### Review Article

**Application of the Pyphagor’s Theorem for Correction of ***K*_{i} and *K*_{a} constants of enzyme inhibition and activation

*K*and

_{i}*K*constants of enzyme inhibition and activation

_{a}**
VI Krupyanko**^{1*} and PV Krupyanko^{2}

^{1*}and PV Krupyanko

^{2}

^{1}GK Skryabin Institute of Biochemistry and Physiology of Microorganism, Russian Academy of Sciences, 142290 Pushchino, Moscow Region, Prospect Nauki 5, Russia

^{2}Center for Information Technologies on Transport LLC, 142060, Moskow, Domodedovo, MK-Region Barybino, str. Yuzhnaya 17, Russia

***Address for Correspondence:** Vladimir I Krupyanko, GK Skryabin Institute of Biochemistry and Physiology of Microorganism, Russian Academy of Sciences, Town Pushchino, Prospekt Nauki 5, Moscow Region, Russia, 142290, Tel: (495) 625-74-48; Fax: 956-33-70; Email: krupyanko@ibpm.pushchino.ru

**Dates:** **Submitted:** 19 September 2018; **Approved:** 29 October 2018; **Published:** 30 October 2018

**How to cite this article:** Krupyanko VI, Krupyanko PV. Application of the Pyphagor’s Theorem for Correction of *K _{i}* and

*K*constants of enzyme inhibition and activation. Arch Pharm Pharma Sci. 2018; 2: 060-068. DOI: 10.29328/journal.apps.1001011

_{a}**Copyright License:** © 2018 Krupyanko VI, et al. This is an open access article distributed under the Creative Commons Attribution *L _{i}*cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Keywords:** Quadratic forms of equations for correction of *K _{i}* and

*K*constants

_{a}### Abstract

The analysis of dependence of the length projection of *L _{i}* vectors of biparametrical inhibited and activated (

*L*) enzymatic reactions from the length projection of vectors of monoparametrical inhibited and activated enzymatic reactions on the basic σ0 plane in three-dimensional coordinate system, allows to deduct the quadratic forms of equations for the correction of the constants of inhibition (

_{a}*k*) and activation (

_{i}*K*) of enzymes. Examples of correction of constants are given.

_{a}### Introduction

The study inhibition of enzymes helps to synthesize the drugs from poisoning of living organisms.

In previous articles [1-9], devoted to construction of a vector method representation of enzymatic reactions in the three-dimensional K’m 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*) reaction rates was deduced. In articles [2-9] the equations of traditional form (

_{i}*t.f.*) for calculation of the constants of activation (

*K*) and absent in practice the equations of nontrivial types of biparametrical constants of inhibition (

_{a}*k*) of enzymes (Table 1), was deduced [5].

_{i}This work is devoted to deduction of quadratic form (*q.f.*) of the equations for correction of biparametrical constants of inhibition *K _{i}* and activation

*K*of enzymes (Table 1,

_{a}*q.f.*), opening additional ability in the analysis of enzyme action what help of these equations.

The examples of comparative using traditional and quadratic form of equations for correction of *K _{i}* and

*K*constants of enzyme inhibition and activation are given.

_{a}**Deduction of traditional form of equations**

From Figures 1 and 2 it easy to see, that (${l}_{Ii}$
) length of (*L _{Ii}*) projection of

*L*vector of biparametrically coordinated,

_{Ii}*I*type (or mixed type [10-12] of enzyme inhibition) on

_{i}*P*semiaxis will be determined by difference: (

_{i}*i*-0) parameters, The basic σ0 plane (Figure 2), actually is orthogonal projection of three-dimensional L vectors of (Figure 1), i.e. the scalar magnitudes (orthogonal between them self)

*L*and

_{IIIi}*L*projections of monoparametrical

_{IVi}*L*and

_{IIIi}*L*vectors of II

_{IVi}*I*and IVi type of enzyme inhibition, (which also are the coordinate of these vectors) but in the same time they taking adjacent place relative to orthogonal

_{i}*L*projection of

_{Ii}*L*vector (Figure 2), determined by equation:

_{Ii}${l}_{Ii}=\sqrt{{({l}_{IIIi})}^{2}+{({l}_{IVi})}^{2}}\text{(1)}$

It is analogous for length of adjacent projections of LI*I _{i}*,

*L*… and

_{Vi}*L*,

_{Ia}*L*,

_{IIa}*L*… for all other LI

_{Va}*I*,

_{i}*L*… ,

_{Vi}*L*,

_{Ia}*L*,

_{IIa}*L*… three-dimensional vectors of biparametrical reactions (Figure 2).

