Mathematical Modeling of the Dehydrogenase Catalyzed Hexanol Oxidation with Coenzyme Regeneration by NADH Oxidase

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1 Maheaal Modelng of he Dehydrogenase Caalyzed Hexanol Oxdaon wh Coenzye Regeneraon by H Oxdase T. Iswarya, L. Rajendran, E. Sahappan,,*,Deparen of Maheas, Alagappa Gov. Ars College, arakud, Talnadu. Deparen of Maheas, Aadey of Mare Eduaon and Tranng (AMET) anahur, Chenna, Talnadu. swarya5@gal.o; raj_ss@redffal.o; kesahappan@gal.o Absra: A aheaal odelng of he dehydrogenase aalyzed oxdaon wh oenzye regeneraon by H s dsussed. Ths odel n based on he syse of non-lnear rae equaons onanng double subsrae Mhaels-Menen knes wh opeve produ nhbon. A reaon rae of oenzye regeneraon aalyzed by H s desrbed by Mhaels-Menen knes wh + nhbon. In hs paper a new approah of hooopy perurbaon ehod (NHPM) s appled o solve he syse of non-lnear rae equaons. Sple analyal expressons of onenraons of,, +, H and hexano ad are obaned. Nueral sulaon (Malab) for he onenraon profles are opared wh he analyal resuls. The kne paraeers of produ nhbon and Mhaels-Menen knes are esaed fro he rae equaons. eywords: Maheaal Modellng; Non-lnear dfferenal equaons; Coenzye regeneraon; Enzye knes; H oxdase.. INTRODUCTION The use of he + dependen dehydrogenase n he organ synhess s led by he pre of oenzye, sne should be added n a sohoer aoun. So an effen and eonoal ofaor regeneraon syse s neessary [,]. Reenly vrsalov presek e al. [] desrbed he aheaal odellng of oparson of he l-al ad produon by solaed fuarase and fuarase n pereablzed baker s yeas ells. Coenzye regeneraon aalyzed by H oxdase fro Laoballus brevs n he reaon of l-ano ad oxdaon was suded by Fndrk e al. []. Fndrk and Coworkers [5] developed he aheaal odellng of L-Dopa enzya oxdaon aalyzed by l-ano ad oxdases fro roalus adaaneus and rhodoous opaus. rsalov Presek e al. [6] Forulae he aheaal odel for sulaes ell growh and enzye produon durng he aerob and ro-aerob ulure of he yeas Saharoyes erevsae. Several ehods are avalable for he purpose [7,8] (heal, bologal, phooheal, eleroheal or enzya) of H oxdaon. The enzye H oxdase solaed fro Laoballus brevs [] wll be eployed for he + regeneraon n he reaon of oxdaon o aalyzed by wo dfferen YADHs solaed fro baker s yeas and n pereablzed baker s yeas ells. Pavhra e al. [6] s derved he analyal soluon of nonlnear rae equaon n he ransforaon of D - Mehonne no L-Mehonne n bah reaor usng he new hooopy perurbaon ehod. Reenly Presek e al. [] he proposed aheaal odels for dehydrogenase aalyzed oxdaon wh oenzye regeneraon by H oxdaon. To our knowledge, analyal expressons for he onenraon of enzyes have no been deerned for hs odel. In hs paper, we obaned he analyal expressons of onenraons of,, +, H and hexano ad usng NHPM. Page No:78

2 Ths s an effeve ool for solvng he nonlnear probles n heal senes [7]. These analyal resuls are helpful o undersand he ehans and physal effes of paraeers hrough he odel proble. I s also useful o valdae he nueral resuls and he experenal daa.. Maheaal forulaon of he Proble The ass balane equaons for he oxdaon aalyzed by solaed YADH, wh oenzye regeneraon aalyzed by H oxdase n he bah reaor (Fg A), are gven by he followng equaons (Eqns. () ()) []: d r r () d r r () d d H where r r r r r r,, and H are he onenraons of,, +, H and r, r, r are he reaon raes. The aheaal odel of boaaly oxdaon by pereablzed baker s yeas ell s based on he reaon shee represened n Fgure B. The ass balane equaons for he,,, n hs odel are presened as follows: H hexanoad () () d r r r (5) d d H d r r r r r r r r hexanoad r (8) The reaon rae of oxdaon and reduon aalyzed by solaed YADH or pereablzed baker s yeas ells s gven n he eqns. (9) and (). Also eqn. () represens he rae of reaon n oenzye regeneraon aalyzed by H. (6) (7) r r r H H H oxdase H boaalys boaalys H H H H H (9) () () Page No:785

