CHAPTER 2: ENZYME KINETICS 2.3 In some enzyme- catalyze !eact"on# m$lt"%le m$lt"%le com%le&es a!e "n'ol'e as (ollo)s: (ollo)s: S * E + , ES , ES + , ES 2 , E 2 / P * E RE01IRE: e'elo% a !ate e&%!ess"on e&%!ess"on $s"n a. M"c4a M"c4ael el"s "s Ment Menten en a%%! a%%!oa oac4 c4 5. T4e 6!"s Halane a%%!oac4 S781TI7N : rp =
−dcs dcp = = ksCes− k 4 CpCe dt
dt
Ceo 9 Ce 9 Ces Ce 9 Ceo ; Ces !% 9 <3Ces ; <=C%Ceo * <=C%Ces !% 9 , <3 * <= Ces ; <=C%Ceo < CsCe 9 <2Ces <,Cs,Ceo-Ces 9 <2Ces <CsCeo-<CsCes9<2Ces <2Ces * <CsCes 9 <CsCeo
Ces=
k 1 CsCeo k 2 + k 1 Cs
Ces=
CeoCs k 2 +Cs k 1
rp =( k 3 + k 4 4 Cp )
CeoCs k 2 +Cs - <=Ceo k 1
(
)
( ks + k 4 Cp ) (CeoCs ) − k 2 Cs (k 4 4 CpCeo) !% 9
k 1
k 2 +Cs k 1
k 3 CeoCs + k 4 4 CeoCpCs − rp =
k 2 k 4 4 CpCeo − k 4 CpCesCs k 1
k 2 +Cs k 1
k 3 CeoCs − rp =
k 2 k 4 CpCeo k 1
k 2 +Cs k 1
(
Ceo k 3 cs − rp =
k 2 k 4 Cp k 1
)
k 2 + Cs k 1
Ce9Ceo9Ces KCs , Ceo-Ces 9 <2Ces K2Ces * <CsCes 9 <CsCeo k 1 CsCeo k 2 +Cs k 1 CsCeo k 2 + Cs k 1 ANS>ER k 3 CeoCs rmaxCx = k 2 Km + Cs +Cs k 1
T4e 6!"s Halane
−dcs dcp = =k 3 Ces dt
dt
Rate ete!m"n"n e?$at"on: −dcs =k 1 CsCe −k 2 Ces− k 3 Ces =0 dt Ceo 9 Ce * Ces KCs , Ceo-Ces 9<2Ces-<3Ces KCsCeo-<CsCes-<3 Ces9@ KCsCes*<2Ces*<3Ces 9 <CsCeo Ces=
k 1 CsCeo k 1 Cs + k 2 + k 3
Ces=
CsCeo k 3 + k 2 + Cs k 1
−dcs dcp k 3 CsCeo = = dt dt k 3 + k 2 + Cs k 1
!%9
rmaxCs KmCs
CHAPTER 2: ENZYME KINETICS 2.= Ea"e ,=2 meas$!e t4e "n"t"al !eact"on !ate o( 4y!olys"s o( acetylc4ol"ne ,s$5st!ate 5y o se!$m ,so$!ce o( enzyme an o5ta"ne t4e (ollo)"n ata: S$5st!ate Concent!at"on molB8 @.@@32 @.@@= @.@@2 @.@@D@ @.@@F
In"t"al React"on Rate molB8m"n @. @.=D @.=3 @. @.2@@
E'al$ate t4e M"c4ael"s-Menten <"net"c %a!amete!s 5y em%loy"n ,a t4e 8anm$"! %lot# ,5 t4e 8"ne)ea'e!-6$!< %lot# an ,c t4e Ea"e-Ho(stee %lot.
G"'en: !e(e! to ta5le Re?$"!e: M"c4ael"s-Menten <"net"c %a!amete!s Sol$t"on: a. 8anm$"! CS r
=
KM
1 + C S r max r max
&# C S @.@@32 @.@@= @.@@2 @.@@D@ @.@@F
y# C S /r @.@@32B@. @.@@=B@.=D @.@@2B@.=3 @.@@D@B@. @.@@
[email protected]@@
5. 8"ne)ea'e!-6$!< 1 r
=
1 r max
+
KM
1
r max C S
!ma& 9 @.3@D molB8m"n KM 9 F.2 & @-3 molB8
&# 1/C S B@.@@32 B@.@@= B@.@@2 B@.@@D@ B@.@@F
y# 1/r B@. B@.=D B@.=3 B@.