_{Va}Having expressed from Eqn. (2)

${l}_{IIIi}=\frac{{V}^{0}-{V}^{\text{'}}}{{V}^{\text{'}}}=\frac{i}{{K}_{IIIi}}\text{(2)}$

the *l _{IIIi}* length of dimensionless of

*L*projection of L

_{IIIi}*vector on*

_{IIIi}*P*0

*semiaxis of*

_{V’}*K’*coordinate (Figure 1) and from Eqn. (3)

_{m}V’ I${l}_{IVi}=\frac{{K}_{m}^{\text{'}}-{K}_{m}^{0}}{{K}_{m}^{0}}=\frac{i}{{K}_{IVi}}\text{(3)}$

the *l _{IVi}* length of the second adjacent dimensionless of

*L*vector projection on

_{IVi}*PK’*semiaxis and substituted them in Eqn. (4):

_{m}${K}_{Ii}={\mathrm{Pr}}_{Pi}{L}_{Ii}/{\mathrm{Pr}}_{{\sigma}_{0}}{L}_{Ii},\text{(4)}$

we shall obtain traditional form (*t.f.*) of equation for calculation of the ${K}_{Ii}$
constant of biparametrically coordinated, *I _{i}* type, inhibition of enzymes, taking in to consideration the l

*I*length of orthogonal projection of

_{i}*L*vector on basic σ0 plane of figure 1:

_{Ii}${K}_{Ii}=\frac{i}{{\left({\left(\frac{{K}_{m}^{\text{'}}-{K}_{m}^{0}}{{K}_{m}^{0}}\right)}^{2}+{\left(\frac{{V}^{0}-{V}^{\text{'}}}{{V}^{\text{'}}}\right)}^{2}\right)}^{0.5}}\text{(5)}$

Similarly for deduction of all biparametrical equations of table 1 [5,7,8].

**Deduction of quadratic form of equations**

From analysis of equations (1 – 4) one can easily see that substitution in Eqn. (4) of the dimensionless coordinates of the lengths of ${L}_{IIIi}$ and ${L}_{IVi}$ vector projections is equal to substitution in this equation of the $i/{K}_{IIIi}$ and $i/{K}_{IVi}$ parameters

${l}_{Ii}=\sqrt{{\left(\frac{i}{{K}_{IIIi}}\right)}^{2}+{\left(\frac{i}{{K}_{IVi}}\right)}^{2}}\text{(6)}$

then it is not difficult to become the quadratic forms of equations for correction of *K _{i}* and

*K*constants of biparametrical types of inhibition and activation of enzymes (Table 1).

_{a}For example, such as:

${l}_{Ii}=\frac{{i}_{Ii}}{{K}_{Ii}}\text{(7)}$

this substitution will leads to equation:

${K}_{Ii}=i/{l}_{Ii}=i/(i/{\left(\frac{1}{{K}_{IIIi}^{2}}+\frac{1}{{K}_{IVi}^{2}}\right)}^{0,5})\text{}=\text{1}/{\left(\frac{1}{{K}_{IIIi}^{2}}+\frac{1}{{K}_{IVi}^{2}}\right)}^{0,5}\text{(8)}$

or, in quadratic form:

$\frac{1}{{K}_{Ii}^{2}}=\frac{1}{{K}_{IIIi}^{2}}+\frac{1}{{K}_{IVi}^{2}},\text{(9)}$

convenient for correction of constant inhibition of enzymes (Eqn. 1, *q.f.*, Table 1).