3 r yeas The nal ondons for above equaons ()-(8) are gven below: A,,, hexanoad, hexanoad () () Fgure. Shee of boaaly oxdaon by: (A) solaed alohol dehydrogenase; (B) pereablzed baker s yeas ell n a oupled syse wh H oxdase fro Laoballus brevs [7]. where,,,, and hexanoad are he onenraons of,, H, + and H hexano ad,, are he axu reaon rae,,,, orrespondng Mhaels-Menen onsan of,, H, + and H,,,, Correspondng produ nhbon onsan of enzye. H H, are he. Analyal expressons of onenraon of,, +, H and Hexano ad: The onenraon of enzyes are obaned (Appendx - A) by solvng he ass balane eqns. ()-() for he oxdaon aalyzed by solaed YADH, wh oenzye regeneraon aalyzed by H oxdase n he bah reaor as follows: ( ) e () ( ) e (5) 6 e ( ) (6) Page No:786

4 H ( ) H e 7 (7) 7 7 Slarly solvng he eqns. (5)-(8) usng NHPM ehod we an oban he onenraon of enzyes n boaaly oxdaon by pereablzed baker s yeas ell as follows: 8 e (8) 8 ( ) 6 e ( ) (9) ( ) e () ) () H 7 H 7 7 hexano hexanoad ( yeas ad where, he onsan H, H, 5, 6 [ ] () () 7 ] 8 [ H (6) (7) 9 5 yeas (8) yeas (9) () boaalys H H H oxdase boaalys H H H H H Nueral Sulaon The non-lnear dfferenal equaons ()-(8) are also solved usng nueral ehods. The funon pdex n Slab sofware whh s he funon of solvng he nal value probles for ordnary dfferenal s used o solve hs equaon. The Slab progra s also gven n Appendx C. We run slab. on apple a ore 5. Slarly, our analyal resuls for he onenraon of,, + and H are opared wh nueral resuls and avalable experenal resuls n Fgs - 9. Sasfaory agreeen s noed for all values of e. Esaon of ne Paraeers () Esaon of Mhaels-Menen onsans fro rae equaon r To hek he valdy of he odel agans he experenal daa, he odel equaon, whh onans four kne paraeers, s ransfored so ha a lnear plo of he daa an be ade. H () () (5) Page No:787

5 The plo has yelded reasonable lneary [8], and he paraeer values an be esaed fro he plo. The four paraeers n Eqn.(9) an be evaluaed by eans of lnear f. Eqn. (9) an be rewren as a r a boaalys boaalys As shown n Fgure 9, Plo of / and y-nerep boaalys a a a a a r versus / gves he slope H where, a () H Fro he Eqn. () we an oban he value of kne paraeers a () boaalys, () Esaon of Mhaels-Menen onsans fro rae equaon r The rae eqn. () an be rewren as b r boaalys H b boaalys As shown n Fgure 9, Plo of / r versus / gves he slope b boaalys H and y-nerep H where, b H b Fro hese plos, we an oban he value of kne paraeers boaalys H (), and H H, (5) (6) (7), and H.. () Esaon of Mhaels-Menen onsans fro rae equaon r The rae eqn.() an be wren anoher way, equvalenly H r H oxdase As shown n Fgure, Plo of / H oxdase H r versus / gves he slope (8) Page No:788