[email protected]@@
!ma& 9 @.2F2 molB8m"n KM 9 =.3@3 & @-3 molB8
y# r @. @.=D @.=3 @. @.2@@
!ma& 9 @.2=F molB8m"n KM 9 =.23 & @-3 molB8
c. Ea"e-Ho(stee r = r max - K M
&# r/C S @.B@.@@32 @.=DB@.@@= @.=3B@.@@2 @.B@.@@D@ @.2@@B@.@@F
r CS
CHAPTER 2: ENZYME KINETICS 2. T4e K M 'al$e o( an enzyme "s
4at "s t4e ma&"m$m !eact"on !ateJ 5. >4at "s t4e concent!at"on o( t4e s$5st!ate a(te! F m"n$tesJ G"'en: K M 9 @.@ molB8
t9F m"n$tes
Cso9 3.=&@-= molB8
@ %e!cent )as cons$me
Re?$"!e: a. ! ma& 5. Cs t9F m"n$tes
Sol$t"on: K M ln
0.01
Cso +( Cso−Cs )= r max t Cs
mol ln L
1 0.9
+ (1 −0.9 ) ( 3.4 x 10 −4 )
r max 9 3.2F=&@-
mol = r max ( 5 x 60 ) s L
t9 F m"n$tes K M ln
Cso +( Cso−Cs )= r max t Cs
0.01
mol 3.4 x 10 −4 mol kmol + ( 3.4 x 10− 4 −Cs ) =(3.6254 x 10−6 ) ( 15 x 60 ) s ln L Cs L m3 s
Cs9 2.=2 &@-= molB 8
CHAPTER 2: ENZYME KINETICS 2.D A s$5st!ate "s con'e!te to a %!o$ct 5y t4e catalyt"c act"on o( an enzyme. Ass$me t4at t4e M"c4ael"s-Menten <"net"c %a!amete!s (o! enzyme !eact"on a!e: K M 9 @.@3 molB8 ! ma&9 3 molB 8 m"n a. >4at s4o$l 5e t4e s"ze o( a steay-state CSTR to con'e!t F %e!cent o( "ncom"n s$5st!ate ,Cso9 @ molB8 )"t4 a (lo) !ate o( @ 8B4J 5. >4at s4o$l 5e t4e s"ze o( t4e !eacto! "( yo$ em%loy a %l$-(lo) !eacto! "nstea o( t4e CSTR "n %a!t ,aJ
G"'en: K M 9 @.@3 molB8 ! ma&9 3 molB 8 m"n Cso9 @ molB8 9 @ 8B4 SolLn: a F 1 rmax Cs = = V τ ( Cso −Cs)( Km + Cs )
Re?L: aCSTR 5 PR
V =
F rmax Cs (Cso−Cs )( Km+ Cs)
( ) ( )( 10 L
V =
h
(
1 hr 60 min
)
mol mol ) 0.05 × 10 Lmin L 10 mol 0.5 mol 0.03 mol 0.5 mol ( − )( + ) L L L L 13
CSTR 9@.2 L
5 Cso−Cs t =− Km+ rmax ( Cso Cs ) ln ( ln Cs Cso
( )
10 −0.5 ln
( ) 10
t
=−( 0.03 ) +( 13)(
0.5
ln
( ) 10
)
)
0.5
t 9 @.3 m"n t9B 9t 9,@.3 m"n, 4B @ m"n,@ 8B4 PR 9 @.22 8
CHAPTER 2: ENZYME KINETICS 2. A s$5st!ate "s ecom%ose "n t4e %!esence o( an enzyme acco!"n to t4e m"c4ael"s menten e?$at"on )"t4 t4e (ollo)"n <"net"c %a!amete!s: grams liter
Km9@ Rma& 9
7
g L −min
I( )e o%e!ate t)o -8 CSTR n se!"es at steay state# )4t )"ll 5e t4e concent!at"on o( s$5st!ate lea'"n t4e secon !eacto!J T4e (lo) !ate "s @.F 8Bm"n. T4e "nlet s$5st!ate concent!at"on "s F@B8 an t4e enzyme concent!at"on "n t4e t)o !eacto!s "s ma"nta"ne "n t4e sa 'al$e all o( t4e t"me. Is t4e t)o !eacto! system mo!e e(("c"ent t4an one !eacto! )4ose 'ol$me "s e?$al to t4e s$m o( t4e t)o !eacto!sJ GIEN: Cso F@B8
