Mathematical Expressions for Estimation of Errors in the Formulas which are used to obtaining intermediate values of Biological Activity in QSAR

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1 Iteratioal Joural o Iovatio ad Applied Studies ISSN Vol. No. 3 Mar. 03, pp Iovative Space o Scietiic Researc Jourals ttp// Matematical Expressios or Estimatio o Errors i te Formulas wic are used to obtaiig itermediate values o Biological Activity i QSAR Nizam Uddi M. B. Kalsa College, Idore, Madya Prades, Idia Copyrigt 03 ISSR Jourals. Tis is a ope access article distributed uder te Creative Commos Attributio Licese, wic permits urestricted use, distributio, ad reproductio i ay medium, provided te origial work is properly cited. ABSTRACT Quatitative structure-activity relatiosips (QSAR) attempts to id cosistet relatiosips betwee te variatios i te values o molecular properties ad te biological activity or a series o compouds. Tese pysicocemical descriptors, wic iclude parameters to accout or ydropobicity, topology, electroic properties, ad steric eects, are determied empirically or, more recetly, by computatioal metods. Quatitative structure -activity relatiosips (Q SAR) geerally take te orm o a liear equatio were te biological activity is depedet variable. Biological activity is depeded o te parameters ad te coeiciets. Parameters are computed or eac molecule i te series. Coeiciets are calculated by ittig variatios i te parameters. Itermediate values o te biological activity are obtaied by some ormulas. Tese ormulas are worked i tabulated values o biological activity i Quatitative structure - activity relatiosips. Tese ormulas are worked i te coditios ad all coditios are based o te positio o te poit lies i te table. Derived ormulas usig Newto s metod or iterpolatio are worked i coditios wic are depedig o te poit lies. I te poit lies i te upper al te used Newto s orward iterpolatio ormula. I te poit lies i te lower al te we used Newto s backward iterpolatio ormula. Ad we te iterval is ot equally spaced te used Newto s divide dierece iterpolatio ormula. We te tabulated values o te uctio are ot equidistat te used Lagragia polyomial. Matematical expressios are derived or estimatio o errors usig itermediate values ad ormulas. KEYWORDS Biological activity, Estimatio o Errors, Itermediate values, Matematical Expressios, QSAR. INTRODUCTION Quatitative structure-activity relatiosips (QSAR) represet a attempt to correlate structural or property descriptors o compouds wit activities. Tese pysicocemical descriptors, wic iclude parameters to accout or ydropobicity, topology, electroic properties, ad steric eects, are determied empirically or, more recetly, by computatioal metods. Activities used i QSAR iclude cemical measuremets ad biological assays []-[5]. A QSAR geerally takes te orm o a liear equatio Biological Activity = Costat + (C P ) + (C P ) + (C 3 P 3 ) +... () Were te parameters P trougpare computed or eac molecule i te series ad te coeiciets C C are calculated by ittig variatios i te parameters ad te biological activity []. troug I (CP) = Costat + (C P ) + (C P ) + (C 3 P 3 )+... From equatio (), we got Correspodig Autor Nizam Uddi (izamuddi4researc@gmail.com) 7

2 Nizam Uddi Suppose BA () CP CP te we ca write more simple orm BA () Were BA is biological activity ad is variable rom above uctio []. Some ormulas are derived o te basis o tis uctio usig Newto s metod or iterpolatio ad Lagragia polyomial. Tese ormulas are used to obtaiig itermediate values o te biological activity. Derived ormulas usig Newto s metod or iterpolatio are worked i coditios wic are depedig o te poit lies. I te poit lies i te upper al te used Newto s orward iterpolatio ormula. I te poit lies i te lower al te we used Newto s backward iterpolatio ormula. Ad we te iterval is ot equally spaced te used Newto s divide dierece iterpolatio ormula. We te tabulated values o te uctio are ot equidistat te used Lagragia polyomial [6]-[]. IF THE POINT LIES IN THE UPPER HALF Let ( ) BA be a uctio deied by poits ( BA 0, 0), ( BA, )...( BA, ),, Were BA is biological activity ad is te variable. We are equally spaced wit iterval. Ad I te poit lies i te upper al te we used ollowig ormula [] [7]-[9] Δ BA0 Δ BA0 BA() BA0 ΔBA(q)(q)[(q 0 )]... [(q)(q )...(q )]!! [Were is orward dierece operator ad 0 q ]. ESTIMATION OF ERROR Let BA ( ) be a uctio deied by ( ) poits ( BA 0, 0), ( BA, )...( BA, ). We,, 3... ( times.... are equally spaced wit iterval ad tis uctio is cotiuous ad dieretiable ) Let BA ( ) be approximated by a polyomial P( ) o degree ot exceedig a suc tat P ( i) BAi [Were i,,3.... Sice te expressio ( ) P( ) vaises or,, 3... We put ( ) P( ) K( ) Were ( ) ( 0)( )...( ) ] ()..., Ad K is to be determied i suc a way tat equatio ( 3) olds or ay itermediate values o, say [were 0 ' ]. Tereore rom (3) ( ')( P') K ( ') Now we costruct a uctio ( ) suc tat ()()()() 0 P K Were K is give by equatio (5). It is clear tat ()()()() 0...()( ') 0 3 (6) (3) (4) ' Let ( ) vaises ( ) times i te iterval 0 ; cosequetly, by te repeated applicatio o Rolle s Teorem [3] [4], '( ) must vais ( ) times, ''( ) must vais times etc i te iterval 0. (5) ( Particularly, ) ( ) must vais oce i te iterval 0. Let tis poit be U, Now dieretiatig equatio (6) ( ) times wit respect to ad puttig U, we got W 0. ( ) ()( U K)! 0 ISSN Vol. No. 3, Mar