6 H and y-nerep H oxdase H oxdase (9) If we know he axu enzye reaon rae,,, we an oban he oher paraeers (Mhaels-Menen onsan of H) and H (produ nhbon onsan of + ) As ends o nfny he equaon redues o he for of Mhaels-Menen knes for whh he Lneweaver Burk plo 7 s oonly used o deerne he paraeer values. Resul and Dsusson Eqns. () - () he new and sple approxae analyal expresson of he onenraons of,, +, H and hexano ad. Fro Fgure (a)-(), s observed ha when onenraon of nreases when he paraeer boaalys,, nreases. Fro Fgure (d), (e), s explored H ha when onenraon of dereases when he paraeer, nreases. Fro Fgure (a)-(), s onluded ha when onenraon of dereases when he paraeer boaalys,, nreases when he paraeer boaalys nreases. Fro Fgure (d), (e), s nferred ha when onenraon of H, dereases. Fro Fg (a) -(d), sgnfes ha when onenraon of +, H oxdase,, nreases when he paraeer nreases. Fro Fg (e), (f), s explored ha when onenraon of + H dereases when he paraeer, nreases. Fro Fgure 5 (a)-(), s observed ha when onenraon of H dereases when he paraeer boaalys, H oxdase, dereases when he paraeer nreases. Fro Fgure 5 (d), (e), s explored ha when onenraon of H H hange n he onenraon of H when he paraeer, dereases. Fro Fgure 5 (f) s onluded ha here s no nreases. Fro Fgure 6(a)-(), s observed ha when onenraon of dereases when he paraeer boaalys, H oxdase, dereases when he paraeer nreases. Fro Fgure 6 (d)-(f), s explored ha when onenraon of H, dereases. Fro Fgure 6 (g) s onluded ha here s no hange n he onenraon of when he paraeer nreases. Fro Fgure 7 (a)-(e), s observed ha when onenraon of + dereases when he paraeer boaalys H,,,, k yeas + dereases when he paraeer nreases. Fro Fgure 7 (f), (g), s onluded ha when onenraon of k, dereases. Fro Fgure 8 (a)-(), s observed ha when onenraon of H dereases when he paraeer boaalys, H oxdase, dereases when he paraeer nreases. Fro Fgure 8 (d)-(f), s onluded ha when onenraon of H, k, dereases. Fro Fgure 8 (g), (h) s onluded ha here s no yeas, hange n he onenraon of H when he paraeer nreases. Fro Fgure 9 (a), (b) s observed ha when onenraon of hexano ad dereases when he paraeer k, yeas dereases. Page No:789

7 Conlusons A non-lnear e dependen reaon rae equaons n enzye knes has been solved analyally by usng he new Hooopy perurbaon ehod. In hs paper, we have presened approxae analyal expresson of he onenraon of,, +, H and hexano ad. The analyal expressons are opared o he nueral sulaon usng Slab sofware. Good agreeen s noed. Theoreal evaluaon of he kne paraeer s also repored. Page No:79

8 Fgure. Coparson beween he analyal expresson of onenraon of (Eqn. (5)) wh sulaon resuls for varous experenal values of paraeers (Appendx C). Doed lnes are represens n analyal resul and he sold lne represens nueral resuls. Page No:79

9 Fgure. Coparson beween he analyal expresson of onenraon of + (Eqn. (6)) wh sulaon resuls for varous experenal values of paraeers (Appendx C). Doed lnes are represens n analyal resul and he sold lne represens nueral resuls. Fgure 5. Coparson beween he analyal expresson of onenraon of H (Eqn. (7)) wh sulaon resuls for varous experenal values of paraeers (Appendx C). Doed lnes are represens n analyal resul and he sold lne represens nueral resuls. Page No:79

10 Fgure 6. Coparson beween he analyal expresson of onenraon of (Eqn. (8)) wh sulaon resuls for varous experenal values of paraeers (Appendx C). Doed lnes are represens n analyal resul and he sold lne represens nueral resuls. Page No:79

11 Fgure 9. Coparson beween he analyal expresson of onenraon of hexano (Eqn. ()) wh sulaon resuls for varous experenal values of paraeers (Appendx C). Doed lnes are represens n analyal resul and he sold lne represens nueral resuls. Page No:79