Km9 @B8
!ma&9 B8-m"n 9 @.F 8Bm"n RE01IRE: a. Cs2 5. Is t)o !eacto! mo!e e(("c"ent t4an !eacto! )"t4 'ol$me 9 28
Cs9-
rmax Csτ Cso −Cs
o! t4e ("!st !eacto! sol'e Cs
( ) 7g
L
(Cs 2 ) (
1l 0.5 g
)
min ❑
Cs9 ,-@B8 * 50 g
L
−Cs 1
Cs9 3D.DF@B8 At t4e secon !eacto! cso 93D.DF@B8
( ) 7g
L
Cs29 ,-@B8 *
(Cs 2 ) (
1l 0.5 g
)
min ❑ 38.8650 −Cs 2
Cs29 2D.F@2@B8 >4"c4 "s mo!e e(("c"ent
O con'e!s"on 9
Cso −Cs x 100 Cso
50 −28.5012
Ocon'e!s"on 9
50
x 100
Ocon'e!s"on 9 =2. O ,(o! 2 CSTR IN SERIES o! CSTR )"t4 'ol$me 928
( )( 7g
L
Cs 2 ) (
2l 0.5 g
)
min ❑
Cs9 ,-@B8 * 50 g
L
−Cs 1
Cs9 2.F B8
%conversion =
50 −29.1517 50
x 100
Con'e!s"on 9 =. OO , (o! CSTR )"t4 28 'ol$me
CHAPTER 2: ENZYME KINETICS 2.= Ea"e ,=2 meas$!e t4e "n"t"al !eact"on !ate o( 4y!olys"s o( acetylc4ol"ne ,s$5st!ate 5y o se!$m ,so$!ce o( enzyme "n t4e a5sence an %!esence o( %!ost"m"ne ,"n4"5"to!# .F & @- molB8 an o5ta"ne t4e (ollo)"n ata: S$5st!ate Concent!at"on ,molB8 @.@@32 @.@@= @.@@2 @.@@D@ @.@@F
In"t"al React"on Rate A5sence o( P!ost"m"ne @. @.=D @.=3 @. @.2@@
a. Is %!ost"m"ne com%et"t"'e o! noncom%et"t"'e "n4"5"to!J
Rate ,molB8.m"n P!esence o( P!ost"m"ne @.@F @.@ @.@ @. @.2F
5. E'al$ate t4e M"c4ael"s-Menten <"net"c %a!amete!s "n t4e %!esence o( "n4"5"to! 5y em%loy"n t4e 8anm$"! %lot. 0.08 f(x) = 2. 99x + 0.05
0.07 0.06 0.05
Cs/r
f(x) = 3. 31x + 0.02
0.04 0.03 0.02 0.01 0 0
0
0
0.01
0.01
0.01
0.01
Cs Linear ()
a. P!ost"m"ne "s a com%et"t"'e "n4"5"to!. 5. R ma&9 Bm 9 @.33= ,)"t4 "n4"5"to! concent!at"on R ma&9 Bm 9 @.3@D ,no "n4"5"to! concent!at"on K m 9 5Rma& 9 F.==&@-3 K I 9 5Rma& 9 @.@=
Linear ()
0.01
0.01
CHAPTER 2: ENZYME KINETICS 2. T4e "n"t"tal !ate o( !eact"on (o! t4e enzymat"c clea'ae o( eo&y$anos"ne t!"%4os%4ate )as meas$!e as a ($nct"on o( "n"t"al s$5st!ate concent!at"on as (ollo)s ,Ko!n5e! et al.# . 6"ol. C4em.# 233# F# FD: G"'en: S$5st!ate Concent!at"on In"t"al React"on Rate μmol/L μmol/L min . @.3@ 3.F @.2F . @. a. Calc$late t4e M"c4ael"s-Menten constants o( t4e a5o'e !eact"on. 5. >4en t4e "n4"5"to! )as ae# t4e "n"t"al !eact"on !ate )as ec!ease as (ollo)s: S$5st!ate μmol/L . 3.F .
