VI Krupyanko 1 * and PV Krupyanko 2. The analysis of dependence of the length projection of L i. ) and inhibited (v i

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1 Open Access Archives of Pharacy and Pharaceutical Sciences Review Article Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation VI rupyanko * and PV rupyanko G Skryabin Institute of Biocheistry and Physiology of Microorganis, Russian Acadey of Sciences, 49 Pushchino, Moscow Region, Prospect Nauki 5, Russia Center for Inforation Technologies on Transport LLC, 46, Moskow, Doodedovo, M- Region Barybino, str. Yuzhnaya 7, Russia *Address for Correspondence: Vladiir I rupyanko, G Skryabin Institute of Biocheistry and Physiology of Microorganis, Russian Acadey of Sciences, Town Pushchino, Prospekt Nauki 5, Moscow Region, Russia, 49, Tel: (495) ; Fax: ; Eail: krupyanko@ibp.pushchino.ru Subitted: 9 Septeber 8 Approved: 9 October 8 Published: 3 October 8 Copyright: 8 rupyanko VI, et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited eywords: Quadratic fors of equations for correction of i and a constants Abstract The analysis of dependence of the length projection of L i vectors of biparaetrical inhibited and activated (L a ) enzyatic reactions fro the length projection of vectors of onoparaetrical inhibited and activated enzyatic reactions on the basic σ plane in three-diensional V I coordinate syste, allows to deduct the quadratic fors of equations for the correction of the constants of inhibition ( i ) and activation ( a ) of enzyes. Exaples of correction of constants are given. Introduction The study inhibition of enzyes helps to synthesize the drugs fro poisoning of living organiss. In previous articles [-9], devoted to construction of a vector ethod representation of enzyatic reactions in the three-diensional V I coordinate syste the properties of L vectors of enzyatic reactions was analyzed, fro which the paraetriacal classiication of the types of enzyatic reactions and the equations for calculation of initial activated (v a ) and inhibited (v i ) reaction rates was deduced. In articles [-9] the equations of traditional for (t.f.) for calculation of the constants of activation ( a ) and absent in practice the equations of nontrivial types of biparaetrical constants of inhibition ( i ) of enzyes (Table ), was deduced [5]. This work is devoted to deduction of quadratic for (q.f.) of the equations for correction of biparaetrical constants of inhibition i and activation a of enzyes (Table, q.f.), opening additional ability in the analysis of enzye action what help of these equations. The exaples of coparative using traditional and quadratic for of equations for correction of i and a constants of enzye inhibition and activation are given. Deduction of traditional for of equations Fro Figures and it easy to see, that (l I ) length of (L ) projection of L vector of biparaetrically coordinated, I i type (or ixed type [-] of enzye inhibition) on P i seiaxis will be deterined by difference: (i- ) paraeters, The basic σ plane (Figure ), actually is orthogonal projection of three-diensional L vectors of (Figure ), i.e. the scalar agnitudes (orthogonal between the self) L II and L projections of onoparaetrical L II and L vectors of III i and IV i type of enzye inhibition, (which How to cite this article: rupyanko VI, rupyanko PV. Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation. Arch Phar Phara Sci. 8; :

2 Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation Table : Equations for calculation of i and a constants (in traditional for). *The sybol of a graph in Figure. -5 corresponds to the type of reaction under study. For exaple: the line () characterizes the position of initial (nonactivated) enzyatic reaction, line I the position of a graph representing the I a type of activated enzyatic reaction etc. Published: October 3, 8 6

3 IVa IIIa II Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation Table continued V L Ia L Ia L IIIa L IIIa L IVa l P L L IVa L II L L L I L II L a, i v Figure : Three-diensional (packed) V I syste of rectangular coordinates with coincident P i and P a seiaxes of olar concentrations of inhibitor i and activator a. The sybols of kinetic paraeters:, V, three-diensional vectors: L, L L Ia, L IVa, L, L, and their projections L, L L Ia, L IVa on the basic σ plane as well as the sybols of projections of directing planes σ, σ II, σ IVα, σ IIIα on the P, P, P and PV coordinate seiaxes the sae as in the text. V Published: October 3, 8 6