12 Fgure. Esaon of kne paraeers,, and H nueral value of he kne paraeers used for he above fgure s gven n Table. usng eqn. ().The Fgure. Esaon of kne paraeers H,, and usng eqn. (5).The nueral value of he kne paraeers used for he above fgure s gven n Table. Page No:795

13 Fgure. Esaon of kne paraeers H and usng eqn. (8).The nueral value of he kne paraeers used for he above fgure s gven n Table. APPENDIX A: Approxae Analyal Soluons of he Equaon () Usng New Hooopy Perurbaon Mehod Mass balanes equaons for s r r d (A) H H boaalys H H H boaalys d (A) The new Hooopy of he above Eqn. (A) an be wren as follows: ) ( A d p d p H H boaalys H H H boaalys H H boaalys H H H boaalys Page No:796

14 The zeroh eraon of he above equaon an be wren as H H boaalys H H H boaalys d (A) : H H boaalys H H H boaalys d p (A5) Solvng he above equaon usng he nal ondon, we ge e ) ( (A6) where, H, (A7) (A8) H H boaalys 9) ( A H H H boaalys APPENDIX B: Slab Progra To Fnd The Nueral Soluon of Eqns. ()-(). funon an; opons=odese('reltol',e-6,'sas','on'); %nal ondons x=[5;;5;.]; span=[,]; [,x]=ode5(@tesfunon,span,x,opons); o fgure hold on %plo(,x(:,)) %plo(,x(:,)) %plo(,x(:,)) plo(,x(:,)) reurn funon[dx_]=tesfunon(,x) v=.5;g=;k=.67;k=.8;k=.85;=;=5;=;=5;v=59.5;k=.99;k5=.7 88;k6=.5; Page No:797

15 k7=.788;k=.85;v=5.98;p=.;k8=.85;k9=.7;k=5;g=.5;5=.; a=(v*g)/((+k*(+(/k)))*(+k*(+(/k)))); b=(v*g)/((+k*(+(/k5)))*(+k6*(+(/k7)))); f=(v*p)/(*(+(/k8))+k9); dx_()=(a**)-(b*x()*)-(k*g*x()*); dx_()=-(a**x())+(b**)+(f*)-(k*g**x()); dx_()=(a**)-(b**x())-(f*x())+(k*g**); dx_()=k*g**; dx_=dx_'; reurn Table : Experenal alues of Paraeer Use n Ths Work and Fndrk e al. and o fnd a esaon of kne paraeer. Paraeer Experenal alue [] Esaon of kne paraeer..58 (ol d - n - g - ) (ol d - n - g - ) (ol d - n - g - ) (ol d - ).67(ol d - ).85 (ol d - ).8999 (ol d - ).8 (ol d - ).8999 (ol d - ) H.9 (ol d - ).9 (ol d - ).99 (ol d - ) (ol d - ) H.5 (ol d - ).5 (ol d - ).788 (ol d - ) (ol d - ) 5.95 (ol d - ) 5.95 (ol d - ) H.7 (ol d - ).75 (ol d - ).7 (ol d - ).7 (ol d - ) Page No:798