It is analogous for all the other equations of biparametrical types of inhibition (Eqns. 2, 5 – 7), and activation (Eqns. 9 – 11 and 14, 15) of enzymes, (Table 1, *q.f.*) taking into account, orthogonal projections of tree-dimensional L vectors on the basic σ0 plane of (Figure 1) by data analysis of correspond position two-dimensional scalar *L* projections of L vectors on these vectors in *K’ _{mV}*’ coordinate system (Figure 2). For example, the orthogonal projection length of

*L*vector of, ${I}_{a}$ type, activation will be determined by analogous common equation (1, text) of enzyme activation that is located in the σ0 plane of scalar

_{Ia}*K’*’ coordinate system (Figure 2, in IInd quadrant) and edged by two ${L}_{IIIa}$ and ${L}_{IVa}$ lengths of edged projection of this vector on the

_{mV}*PV’*and

*P*0

*semiaxes (*

_{Km}*l*= $\sqrt{{({l}_{IIIa})}^{2}+{({l}_{IVa})}^{2}}$ ),

_{Ia}a) in equation of lI*I _{i}* length projection – by two

*l*and lII

_{IVa}*I*lengths of edged vector projections (${l}_{IIi}=\sqrt{{({l}_{IIIi})}^{2}+{({l}_{IVa})}^{2}}$ );

_{i}b) in equation of *l _{Vi}* length projection –

*l*and

_{IVi}*l*lengths of edged vector projections (${l}_{Vi}=\sqrt{{({l}_{IIIa})}^{2}+{({l}_{IVi})}^{2}}$ ) and so on.

_{IIIa}**Examples of constants correction **

**Example 1:** Calculation of ${K}_{Ii}$
constant inhibition.

The inhibitory effect of Tungstic acid anions WO ${}_{4}^{2-}$
(0.510^{-4} M)

on the initial rate of pNPP cleavage by calf alkaline phosphatase figure 3 shows that the presence 0.510^{-4} M of these anions in the enzyme-substrate system makes the binding of the enzyme to the substrate cleaved (${K}_{m}^{0}$
= 4.4510^{-5} M, ${K}_{m}^{\text{'}}$
= 6.5610^{-5} M) difficult and leads to a decrease in the maximum reaction rate (*V ^{0}* = 2.56,

*V’*= 1.74 µmol/(min per µg protein). This meets all the features ( ${K}_{m}^{\text{'}}>{K}_{m}^{0},{V}^{\text{'}}<{V}^{0},i>\text{}0$ ) of the biparametrically coordinated,

*I*type, of enzyme inhibition (Table 1, line 1). Hence, to calculate the

_{i}*K*constant of this enzyme inhibition it is necessary to use Eqn. (5, text), or (Eqn. 1,

_{Ij}*t.f.*, Table 1).

Substitution in this equation of the parameters ${K}_{m}^{\text{'}}>{K}_{m}^{0},{V}^{\text{'}}<{V}^{0},i>\text{}0$ and $i$ obtained by data analysis of (Figure 3) allows the calculation of this constant of enzyme inhibition:

${K}_{Ii}=\frac{0.5\cdot {10}^{-4}M}{{\left({\left(\frac{6.56-4.45}{4.45}\right)}^{2}+{\left(\frac{2.56-1.74}{1.74}\right)}^{2}\right)}^{0.5}}=\text{7}.\text{48}\times \text{1}0-\text{5M}.\left(\text{1}0\right)$

Substitution of these parameters rewritten to forms with (${K}_{IIIi}$
= 1.062 10^{-4} M, ${K}_{IVi}$
= 1.055 10^{-4} M) in (Eqn. 1, *q.f.*, Table 1)

$\frac{1}{{K}_{Ii}^{2}}=\frac{1}{{K}_{IIIi}^{2}}+\frac{1}{{K}_{IVi}^{2}}=\left(\frac{1}{{1.062}^{2}}+\frac{1}{{1.055}^{2}}\right),\left(\text{11}\right)$

result in to the same value of the constant of enzyme inhibition:

${K}_{Ii}=\frac{1}{{\left(\frac{1}{{K}_{IIIi}^{2}}+\frac{1}{{K}_{IVi}^{2}}\right)}^{0.5}}={\left(\frac{({K}_{IVi}^{2}\cdot {K}_{IIIi}^{2})\cdot {({10}^{-4})}^{2}\cdot {({10}^{-4})}^{2}}{({K}_{IIIi}^{2}+{K}_{IVI}^{2})\cdot {({10}^{-4})}^{2}}\cdot \frac{{M}^{4}}{{M}^{2}}\right)}^{0.5}=\sqrt{0.5602},\sqrt{{({10}^{-4})}^{2}}\cdot \sqrt{{M}^{2}}=\text{}0.\text{7485}\xb7\text{1}{0}^{-\text{4}}\xb7\u041c.\left(\text{12}\right)$

From Eqns. (10 – 12) it follows that dimension of K*I _{i}* constants in all cases, are the molar concentration of inhibitor:

${K}_{Ii}=\sqrt{{i}^{4}/{i}^{2}}=i[\u041c].\left(\text{13}\right)$

**Сorrection.** Determine the value of the *K _{IVi}* constant of this experiment (Figue 3) by values of K

*I*and KII

_{i}*I*constants.

_{i}From equation (11), rewritten to the form,

$(\frac{1}{{K}_{Ii}^{2}}=\frac{1}{{K}_{IIIi}^{2}}+\frac{1}{{K}_{IVi}^{2}})\text{}=\text{}(\frac{1}{{0.7485}^{2}}=\frac{1}{{K}_{IVi}^{2}}+\frac{1}{{1.062}^{2}}),\left(\text{14}\right)$

it follows that:

${K}_{IVi}={(\frac{{K}_{Ii}^{2}\cdot {K}_{IIIi}^{2}}{{K}_{IIIi}^{2}-{K}_{Ii}^{2}})}^{0.\text{5}}\text{1}0-\text{4M}.\left(\text{15}\right)$

Substitution the necessary parameters from (Eqn. 14) to (Eqn. 15), we find that:

${K}_{IVi}=\text{}{(\frac{{0.7485}^{2}\cdot {1.062}^{2}}{{1.062}^{2}-{0.7485}^{2}})}^{0.\text{5}}\xb7\text{1}{0}^{-\text{4}}\text{M}=\text{}{(\frac{0.5595\cdot 1.1278}{1.1276-0.5595})}^{0.\text{5}}\xb7\text{1}{0}^{-\text{4}}\text{M}=\text{1}.{\text{11295}}^{0,\text{5}}\xb7\text{1}{0}^{-\text{4}}\text{M}=\text{1}.0\text{549}\xb7\text{1}{0}^{-\text{4}}\text{M},\left(\text{16}\right)$

which is in good agreement with the experimental value of this constant (Eqn. 10).

**Example 2:** Calculation of *K _{Vi}* 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^{-3} М PDTA the parameters ${K}_{m}^{0}$
= 4.69·10^{-5} М and ${V}^{0}$
= 2.921 µmol/(min per µg protein) change as follows: ${K}_{m}^{\text{'}}=$
11.26·10^{-5} М and ${V}^{\text{'}}$
= 3.616 µmol/(min per µg protein) (Figure 4). This corresponds to the, Vi type, of enzyme pseudoinhibition ( ${K}_{m}^{\text{'}}$
> ${K}_{m}^{0}$
, ${V}^{\text{'}}$
> ${V}^{0}$
, $i$
> 0) (Table 1, line 5) and Eqn. (5, *t.f.*) is applicable for calculation of the *K _{Vi}* constant of enzyme inhibition. Substitution all necessary parameters in this equation allow calculation of this constant of enzyme inhibition:

${K}_{Vi}=\frac{1\cdot {10}^{-3}M}{{\left({\left(\frac{11.26-4.69}{4.69}\right)}^{2}+{\left(\frac{3.616-2.92}{2.92}\right)}^{2}\right)}^{0.5}}=\text{7}.0\text{4}\xb7\text{1}{0}^{-\text{4}}\text{\u041c}.\left(\text{17}\right)$