4 VIIa/Vi Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation V L Ia L IIIa L VIa L IIa LVIIa L Vi L IVa P L L Va L LVI I L V L II L Va/VI V Figure : Two-diensional (scalar) V coordinate syste. The sybols of kinetic paraeters:, V,..., the projections L, L LIa, LIVa of three-diensional vectors: L, L LIa, LIVa on the basic σ plane and sybols of P, P, P and PV coordinate seiaxes the sae as in Figure and in the text. V also are the coordinate of these vectors) but in the sae tie they taking adjacent place relative to orthogonal L projection of L vector (Figure ), deterined by equation: l ( l II ) ( l ) () It is analogous for length of adjacent projections of L I, L Vi and L Ia, L IIa, L Va for all other L I, L Vi, L Ia, L IIa, L Va three-diensional vectors of biparaetrical reactions (Figure ). Having expressed fro Eqn. () l II V V V i II, () the l II length of diensionless of L II projection of L II vector on P V seiaxis of V I coordinate (Figure ) and fro Eqn. (3) l i (3) the l length of the second adjacent diensionless of L vector projection on P seiaxis and substituted the in Eqn. (4): Pr Pi L / Pr s L, (4) we shall obtain traditional for (t.f.) of equation for calculation of the constant of biparaetrically coordinated, I i type, inhibition of enzyes, taking in to consideration the l length of orthogonal projection of L vector on basic σ plane of igure : i.. 5 V V V Siilarly for deduction of all biparaetrical equations of table [5,7,8]. Deduction of quadratic for of equations Fro analysis of equations ( 4) one can easily see that substitution in Eqn. (4) of the diensionless coordinates of the lengths of L II and L vector projections is equal to substitution in this equation of the i / II and i / paraeters II æ i ö æ i ö l ç è çè, (6) then it is not dificult to becoe the quadratic fors of equations for correction (5) Published: October 3, 8 63

5 Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation of i and a constants of biparaetrical types of inhibition and activation of enzyes (Table ). For exaple, such as: i l, (7) this substitution will leads to equation:,5,5 i/l i/(i/ ) / II, (8) II or, in quadratic for:, (9) II convenient for correction of constant inhibition of enzyes (Eqn., q.f., Table ). It is analogous for all the other equations of biparaetrical types of inhibition (Eqns., 5 7), and activation (Eqns. 9 and 4, 5) of enzyes, (Table, q.f.) taking into account, orthogonal projections of tree-diensional L vectors on the basic σ plane of (Figure ) by data analysis of correspond position two-diensional scalar L projections of L vectors on these vectors in V coordinate syste (Figure ). For exaple, the orthogonal projection length of L Ia vector of, I a type, activation will be deterined by analogous coon equation (, text) of enzye activation that is located in the σ plane of scalar V coordinate syste (Figure, in II nd quadrant) and edged by two L IIIa and L IVa lengths of edged projection of this vector on the PV and P seiaxes (l Ia l ) ( l ) ), ( IIIa IVa a) in equation of l I length projection by two l IVa and l II lengths of edged vector projections ( l I l ) ( l ) ); ( Vi ( II IVa b) in equation of l Vi length projection l and l IIIa lengths of edged vector projections l ) ( l ) ) and so on. ( IIIa l Exaples of constants correction Exaple : Calculation of constant inhibition. The inhibitory effect of Tungstic acid anions WO 4 (.5-4 M) on the initial rate of pnpp cleavage by calf alkaline phosphatase igure 3 shows that the presence.5-4 M of these anions in the enzye-substrate syste akes the binding of the enzye to the substrate cleaved ( M, M) dificult and leads to a decrease in the axiu reaction rate (V.56, V.74 μol/(in per μg protein). This eets all the features ( >, V < V, i > ) of the biparaetrically coordinated, I i type, of enzye inhibition (Table, line ). Hence, to calculate the Ij constant of this enzye inhibition it is necessary to use Eqn. (5, text), or (Eqn., t.f., Table ). Substitution in this equation of the paraeters,, V, V and i obtained by data analysis of (Figure 3) allows the calculation of this constant of enzye inhibition:.5 M æ ö æ - ö æ - ö ç çè 4.45 çè.74 è M. () Substitution of these paraeters rewritten to fors with ( II.6-4 M,.55-4 M) in (Eqn., q.f., Table ) II, ().6.55 Published: October 3, 8 64