In4"5"to! Μmol/L = = =
In"t"al React"on Rate Μmol/L min @. @.@D @.@
Is t4"s com%et"t"'e "n4"5"t"on o! noncom%et"t"'e "n4"5"t"onJ $st"(y yo$! ans)e! 5y s4o)"n t4e e((ect o( t4e "n4"5"to! !a%4"cally. QCont!"5$te 5y P!o(esso! Ga!y . 6ennett# T4e $n"'e!s"ty o( Toleo# Toleo# 7H Re?$"!e: MM constants
>"t4o$t In4"5"to! a 8anm$"! C s k m r
=
r max
+
1
r max
(C s )
! 9 @.D molB8-m"n !,ma& 9 @.=2F molB8-m"n
=
r max
+
(
r max C s
)
! 9 @. molB8-m"n
>"t4 "n4"5"to! a 8anm$"! C s k m r
=
r max
+
1
r max
(C s )
! 9 @. molB8-m"n !,ma& 9 @.F molB8-m"n
=
r max
+
(
r max C s
)
! 9 @.D molB8-m"n
!,ma& 9 @.=F molB8-m"n
!,ma& 9 @.= molB8-m"n
! 9 @.D molB8-m"n !,ma& 9 @.=33 molB8-m"n
! 9 @.F= molB8-m"n !,ma& 9 @.=F molB8-m"n
T4e!e(o!e# 8anm$"! "sot4e!m 5est ("t t4e ata )"t4 ! 9 @.D (o! )"t4o$t"n4"5"to!.
CHAPTER 2: ENZYME KINETICS 2.D T4e enzyme# cat4e%s"n# 4y!olyzes 8-l$tamyl-8-ty!os"ne to ca!5o5enzo&y-8-l$tam"c ac" an 8-ty!os"ne. It 4as 5een (o$n ,!antz an Ste%4enson# J.
Biol. Chem.#
# 3F# = t4at
t4e l$tam"c ac" (o!me "n t4e 4y!olys"s# "n4"5"ts ,com%et"t"'ely t4e %!o!ess o( t4e !eact"on 5y (o!m"n a com%le& )"t4 cat4e%s"n. T4e co$!se o( t4e !eact"on "s (ollo)e 5y a"n ty!os"ne eca!5o&ylase )4"c4 e'ol'es C72.