Substitution all necessary parameters from of recalculated parameters of (Figure 4) to (Eqn. 5, Table 1, *q.f.*) – result in to value of *K _{Vi}* constant inhibition:

$\frac{1}{{K}_{Vi}^{2}}=\text{}(\frac{1}{{K}_{IIIa}^{2}}+\frac{1}{{K}_{IVi}^{2}}),\left(\text{18}\right)$

rewritten to the forms (*K _{IVi}* = 0.714 10

^{-3}М and

*= 4.203 10*

_{KIIIa}^{-3}М)

from which it follows that

$\begin{array}{l}{K}_{Vi}=\frac{1}{{\left(\frac{1}{{K}_{IIIai}^{2}}+\frac{1}{{K}_{IVi}^{2}}\right)}^{0.5}}={\left(\frac{({K}_{IVi}^{2}\cdot {K}_{IIIa}^{2})\cdot {({10}^{-4})}^{2}\cdot {({10}^{-4})}^{2}}{({K}_{IIIai}^{2}+{K}_{IVI}^{2})\cdot {({10}^{-4})}^{2}}\cdot \frac{{M}^{4}}{{M}^{2}}\right)}^{0.5}=\\ {\left(\frac{0.51\cdot 17.67}{0.51+17.67}\right)}^{0.5}={\left(\frac{9.093}{18.175}\right)}^{0.5}=\sqrt{0.496}=0.\text{7}0\text{41}\xb7\text{1}{0}^{-\text{3}}\text{\u041c}\text{.}\left(\text{19}\right)\end{array}$

**Example 3:** Calculation of *K _{Va}* constant activation.

The results of study presented in figure 5 show that: the parameters of initial non-activated reaction of pNPP cleavage by alkaline phosphatase ${K}_{m}^{0}$
= 5.45·10^{-5} М, *V ^{0}* = 9.363 µmol/(min µg protein) in the presence of 0.001 M of activator change as follows: ${K}_{m}^{\text{'}}$
= 3.47·10

^{-5}М,

*V’*= 8.803 µmol/(min µg protein), which satisfies all the features of type

*Va*of enzyme pseudoactivation (line 11, Table 1).

Substitution of the experimental parameters of (Figure 5, in Eqn. 11, *t.f.*, Table 1) gives the following value of *K _{Va}* constant:

${K}_{Va}=\frac{1\cdot {10}^{-3}M}{\sqrt{{\left(\frac{5.45-3.47}{3.47}\right)}^{2}+{\left(\frac{9.363-8.803}{8.803}\right)}^{2}}}=\text{1}.\text{74}\xb7\text{1}{0}^{-\text{3}}\text{\u041c},\left(\text{2}0\right)$

or according Eq. 11, Table 1)

$\begin{array}{l}{K}_{Va}=\frac{1}{{\left(\frac{1}{{K}_{IIIa}^{2}}+\frac{1}{{K}_{IVi}^{2}}\right)}^{0.5}}={\left(\frac{({K}_{IVi}^{2}\cdot {K}_{IIIa}^{2})\cdot {({10}^{-4})}^{2}\cdot {({10}^{-4})}^{2}}{({K}_{IIIa}^{2}+{K}_{I{V}_{i}}^{2})\cdot {({10}^{-4})}^{2}}\cdot \frac{{M}^{4}}{{M}^{2}}\right)}^{0.5}=\\ {\left(\frac{{1.753}^{2}\cdot {15.72}^{2}}{{1.753}^{2}+{15.72}^{2}}\right)}^{0.5}\text{1}{0}^{-\text{3}}\text{M}={\left(\frac{759.39}{258.19}\right)}^{0.5}=\text{1}.\text{742}\xb7\text{1}{0}^{-\text{3}}\text{\u041c}.\left(\text{21}\right)\end{array}$

Example 4: Calculate the value of KII*I _{i}* constant of experiment (Figure 3), by value of K

*I*and

_{i}*K*constants.