6 Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation 3 /v /[pnpp], 4 M - Figure 3: Inhibitory effect of anions WO 4 calf alkaline phosphatase. on the initial rate v, ol/(in per g protein) of pnpp cleavage by Note: line the concentration of WO 4 is.5-4 M; line () the inhibitor is absent. result in to the sae value of the constant of enzye inhibition:. 5 II.56 4 ( ) - æ( ) ( ) ( ) M ö ç è II -4 ( II IVI ) ( ) M.5 M М. () Fro Eqns. ( ) it follows that diension of constants in all cases, are the olar concentration of inhibitor: i 4 / i i [М]. (3) Сorrection. Deterine the value of the constant of this experient (Figue 3) by values of and II constants. Fro equation (), rewritten to the for, ( II ) ( ), (4) it follows that: II ( II ).5-4 M. (5) Substitution the necessary paraeters fro (Eqn. 4) to (Eqn. 5), we ind that: ) M ( ).5-4 M (.95,5-4 M M, (6) which is in good agreeent with the experiental value of this constant (Eqn. ). Exaple : Calculation of 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 -3 М PDTA the paraeters М and V.9 μol/(in per μg protein) change as follows:.6-5 М and V 3.66 μol/(in per μg protein) (Figure 4). This corresponds to the, V i type, of enzye pseudoinhibition ( >, V >, i > ) (Table, line 5) and Eqn. (5, t.f.) is applicable for calculation of the Vi constant of enzye Published: October 3, 8 65

7 Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation /v,5,,5, /[pnpp], 4 M - Figure 4: Inhibitory effect of PDTA on the initial rate v, ol/(in per g protein) of pnpp cleavage by canine alkaline phosphatase. Note: line the concentration of PDTA is -3 M; line () the inhibitor is absent. inhibition. Substitution all necessary paraeters in this equation allow calculation of this constant of enzye inhibition: Vi M æ ö æ - ö æ - ö ç çè 4.69 çè.9 è М. (7) Substitution all necessary paraeters fro of recalculated paraeters of (Figure 4) to (Eqn. 5, Table, q.f.) result in to value of Vi constant inhibition: ( ), (8) Vi IIIa rewritten to the fors (.74-3 М and IIIa М) fro which it follows that.5 æ ( IIIa) ( ) ( ) M ö Vi ( IIIai IVI ) ( ) M ç è IIIai æ ö ç çè æ 9.93 ö ç çè ۰-3 М. (9) Exaple 3: Calculation of Va constant activation. The results of study presented in igure 5 show that: the paraeters of initial nonactivated reaction of pnpp cleavage by alkaline phosphatase М, V μol/(in μg protein) in the presence of. M of activator change as follows: М, V 8.83 μol/(in μg protein), which satisies all the features of type V a of enzye pseudoactivation (line, Table ). Substitution of the experiental paraeters of (Figure 5, in Eqn., t.f., Table ) gives the following value of Va constant: Va -3 M æ ö æ ö - - ç è 3.47 èç 8.83 or according Eq., Table ) æ ( IIIa) ( ) ( ) M ö Va. 5-4 ç ( IIIa IV ) ( ) M i çè IIIa.74-3 М, () Published: October 3,

