Archives of Pharmacy and Pharmaceutical Sciences

ISSN: 2639-992X

Review Article

Application of the Pyphagor’s Theorem for Correction of K_i and K_a constants of enzyme inhibition and activation

VI Krupyanko^1* and PV Krupyanko²

¹GK Skryabin Institute of Biochemistry and Physiology of Microorganism, Russian Academy of Sciences, 142290 Pushchino, Moscow Region, Prospect Nauki 5, Russia
²Center 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_a constants of enzyme inhibition and activation. Arch Pharm Pharma Sci. 2018; 2: 060-068. DOI: 10.29328/journal.apps.1001011

Copyright License: © 2018 Krupyanko VI, et al. This is an open access article distributed under the Creative Commons Attribution L_icense, 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_a constants

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_i) reaction rates was deduced. In articles [2-9] the equations of traditional form (t.f.) for calculation of the constants of activation (K_a) and absent in practice the equations of nontrivial types of biparametrical constants of inhibition (k_i) of enzymes (Table 1), was deduced [5].

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_a of enzymes (Table 1, 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_a constants of enzyme inhibition and activation are given.

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_Ii vector of biparametrically coordinated, I_i type (or mixed type [10-12] of enzyme inhibition) on P_i semiaxis will be determined by difference: (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_IIIi and L_IVi projections of monoparametrical L_IIIi and L_IVi vectors of III_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 L_Ii projection of L_Ii vector (Figure 2), determined by equation:

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

It is analogous for length of adjacent projections of LII_i, L_Vi … and L_Ia, L_IIa, L_Va … for all other LII_i, L_Vi … , L_Ia, L_IIa, L_Va … three-dimensional vectors of biparametrical reactions (Figure 2).

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_IIIi projection of L_IIIi vector on P0_V’ semiaxis of K’_m V’ I coordinate (Figure 1) and from Eqn. (3)

$l_{IVi}=\frac{K_m^{\text{'}}-K_m^0}{K_m^0}=\frac{i}{K_{IVi}}\text{ (3)}$

the l_IVi length of the second adjacent dimensionless of L_IVi vector projection on PK’_m semiaxis and substituted them in Eqn. (4):

$K_{Ii}={\mathrm{Pr}}_{Pi}{L}_{Ii}/{\mathrm{Pr}}_{{\sigma }_0}{L}_{Ii},\text{ (4)}$

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 lI_i length of orthogonal projection of L_Ii vector on basic σ0 plane of figure 1:

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

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

For example, such as:

$l_{Ii}=\frac{i_{Ii}}{K_{Ii}}\text{ (7)}$

this substitution will leads to equation:

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

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_Ia 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 K’_mV’ 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 PV’ and P0_Km semiaxes (l_Ia = $\sqrt{{(l_{IIIa})}^2+{(l_{IVa})}^2}$ ),

a) in equation of lII_i length projection – by two l_IVa and lIII_i lengths of edged vector projections ($l_{IIi}=\sqrt{{(l_{IIIi})}^2+{(l_{IVa})}^2}$ );

b) in equation of l_Vi length projection – l_IVi and l_IIIa lengths of edged vector projections ($l_{Vi}=\sqrt{{(l_{IIIa})}^2+{(l_{IVi})}^2}$ ) and so on.

Examples of constants correction

Example 1: Calculation of $K_{Ii}$ constant inhibition.

The inhibitory effect of Tungstic acid anions WO ${}_4^{2-}$ (0.510^-4 M)

on the initial rate of pNPP cleavage by calf alkaline phosphatase figure 3 shows that the presence 0.510^-4 M of these anions in the enzyme-substrate system makes the binding of the enzyme to the substrate cleaved ($K_m^0$ = 4.4510^-5 M, $K_m^{\text{'}}$ = 6.5610^-5 M) difficult and leads to a decrease in the maximum reaction rate (V⁰ = 2.56, V’ = 1.74 µmol/(min per µg protein). This meets all the features ( $K_m^{\text{'}}>K_m^0,V^{\text{'}}<V^0,i>0$ ) of the biparametrically coordinated, I_i type, of enzyme inhibition (Table 1, line 1). Hence, to calculate the K_Ij constant of this enzyme inhibition it is necessary to use Eqn. (5, text), or (Eqn. 1, t.f., Table 1).

Substitution in this equation of the parameters $K_m^{\text{'}}>K_m^0,V^{\text{'}}<V^0,i>0$ 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:

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

From Eqns. (10 – 12) it follows that dimension of KI_i constants in all cases, are the molar concentration of inhibitor:

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

Сorrection. Determine the value of the K_IVi constant of this experiment (Figue 3) by values of KI_i and KIII_i constants.

From equation (11), rewritten to the form,

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

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:

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

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:

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

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:

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

rewritten to the forms (K_IVi = 0.714 10^-3 М and _KIIIa = 4.203 10^-3 М)

from which it follows that

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

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, V⁰ = 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:

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

or according Eq. 11, Table 1)

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

Example 4: Calculate the value of KIII_i constant of experiment (Figure 3), by value of KI_i and K_IVi constants.

From equation (1, Table 1, t.f.), rewritten to the form (22)

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

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:

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

Example 5: Calculate the value of _KIIIa constant of experiment (Figure 5), by value of K_Va and K_IVi constants.

From equation (11, Table 1, t.f.), rewritten to the form (22)

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

it follows that:

$K_{IIIa}={(\frac{K_{Va}^2\cdot K_{IVi}^2}{K_{IVi}^2-K_{Va}^2}\cdot M^2)}^{0.\text{5}}={(\frac{{1.753}^2\cdot {1.742}^2}{{1.753}^2-{1.742}^2})}^{\text{2}}\text{ 1}{0}^{-\text{3}}\text{ M }={(\frac{9.435}{0.038})}^{0.\text{5}}\text{ 1}{0}^{-\text{3}}\text{ M}=\text{ 245}.{\text{4}}^{0.\text{5}}\text{ 1}{0}^{-\text{3}}\text{ M }=\text{15}.\text{66 1}{0}^{-\text{3}}\text{ M} \left(\text{26}\right)$

It is no desirable to put K_Va = 1.74∙10^-3 М from (Eqn. 20) in (Eqn. 25), because calculation leads to _KIIIa = 14.43 10^-3M (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).

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_a constants (Examples 1 – 4).