_{IVi}From equation (1, Table 1, *t.f.*), rewritten to the form (22)

$(\frac{1}{{K}_{Ii}^{2}}=\frac{1}{{K}_{IIIi}^{2}}+\frac{1}{{K}_{IVi}^{2}})\text{}=(\frac{1}{{0.7485}^{2}}=\frac{1}{{K}_{IIIi}^{2}}+\frac{1}{{1.055}^{2}}),\left(\text{22}\right)$

it follows that:

${K}_{IVi}={(\frac{{K}_{Ii}^{2}\cdot {K}_{IIIi}^{2}}{{K}_{IIIi}^{2}-{K}_{Ii}^{2}}\cdot {M}^{2})}^{0,\text{5}}.\left(\text{23}\right)$

Having substituted all necessary parameters from (Eqn. 22) into (Eqn. 23), the next value of this constant is received:

${K}_{IIIi}={[(\frac{{0.748}^{2}\cdot {1.055}^{2}}{{1.055}^{2}-{0.748}^{2}}\cdot {({10}^{-4})}^{2}\cdot {M}^{2})]}^{0,\text{5}}={(\frac{0.5595\cdot 1.113}{1.113-0.5595})}^{0,\text{5}}\xb7\text{1}{0}^{-\text{4}}\text{M}=\text{1}.{\text{125}}^{0.\text{5}}\xb7\text{1}{0}^{-\text{4}}\text{M}=\text{1}.0\text{61}\xb7\text{1}{0}^{-\text{4}}\text{M}.\left(\text{24}\right)$

Example 5: Calculate the value of * _{KIIIa}* constant of experiment (Figure 5), by value of

*K*and

_{Va}*K*constants.

_{IVi}From equation (11, Table 1, *t.f.*), rewritten to the form (22)

$(\frac{1}{{K}_{Va}^{2}}=\frac{1}{{K}_{IIIa}^{2}}+\frac{1}{{K}_{IVi}^{2}})\text{}=(\frac{1}{{1.742}^{2}}=\frac{1}{{K}_{IIIa}^{2}}+\frac{1}{{1.753}^{2}}),\left(\text{25}\right)$

it follows that:

${K}_{IIIa}={(\frac{{K}_{Va}^{2}\cdot {K}_{IVi}^{2}}{{K}_{IVi}^{2}-{K}_{Va}^{2}}\cdot {M}^{2})}^{0.\text{5}}={(\frac{{1.753}^{2}\cdot {1.742}^{2}}{{1.753}^{2}-{1.742}^{2}})}^{\text{2}}\text{1}{0}^{-\text{3}}\text{M}={(\frac{9.435}{0.038})}^{0.\text{5}}\text{1}{0}^{-\text{3}}\text{M}=\text{245}.{\text{4}}^{0.\text{5}}\text{1}{0}^{-\text{3}}\text{M}=\text{15}.\text{661}{0}^{-\text{3}}\text{M}\left(\text{26}\right)$

It is no desirable to put *K _{Va}* = 1.74∙10

^{-3}М from (Eqn. 20) in (Eqn. 25), because calculation leads to

*= 14.43 10*

_{KIIIa}^{-3}M (instead 15.66 10

^{-3}M), such as the first constant is not in Pythagorean´s «bundle» (Egn. 25).

It is analogous for all biparametrical types of catalyzed reactions (Table 1).

### Discussion

The analysis of data obtained shows that:

1) The values of the constants of biparametrical types of inhibition (Eqns. 1, 2, 5 – 7) and activation (Eqns. 9 – 11, 14, 15), are not subjected to additive dependencies on the values of the constants of monoparametrical types of inhibition (Eqns. 3, 4) and activation (Eqns. 12, 13) of the enzymes (Table 1);

${K}_{Ii}\ne {K}_{IVi}+{K}_{IIIi}.\left(\text{27}\right)$

They subjected to geometrical relationships (Pyphagorean theorem):

${(1/{K}_{Ii})}^{2}={(1/{K}_{IVi})}^{2}+{(1/{K}_{IIIi})}^{2},\left(\text{28}\right)$

2) this opens an array of possibilities for calculation and correction of the values of *K _{i}* and

*K*constants (Examples 1 – 4).

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