S$5st!ate μ moleBm8
In4"5"to! μ moleBm8
Rate o( C72 Gene!at"on μ moleBm8.m"n
=. =. =. @.D @.D @.D [email protected] [email protected] [email protected]
@ .F [email protected]@ @ .FD [email protected]@ @ .FD [email protected]
@.@=3= @.@2DF @.@33 @.@3 @.@F2 @.@2 @. @.@@ @.@FD
Calc$late ,a t4e 'al$e o( M"c4ael"s-Menten constants o( t4e enzyme# K s# an ,5 t4e "ssoc"at"on constant o( enzyme-"n4"5"to! com%le&# K I. 600 500
f(x) = 6 .41x + 329.1 R² = 0.99
400 300
f(x) = 6 .51x + 137.08 f(x) R² ==16.37x + 80.2 R² = 1
200 100 0 0
5
10
a. I9@ Rma& 9 Bm 9 @.F Ks 9 5Rma& 9 2.FD3
15
20
25
30
35
I 9 [email protected]@ Rma& 9 Bm 9 @.F@ KI 9 5Rma& 9 F.33D
I9 .FD Rma& 9 Bm 9 @.F3 K I 9 5Rma& 9 2.@2@ CHAPTER 2: ENZYME KINETICS AITI7NA8 PR768EM 4tt%s:BB))2.c4em"st!y.atec4.e$Bl)2B5Co$!seUIn(o!mat"onB3FBst$Ucom%Bc4a%2U.%( T4e (ollo)"n ata )e!e o5ta"ne (o! t4e !eact"on A + 6# catalyze 5y t4e enzyme Aase. T4e !eact"on 'ol$me )as m8 an t4e stoc< concent!at"on o( A )as F.@mM. Se'en se%a!ate !eact"ons )e!e e&am"ne# eac4 conta"n"n a "((e!ent amo$nt o( A. T4e !eact"ons )e!e "n"t"ate 5y a"n 2.@V8 o( a @VM sol$t"on o( Aase. A(te! F m"n$tes# t4e amo$nt o( 6 )as meas$!e. React"on 2 3 = F
ol$me o( A ae,V8 D @ F 2@ =@ @ @@
Amo$nt o( 6 %!esent at F m"n$tes ,nmoles 2 2 3 =3 F 2
,a Calc$late t4e "n"t"al 'eloc"ty o( eac4 !eact"on ,"n $n"ts o( VM.m"n- ,5 ete!m"ne t4e K M an ma& o( Aase (!om a 8"ne)ea'e!-6$!< %lot. ,c Calc$late < cat. S78WN: ,a Xo 9 ,2nmolBFm"n B ,.@m8 & ,@3 m8B8 & ,.@@ VmolBnmol 9 5.2 µM.min-1 React"on 2 3 = F
Xo,VM.m"n- F.2 F.D .D D. .2 2.= =.2
,5 Calc$late QS (o! eac4 !eact"on QA 9 ,.@@Dm8,FmM,m8 & ,@@@ VMB mM 9 =@ VM React"on
QS VM
,& BQS VM-
X,VM.m"n-
2 3 = F
=@ F@ F @@ 2@@ 3@@ F@@
@.@2F @.@2 @.@33 @.@@ @.@@F @.@@33 @.@@2
F.2 F.D .D D. .2 2.= =.2
y-"nt 9 B'ma& 9 @.@ Vmax = 16 µM/min
&-"nt 9 -BK M 9 B@.@2 K M = 83 µM
,c Calc$late QET 9 ,@.@@2m8,@ VM Aase B m8 9 @.@2 VM K cat 9 ma& B QET 9 , VMBm"n B @.@2 VM K cat= 800 /min
,y BX ,m"n-B VM @.2 @.2 @.2D @.@ @.@D @.@D @.@@
CHAPTER 2: ENZYME KINETICS AITI7NA8 PR768EM T4e stat"n !$ lo'ast"an 4el%s lo)e! c4oleste!ol le'el 5y acts as com%et"t"'e "n4"5"to! on t4e HMG-CoA !e$ctase enzyme# )4"c4 no!mally catalyzes an ea!ly ste% "n t4e 5"osynt4es"s %at4)ay c4oleste!ol Re?$"!e: a 7n a s"nle !a%4# s
b)
CHAPTER 2: ENZYME KINETICS AITI7NA8 PR768EM Pest"c"e "n4"5"t"on on enzyme 4as 5een !e%o!te# )4"c4 ca$se t4e enzyme act"'"ty to !e$ce. T4e collecte ata )"t4 an )"t4o$t "n4"5"t"on a!e %!esente 5elo). ete!m"ne t4e ty%e o( "n4"5"t"on an t4e K I (o! t4e "n4"5"to!. QS# M Rate QI9@# MBm"n@3 Rate QI92@nM# MBm"n@3
3.3@@-= F
F.@@@-=
.@@-= DD
.F@-3 2=
2.2@-3 =
3
=
@3
2F
1
Ass$m"n 8"ne)ea'e!-6$!< E?$at"on:
=
V
K m
1
V max S
+
1
V max
QI9@: 8"ne : y9=.32@@@-&*F.@@33@- 1 -
V max 9F.@@33@
V max 9 #DD.@D MBm"n
K m
- V max 9=.32@@@
K m 9 D.3=3@-= M
QI92@nM: 8"ne 2: y9.=@F@-&*F.@@- 1 -
V max 9F.@@
V max 9 3#=.@ MBm"n
K mapp V max
-
9.=@F@
K mapp 9 .==33@-3 M
K m
[ ] I K I
1+
= K mapp D.3=3@-= M
K I 9 2.D@@-D M
Ty%e o( In4"5"t"on: C7MPETITIE K I 9 2.D@@-D M
[
−9
1+
20 × 10
K I
]
M
=¿ .==33@-3 M
CHAPTER 2: ENZYME KINETICS AITI7NA8 PR768EM A ce!ta"n !eact"on 4as an act"'at"on ene!y o( 2F <Bmol. T4e !ate "s @.33Bs at FF C. ete!m"ne t4e 'al$e o( t4e s%ec"("c !ate constant at @@ C. GIEN: Ea92F <Bmol T9FF C K FF C 9 @.33
RE01IRE: T29@@ C K @@ C 9 J
S781TI7N: − a
K =A
e !"