16 Table : Noenlaure Sybol Meanng olar onenraon of (ol d - ) olar onenraon of (ol d - ) olar onenraon of + (ol d - ) H olar onenraon of H (ol d - ) hexanoad olar onenraon of hexano ad (ol d - ) Mhaels-Menen onsan of (ol d - ) Mhaels-Menen onsan of (ol d - ) Mhaels-Menen onsan of + (ol d - ) H Mhaels-Menen onsan of H (ol d - ) nhbon onsan of (ol d - ) H nhbon onsan of H (ol d - ) nhbon onsan of (ol d - ) nhbon onsan of + (ol d - ) r, r, r, r reaon rae (ol d - n - g - ) axal reaon rae (ol d - n - g - ),, ass onenraon (ol d - n - g - ) boaalys Hoxdase enzye volue onenraon Referene []. Presek and D. asrak, Maheaal odellng of he dehydrogenase aalyzed oxdaon wh oenzye regeneraon by H oxdase, Proess Bohesry, vol., (9), pp [] R. Whann and D. asrak, Cofaor regeneraon a he lab sale, ehnology ransfer n boehnology. In: ragl U, edor, Advanes n boheal engneerng/boehnology Sprngr-erlag GbH, vol. 9, (5), pp [] rsalov Presek, B. Zel and Ð. as-rak, Coparson of he l-al ad produon by solaed fuarase and fuarase n pereablzed baker s yeas ells, Enzye and Mrobal Tehnology, vol., (7), pp [] Z. Fndrk, I. ˇSunov and Ð. as -Raˇk, Coenzye regeneraon aalyzed by H oxdase fro Laoballus brevs n he reaon of l-ano ad oxdaon, Boheal Engneerng Journal, vol. 9, (8), pp [5] Z. Fndrk, B. Geueke, W. Huel and Ð. as -Raˇk, Modellng of l-dopa enzya oxdaon aalyzed by l-ano ad oxdases fro Croalus adaaneus and Rhodoous opaus, Boheal Engneerng Journal, vol. 7, (6), pp Page No:799

17 [6] A.. Presek and Ð. as Rak, Modellng of he alohol dehydrogenase produon n baker s yeas, Proess Bohesry, vol., (5), pp [7] M. D. Leonda, Redox enzyes used n hral synheses oupled o oenzye regeneraon, Curr Med Che, vol. 8, (), pp [8] W. A. Donk and H. Zhao, Reen developens n pyrdne nuleode regeneraon, Curr Opn Boehnol, vol., (), pp. 6. [9] R. Whann, C. Wandrey, A. F. Bukann and M. R. ula, Connuous enzya ransforaon n an enzye ebrane reaor wh sulaneous (H) regeneraon, Boehnol Boeng, vol. 67, (), pp [] W. Lu and P.Wang, Cofaor regeneraon for susanable enzya bosynhess, Boehnol Adv, vol. 5, (7), pp [] A. Lese, T. Zelnsk, M. R. ula, H. erkel, M. aruz M and U. ragl, A novel reaor onep for he enzya reduon of poorly solubles keones, J Mol Caal B: Enzye, vol., (998), pp [] A.. Presek, Z. Fndrk and B. Zel, Modellng of he boransforaon proesse, Che Bohe Eng, Q, vol., (6), pp. 7. [] S. F. Souza, Ioblzed enzyes n boproess, Curr S Inda, vol. 77, (999), pp [] L. Rajendran and S. Anha, Coens on analyal soluon of aperoer enzya Reaons based on Hooopy perurbaon ehod, Eleroheal Aa, vol., (), pp [5] L. Rajendran, G. Rahaahunssa, The Applaon of He s varaonal eraon ehod o nonlnear boundary value probles n enzye-subsrae reaon dffuson proesses : Par.The seady-sae aperoer response, Journal of Maheaal Chesry, vol. :, (8), pp [6] S. Pavhra, P. G. Jans Ran, T. Iswarya and L. Rajendran, Enzye ne Modellng And Analyal Soluon Of Nonlnear Rae Equaon In The Transforaon Of D -Mehonne Ino L-Mehonne In Bah Reaor Usng The New Hooopy Perurbaon Mehod, Quanova vol. Xy no., (6), pp. -8. [7] H.J. He, Elero heal Aa, vol. 9, (), pp. 67. [8]. Saravanakuar, S. P. Ganesan and L. Rajendran, Theoreal Analyss of Reaon and Dffuson Proesses n a Bofuel Cell Elerode, Fuel Cells, vol., (5), pp. -. Page No:8

Pendulum Dynamics. = Ft tangential direction (2) radial direction (1)

Pendulum Dynamics. = Ft tangential direction (2) radial direction (1) Pendulum Dynams Consder a smple pendulum wh a massless arm of lengh L and a pon mass, m, a he end of he arm. Assumng ha he fron n he sysem s proporonal o he negave of he angenal veloy, Newon s seond law