3 Matematical Expressios or Estimatio o Errors i te Formulas wic are used to obtaiig itermediate values o Biological Activity i QSAR K ( ) () U ( )! (7) Puttig tis value o K i equatio (5), we got ( ) ()( U ')( ') P ( )!( ') Sice ( ) () U ( ')( P')( ') ( )! ' is arbitrary tereore o droppig te prime o U, 0 ' we got ( ) () U ()()() P ( )!, 0 U (8) Now we use Taylor s teorem [5] [6] ()() '() U ''() U...() U... U (9)!! Neglectig te terms cotaiig secod ad iger powers o i equatio (9), we got ()() U '() U U '() U ()() U U (0) '()() U U [()()] d D ()() U U [ D ] du From equatio (0), we got D [Because (U ) is arbitrary] D ( )( ) ()() U U ( ) ( Puttig te values o ) ( U ) i equatio (8), we got () ( )! ( ) ()()() P U ( ) ()()()...() 0 0 ( ) ()()() P U ( ) ( )! 0 I q Te Similarly ( q ) 0 q ()()()( 0 ) 0 q q () Similarly ( q ) ISSN Vol. No. 3, Mar

4 Nizam Uddi Puttig tese values i equatio (), we got ( ) ()( q )( q )( q 3)...() q q ()()() P U ( ) ( )! q( q )( q )( q 3)...() q ( )! ( ) ()()() P U Tis is matematical expressio or estimatio o error, i te poit lies i te upper al. 3 IF THE POINT LIES IN THE LOWER HALF Let ( ) BA be a uctio deied by poits ( BA 0, 0), ( BA, )...( BA, ),, Were BA is biological activity ad is te variable. We are equally spaced wit iterval. Ad I te poit lies i te lower al te we used ollowig ormula [] [7]-[9] BA BA!! BA() BA.BA(r) [r(r )]... [(r){(r )}..{r( -)}] [Were is backward dierece operator ad r ] 3. ESTIMATION OF ERROR Let BA ( ) be a uctio deied by ( ) poits ( BA 0, 0), ( BA, )...( BA, ). We,, are equally spaced wit iterval ad tis uctio is cotiuous ad dieretiable ( ) times. Let BA ( ) be approximated by a polyomial P( ) o degree ot exceedig a suc tat P ( i) BAi [Were i,,3.... ] ()... Sice te expressio ( ) P( ) vaises or,, 3..., We put we put ()()() P K (3) Were ()()()...() 0 (4) Ad K is to be determied i suc a way tat equatio ( 3) olds or ay itermediate values o, say [were 0 ' ]. Tereore rom equatio (3), ( ')( P') K ( ') Now we costruct a uctio ( ) suc tat ()()()() 0 P K Were K is give by equatio (5). It is clear tat ()()()() 0...()( ') 0 3 (6) (5) ' Let ( ) vaises ( ) times i te iterval 0 ; cosequetly, by te repeated applicatio o Rolle s Teorem [3] [4], '( ) must vais ( ) times, ''( ) must vais times etc i te iterval 0. ( Particularly, ) ( ) must vais oce i te iterval 0. Let tis poit be Z, Z 0. ( ) ()( Z K)! 0 ISSN Vol. No. 3, Mar

5 Matematical Expressios or Estimatio o Errors i te Formulas wic are used to obtaiig itermediate values o Biological Activity i QSAR K ( ) () Z ( )! (7) Puttig tis value o K i equatio (5), we got ( ) ()( Z ')( ') P ( )!( ') Sice () Z ( )! ( ) ( ')( P')( ') ' is arbitrary tereore o droppig te prime o Z, 0 ' we got () Z ( )! ( ) ()()() P, 0 Z (8) Now we use Taylor s teorem [5] [6] ()() '() Z ''() Z...() Z... Z (9)!! Neglectig te terms cotaiig secod ad iger powers o i equatio (9), we got ()() Z '() Z Z '() Z ()() Z Z '()() Z Z [ ( ) ( )] d D ()() Z Z [ D ] dz (0) From equatio (0), we got D [Because (Z ) is arbitrary] D ( )( ) ()() Z Z ( ) ( Puttig te values o ) ( W ) i equatio (8), we got () ( ) ()()() P Z ( ) ( )! ()()()...() 0 0 ( ) ()()() P Z ( ) ( )! 0 I r Te Similarly ( r ) 0 r ()()()( 0 ) 0 r r () Similarly ( r ) ISSN Vol. No. 3, Mar