T9FF C: −125 k# / mol 8.314 k$
@.339 A e kmolK ( 55+273 ) K
A=2.@@
T29@@ C: −125 k# / mol
K 100ºC
8.314 k$
9 ,2.@@ e kmolK ( 100+ 273 ) K
K 100ºC 9D2.DD2Bs
CHAPTER 2: ENZYME KINETICS AITI7NA8 PR768EM T4e enzyme ca!5o&y%e%t"ase catalyzes t4e 4y!olys"s o( %e%t"es. T4e (ollo)"n !es$lts )e!e o5ta"ne )4en t4e !ate o( enzymolys"s o( C6GP )as mon"to!e )"t4o$t "n4"5"t"on at QC6GP@9 @.3 molBm3. C&'( mo ) −2 dm 10 Rate# mol
.2F
3.D=
F.D
.3
@.3D
@.=
@.DF
.@@
3
dm s
>4en 2.@@-3 molBm3 %4enyl 5$ty!ate "on )as ae to t4e sol$t"on# t4e !es$lts )e!e: C&'( mo ) −2 dm 10 Rate# mol
.2F
2.F@
=.@@
F.F@
@.2
@.3@
@.3==
@.F=D
3
dm s
In a se%a!ate e&%e!"ment# t4e e((ect o( F.@@-2 molBm3 5enzoate "on )as mon"to!e an t4e !es$lts )e!e: C&'( mo ) −2 dm 10 Rate# mol
.F
2.F@
F.@@
@.@@
@.D3
@.2@
@.23
@.2=
3
dm s
ete!m"ne t4e ty%e o( "n4"5"t"on an K I (o! %4enyl 5$ty!ate an 5enzoate "on.
S781TI7N: 1
Ass$m"n 8"ne)ea'e!-6$!< E?$at"on: 8"ne : )"t4o$t "n4"5"t"on
=
V
K m
1
V max S
+
1
V max
[email protected]2*@.@2& V max 9 .23@ molBm3.s K m 9 @.@2 molBm3
8"ne 2: )"t4 %4enyl 5$ty!ate "on y9.@FD*@.@@& V max 9 @.D=F molBm3.s K m 9 @.@F2 molBm3 8"ne 3: )"t4 5enzoate "on y93.F*@.@3@@& V max 9 @.2F molBm3.s K m 9 D.@22D molBm3
TYPE 7 INHI6ITI7N:
P4enyl 6$ty!ate Ion: C7MPETITIE 6enzoate Ion: 1NC7MPETITIE
7R PHENY8 61TYRATE I7N: K m
[ ] I K I
1+
= K mapp
[
0.0266 1 +
K I 9 .3@-3 molBm3
−3
2.0 × 10
K I
]
=0.0592
7R 6ENZ7ATE I7N: V maxapp=
V max I 1+ K I
¿ @.2F
K I 9 @.@3D molBm3
1.2306 −2
1+
5.0 × 10
K I
