The Quadratic forms of Equations for Calculation of the K i and K a Constants of Enzyme Inhibition and Activation

The analysis of dependence of the length projection of L i vectors of biparametrical inhibited and activated (L a ) enzymatic reactions from the length projection of vectors of monoparametrical inhibited and activated enzymatic reactions on the basic in three-dimensional m K V I coordinate system, allows to deduct the quadratic forms of equations for the calculation of the constants of inhibition i K and activation enzymes. Examples of calculation of constants are given.


Introduction
In previous articles [1][2][3][4][5][6][7][8][9], devoted to construction of a vector method representation of enzymatic reactions in the threedimensional ' ' m K V I 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 (V a ) and inhibited (V i ) reaction rates was suggested. In these article the equations of traditional form (t.f.) for calculation of the constants of activation (K a ) and absent in practice nontrivial types of biparametrical constants of inhibition (K i ) 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 & (Table 1q& 1f).
The examples of comparative using traditional and quadratic form of equations for calculation of K i and K a constants of inhibition and activation are given.

Deduction of Traditional form of Equations
From (Figures 1, 1a and 2) it easy to see, that ( I i l ) length of (L Ii ) projection of L Ii vector of biparametrically coordinated, i I type (or mixed type [10 -12]) of enzyme inhibition on P i semiaxis will be determined by divide (i-0) parameters on ( I i l ) length of (L Ii ) projection of L Ii vector of -by summation of the quadratic (l 2 ) lengths (orthogonal between them self) L IIIi Associa-

Tive Activation
Competi-Tive Activa-Tion   IVa  IIIa  IVi  IIa  IIIa  IVi  I  VIIi  VIIi  Vi  IIi  IVa  IIIa  IVi  II  Va  a  IIIa  IVi  IIIi  I  Vi    PV coordinate semiaxes the same as in Figure. 1 and in the text, the magnitude of ϕ angle about 150.

Ii
IIIi IVi 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).

Example 1: Calculation of Constant Inhibition
The inhibitory effect of Tungstic acid anions Substitution of these parameters in (Eq. 1, q.f., Table 1 result into the same value of the constant of enzyme inhibition: From equations (11) and (12), rewritten to the form, it follows 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

Example 3: Calculation of Constant Activation
The activating effect of Guanosine (Guo) on canine alkaline phosphatase ( Figure 5) shows that in the presence of  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. 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 i K and a K 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].