6 Nizam Uddi Puttig tese values i equatio (), we got ( ) ()( r )( r )( r 3)...() r r ()()() T P T Z ( ) ( )! r( r )( r )( r 3)...() r ( )! ( ) ()()() P Z Tis is matematical expressio or estimatio o error, i te poit lies i te lower al. 4 IF INTERVALS ARE NOT BE EQUALLY SPACED Let ( ) BA be a uctio deied by poits ( BA 0, 0), ( BA, )...( BA, ),, Were BA is biological activity ad is te variable. We are equally spaced wit iterval. Ad I Itervals are ot be equally spaced te we used ollowig ormula we used ollowig ormula [] [7]-[9]. BA() BA Δ BA ( ) BA ( )( )... Δ d Δ d [Were d is divide dierece operator] d BA [( )( )...( )] 4. ESTIMATION OF ERROR Let ( ) be a real-valued uctio deie iterval ad ( ) times dieretiable o ( a, b). I P( ) polyomial. Wic iterpolates ( ) at te ( ) distict poits 0,... ( a, b) exists a, b e P ( ) j j0, te or all a, b Tis is matematical expressio or estimatio o error, i itervals are ot be equally spaced. 5 WHEN THE TABULATED VALUES OF THE FUNCTION ARE NOT EQUIDISTANT Let ( ) BA be a uctio deied by poits ( BA 0, 0), ( BA, )...( BA, ) () is te,tere. Were BA is biological activity ad is te variable. We te tabulated values o te uctio are ot equidistat te we used ollowig ormula [] [7]-[9] BA() () j BA i i j () i j ji 5. ESTIMATION OF ERROR Sice te approximatig polyomial give by Lagragia ormula as te same values as does BA () or te argumets 0,,, 3, 4.., zeros at tese ( ) poits. te error term must ave Tereore () 0 () () () 3.. () must be actors o te error ad we ca write ()()()()...() 0 3 F()()() K ( )! (3) ISSN Vol. No. 3, Mar

7 Matematical Expressios or Estimatio o Errors i te Formulas wic are used to obtaiig itermediate values o Biological Activity i QSAR Let to be ixed i value ad cosider te uctio ()()()()...() x 0 x x x 3 x W()()()() x F x x K ( )! x... ad. Te W() x as zero 0,,, 3... (4) Sice te t ( ) derivative o te t degree polyomial ( ) is zero. ( )( ) W ()()() x F x K (5) As a cosequece o Rolle s Teorem [3] [4], te te rage 0 Tereore substitutig x i equatio (5) t ( ) derivative o (x) W as at least oe real zero x i ( )( ) W ()()() F K ( )( ) K()()() F W F ( ) () Usig tis expressio or K() ad writig out () ( ) Were ( )( )...( ) ( ( 0 )( 0 )...( 0 ) ( 0)( )...(... ( 0)( )...( T 0 T ( 0)( )...( ) ) ( )... ( 0)( )...( ) 0 ) ( ( ) ) )( )...( ) ( )! Tis is matematical expressio or estimatio o error, i te tabulated values o te uctio are ot equidistat. 0 ( ) ( ) 6 CONCLUSION Derived matematical expressios are useul to estimatio o te errors i te ormulas or obtaiig itermediate values o te biological activity i Quatitative structure-activity relatiosips (QSAR). All expressios are worked i limit wic is te last value i te table. We we obtai te itermediate values o te biological activity i Quatitative structure-activity relatiosips te tese matematical expressios are useul to estimate te errors i iterpolated values o te biological activity. REFERENCES [] Nizam Uddi, Formulas or Obtaiig Itermediate Values o Biological Activity i QSAR usig Lagragia polyomial ad Newto s metod, Sciece Isigts A Iteratioal Joural, (4), -3, 0. [] Jo G. Topliss, Quatitative Structure-Activity Relatiosips o Drugs, Academic Press, New York, 983. [3] Frake, R., Teoretical Drug Desig Metods, Elsevier, Amsterdam, 984. [4] Robert F. Gould (ed.), Biological Correlatios -- Te Hasc Approac, Advaces i Cemistry Series, No. 4, America Cemical Society, Wasigto, D.C., 97. [5] Hasc, C., Leo, A., ad Tat, R.W., A Survey o Hammett Substituet Costats ad Resoace ad Field Parameters, Cem. Rev., , 99. [6] Nizam Uddi, Iterpolate te Rate o Ezymatic Reactio Temperature, Substrate Cocetratio ad Ezyme Cocetratio based Formulas usig Newto s Metod, Iteratioal Joural o Researc i Biocemistry ad Biopysics, (), 5-9, 0. [7] Nizam Uddi, Estimatio o Errors Matematical Expressios o Temperature, Substrate Cocetratio ad Ezyme Cocetratio based Formulas or obtaiig itermediate values o te Rate o Ezymatic Reactio, Iteratioal Joural o Iovatio ad Applied Studies, vol., o., pp. 53 7, 03. ISSN Vol. No. 3, Mar

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