg protein). This meets all the features of the biparametrically coordinated, i I type, of enzyme inhibition (Table 1, line 1). Hence, to calculate the Ii K constant of this phosphatase inhibition it is necessary to use Eq. (5, text), or (Eq. 1, Table 1t & 1f).

Substitution in this equation of the parameters and i obtained by data analysis of (Figure 3) allows the calculation of this constant of enzyme inhibition:

Substitution the same parameters (recalculated to values of constants) in equation (1 of Table 1t & 1f), result into next value of this Ii K constant:

Substitution of these parameters in (Eq. 1, q.f., Table 1)

result into the same value of the constant of enzyme inhibition:

From Eqs. (10 -13) it follows that dimension of constants in all cases, are the molar concentration of inhibitor:

Control. Determine the value of the IIIi K constant of this experiment (Figure 3) by values of Ii K and IVi K constants.

From equations (11) and (12), rewritten to the form,

it follows that:

Substitution the necessary parameters from (Eq. 15) to (Eq. 16), we find that:

which is in good agreement with the experimental value of this constant (Eq. 12).

### Example 2: Calculation of Constant Inhibition

The inhibitory effect of Pyrrolidine dithiocarbonic acid (PDTA) on the initial rate of pNPP cleavage by canine alkaline phosphatase shows that in the presence of 1×10−3M PDTA the parameters and V0= 2.921 μmol/(min per μg protein) change as follows and V ' = 3.616 μmol/(min per μg protein) (Figure 4). This corresponds to the, Vi type, of enzyme pseudoinhibition (Table 1, line 5) and Eq. (5, t.f.) is applicable for calculation of the KVi constant of enzyme inhibition. Substitution all necessary parameters in this equation allows calculation of this constant of enzyme inhibition:

Substitution of recalculated parameters of (Figure 4) and to (Eq. 5, Table 1, q.f.) – result into value of Vi K constant inhibition:

### Example 3: Calculation of Constant Activation

The activating effect of Guanosine (Guo) on canine alkaline phosphatase (Figure 5) shows that in the presence of 1×10−3M Guo the parameters of initial reaction of pNPP cleavage, i. e μmol/(min per μg protein), change as follows: μmol/(min per μg protein). This corresponds to the, Ia type, of unassociative enzyme activation.

Hence, to calculate the KIIa constant of enzyme activation, one should use Eq. (14, t.f., Table 1).

Substitution of the obtained values of parameters in this equation allows calculation of this constant of enzyme activation:

Substitution of the: parameters of this experiment (Figure 5) in (Eq. 14, q.f., Table 1), result in to:

as it was to be expected, result in to the same value of activation constant:

From the length parts of equations: (12), (15), (19) and (21) may to see that all they obeys to the signs of Pifagor’s theorem and this may be used as for calculation any of the third constants by the two others known already and for correction the constants, determined by using any other equations.

Example 4: Calculate the value of KIIIi constant of experiment (Figure 3), by value of KIi and KIVi constants. From equation (1, Table 1t & 1f), rewritten to the quadratic form (23).

it follows that:

Having substitution all necessary parameters from (Eq. 23) to (Eq. 24), we shall become, that:

Ii is analogous for all biparametrically types of catalyzed reactions (Table 1).

Introduction in practice of quadratic forms of equations for calculation of Ki and Ka constants, will facilitates for many authors to interpret obtained data of nontrivial types of inhibition and activation by such definition as «essentially competitive inhibition», «similarly to competitive inhibition » and so on [13 -17].

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