Bio-Systems Modeling and Control
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1 Bio-Sytem Modelig ad otrol Leture Ezyme ooperatio i Ezyme Dr. Zvi Roth FAU
2 Ezyme with Multiple Bidig Site May ezyme have more tha oe bidig ite for ubtrate moleule. Example: Hemoglobi Hb, the oxygearryig protei i red blood ell, ha bidig ite for O moleule. I uh ae a will be how reatio veloity may o loger be of the imple Mihaeli-Mete hyperboli urve type. It may have a igmoid hape. Dr. Zvi Roth FAU
3 Ezyme with Two Bidig Site ae I : The two bidig ite are equivalet ad bidig to oe of the ite ha o effet o bidig to the other ite ae II: The two bidig ite are ot equivalet. Oe i preferred ad oe that oe bid with the ubtrate alo alled ligad, the other ite bidig propertie are ehaed. Thi i alled POSITIE OOPERATIITY ae III: The bidig of ligad to the firt ite dimiihe the affiity to the eod ite: NEGATIE OOPERATIITY
4 Ezyme with Two Ative Site S E E P S P Ezyme a exit i three tate: Free moleule E, a omplex with oe oupied ite ad a omplex with two oupied ite. Dr. Zvi Roth FAU
5 Dr. Zvi Roth FAU 5 Ma-Atio Reatio Equatio P S P E E S dt dp e dt de e dt d dt d e dt d Typial iitial oditio: 0 0, e0e 0, 0 0 p00
6 Dr. Zvi Roth FAU 6 Similarly to previou derivatio we obtai e e 0 ad: P S P E E S 0 0 e dt d dt d e dt d I: 0 0, 0 00
7 Dr. Zvi Roth FAU 7 Applyig quai-teady-tate aumptio d /dt d /dt 0, yield: 0 0 e e We a ow fid the peed of reatio, from the dp/dt equatio.
8 Dr. Zvi Roth FAU 8 Quai-Steady-State for, : Reatio eloity 0 0 e e 0 e dt dp
9 Reatio eloity Extreme ae If ative ite at idepedetly ad idetially: e S E E P 0 S P The, ad The above fator of our beaue two idetial bidig ite are ivolved i the reatio, doublig the amout of the reatat. I a free ezyme the probability of bidig ay of the ite i double tha for to beome. Dr. Zvi Roth FAU 9
10 Reatio eloity No ooperativity e0 e e 0 0 If ative ite at idepedetly ad idetially: The, ad Let - / i the equilibrium otat for the idividual bidig ite. Reatio rate imply double, but the reatio baially tay Mihaeli-Mete hyperboli. Dr. Zvi Roth FAU 0
11 Dr. Zvi Roth FAU Extreme ae : Poitive ooperativity 0 e dt dp P S P E E S We have <<
12 Dr. Zvi Roth FAU Extreme ae : Poitive ooperativity 0 e dt dp P S P E E S We have << >> wherea ha ome fiite value
13 Reatio eloity Extreme ae Poitive ooperativity e 0 e 0 max m m If bidig of the firt ubtrate i low, but that with oe ite boud, bidig of the eod i fat thi i alled large poitive ooperativity: The 0 that i very mall, that i very large, while the produt tay otat. other otat fixed The 0 ad while m i otat Neglet but ot Dr. Zvi Roth FAU
14 ommet Reall the two Predatio model? I oe we had By/yA ad it hape i hyperboli Mootoe ireaig of the predatio toward aturatio, wherea the rate of hage of the predatio beome teadily maller. I the other we had By /A y Sigmoid hape tart low, otiue fater, fiih low ad aturate. Dr. Zvi Roth FAU
15 Reatio eloity Extreme ae Strog Poitive ooperativity e0 max m Dr. Zvi Roth FAU 5
16 Negative ooperativity I reality, a ezyme a bid more tha oe ubtrate moleule ad the bidig of the firt a dimiih ot ehae the bidig of the eod. Thi i modeled by reduig. Thi i alled NEGATIE OOPERATIITY 6
17 Negative ooperativity Bidig of the firt ubtrate moleule dereae the rate of ubequet bidig. I uh a ae we uually have a mall value of. Mai effet: A dereae i the maximum peed. Dr. Zvi Roth FAU 7
18 ooperativity Type 8
19 I betwee the two exterme ae: Hill Futio Dr. Zvi Roth FAU 9
20 urve Fitted Hill Equatio max m m equilibrium otat for a -ite ezyme Poitive ooperativity i if >, No ooperativity if, Negative ooperativity i if <. Dr. Zvi Roth FAU 0
21 Hill Futio max m I Hill futio the parameter max, m ad are foud by umerial fit to empirial data. The power i almot alway a o-iteger. For >, the hape reemble igmoid. Dr. Zvi Roth FAU
22 How i the Hill Futio Etimated Empirially? Dr. Zvi Roth FAU l l l max max max max max m m m m m I logarithmi ale / max - deped liearly o.
23 How i the Hill Futio Etimated Empirially? l max l l m I the lab we a vary 0 ad meaure 0. We either ow max by other mea or we gue a value. We the ue EXEL ay to bet fit a lie to the l 0 / max - 0 v. l 0 tryig with may value of max i the lope ad m the iterept Dr. Zvi Roth FAU
24 Hill Equatio Appliatio Hill equatio i ofte ued i multi-tep reatio whoe detailed itermediate tep are ot ow, but for whih ooperative behavior i upeted. Dr. Zvi Roth FAU
25 Hill Plot L S 0
26 Bottom Lie Approximated Model for ooperative Ezyme Reatio dp dt max m The above approximatio i for ae where there are m ative ite for idetial ubtrate. Typially m. Sietit ofte ue uh a model to repreet a hai of ezyme reatio. Dr. Zvi Roth FAU 6
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