8 Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation.5 æ ö ç æ759.39ö ç è M ç çè ۰-3 М. () Exaple 4: Calculate the value of II constant of experient (Figure 3), by value of and constants. Fro equation (, Table, t.f.), rewritten to the for () ( II ) (.7485 II.55 ), () it follows that: II ( M ),5. (3) II Having substituted all necessary paraeters fro (Eqn. ) into (Eqn. 3), the next value of this constant is received: II [( ( ) M )],5 ( ),5-4 M M.6-4 M. (4) Exaple 5: Calculate the value of IIIa constant of experient (Figure 5), by value of Va and constants. Fro equation (, Table, t.f.), rewritten to the for () ( Va ) ( ), (5).74 IIIa.753 IIIa it follows that: Va ( IIIa - Va M ).5 ( ) -3 M ( ).5-3 M M M (6).38 It is no desirable to put Va.74-3 М fro (Eqn. ) in (Eqn. 5), because calculation leads to IIIa M (instead M), such as the irst constant is not in Pythagorean s «bundle» (Egn. 5). It is analogous for all biparaetrical types of catalyzed reactions (Table )..8 /v /[pnpp] ( 4 Ì - ) Figure 5: Activating effect of Guo on the initial rate v, ol/(in per g protein) of pnpp cleavage by canine alkaline phosphatase. Note: line the concentration of Guo is -3 M; line () the activator is absent. Published: October 3, 8 67

9 Application of the Pyphagor s Theore for Correction of i and a constants of enzye inhibition and activation Discussion The analysis of data obtained shows that: ) The values of the constants of biparaetrical types of inhibition (Eqns.,, 5 7) and activation (Eqns. 9, 4, 5), are not subjected to additive dependencies on the values of the constants of onoparaetrical types of inhibition (Eqns. 3, 4) and activation (Eqns., 3) of the enzyes (Table ); II. (7) They subjected to geoetrical relationships (Pyphagorean theore): ( / ) ( / ) ( / II), (8) ) this opens an array of possibilities for calculation and correction of the values of i and a constants (Exaples 4). References. rupyanko VI. A vector ethod of representation of enzyic reactions. 3. Three-diensional syste of the V I coordinates convenient for representation of enzye inhibition and activation. Proc Bioche. 4; 39: Ref.: Кrupyanko VI. Perspectives of Data Analysis of Enzye Inhibition and Activation. Part : Use of the Three-Diensional V I Coordinate Syste for Data Analysis of Enzye Inhibition and Activation. J Bioche Mol Toxicol. 9; 3: 97-. Ref.: 3. Кrupyanko VI. Perspectives of Data Analysis of Enzye Inhibition and Activation. Part : Paraetrical Classification of Types of Enzyatic Reactions. J Bioche Mol Toxicol. 9; 3: -7. Ref.: 4. Кrupyanko VI. Perspectives of data analysis of enzye inhibition and activation, Part 3: Equations for calculation of the initial rates of enzyatic reactions. J Bioche Mol Toxicol. 9; 3: 8-8. Ref.: 5. Кrupyanko VI. Perspectives of Data Analysis of Enzye Inhibition and Activation. Part 4: Equations for Calculation of Constants of Enzye Activation and Inhibition. J Bioche Mol Toxicol. ; 4: Ref.: 6. rupyanko VI. Deterination of intensity of enzye inhibition and activation. Eur Che Bull. 4; 3: Ref.: 7. rupyanko VI. Non-existence of secondary coordinates of intersects. Eur Che Bull. 4; 3: Ref.: 8. rupyanko VI. Correction of Dixon Plots. Eur Che Bull. 5; 4: Ref.: 9. rupyanko VI. Correction of Data Analysis Equations Relating to Two-Substrate Enzye Catalyzed Reactions. Eur Che Bull. 6; 5: Ref.: Webb L. Enzye and etabolic inhibitors. Moscow: Mir Publishers. 963; Ref.: Segel IH. Enzye kinetics. J Wiley: New York. 975; Ref.: Dixon M, Webb EC. Enzyes. Moscow: Mir Publishers. 98; : Published: October 3, 8 68