# kinase by hedongchenchen

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```									               Symbolic Analysis
of

Initial-Rate Kinase Kinetics
Petr Kuzmic, Ph.D.
BioKin, Ltd.
pkuzmic@biokin.com
http://www.biokin.com/seminar/kinase

Outline
• Alternative to algebra: symbolic / numerical analysis.
• Example 1: EuATP inhibition of hexokinase.
• Equilibrium approximation: general numerical treatment.
• Example 2: Mechanism of p56lck protein kinase inhibition.
• Beyond kinases: Analysis of highly complex mechanisms.
• Demonstration of computer program DynaFit.
Difficulty of Enzyme Kinetics
Algebraic Complexity

“The traditional method for introducing enzyme kinetics
We shall try instead to show how the equations are derived
and manipulated. In this way, the nonmathematical person
can learn enough of the subject [...] without having to bother
with the actual detailed algebra.”
(W.W. Cleland, 1970)

   Thirty years later, can a “non-mathematical person”
studying enzyme kinetics avoid “bothering with algebra” ?
2
W. W. Cleland (1970) in “The Enzymes” P. D. Boyer (Ed.), Vol. II
Part 1: Mechanism

Steady-state sequential ordered “Bi Bi” mechanism

k1                           A … ATP
E+A          EA                     B … peptide
k-1

k2           kp             k3
EA + B         EAB           EPQ            EQ + P
k-2           k-p            k-3

k4
k-4
P … phospho-peptide

3
I. Segel (1975) “Enzyme Kinetics” Wiley, New York, p. 560
Part 2: Algebraic Model

k1
KiA     
k1
k3k4k p                    plus five
K mA                                                other constants:
k1 (k3k4  k3k p  k4k p  k4k p )
k1k2  k1k p  k 1k  p  k2k p
KiB                                                 KiQ , KmQ ,
k2 ( k p  k  p )
k4 (k2k3  k 2k p  k3k p )        KiP , KmP ,
K mB    
k2 (k3k4  k3k p  k4k p  k4k p )
Vr
k3k4k p
Vf       [ E ]t
k3k4  k3k p  k4k p  k4k  p

4
I. Segel (1975) “Enzyme Kinetics” Wiley, New York, p. 564
Part 2: Algebraic Model (continued)

Rate equation:

            [ P ][Q ] 
V f Vr  [ A][B ] 


               K eq  
v
V f K mQ [ P ]
Vr KiA K mB  Vr K mB [ A]  Vr K mA[ B ]                   
K eq
V f K mP [Q ]                  V f K mQ [ A][P ] V f [ P ][Q ]
 Vr [ A][B ]                                     
K eq                            KiB K eq             K eq
Vr K mA[ B ][Q ] Vr [ A][B ][P ] V f [ B ][P ][Q ]
                   
KiQ                  KiP               KiB K eq

5
I. Segel (1975) “Enzyme Kinetics” Wiley, New York, p. 563
Symbolic Modeling of Initial Rate Data
Program DynaFit

“Program DYNAFIT for the Analysis of Enzyme Kinetic Data:
Application to HIV Proteinase”
Kuzmic, P. (1996) Anal. Biochem. 237, 260-273.

• Designed for the analysis of (a) reaction progress,
(b) initial velocities, and (c) equilibrium binding data.

• 1998-2000: 48 journal articles cited DynaFit ;
so far used exclusively for the analysis of reaction
progress (see handout for list of references).

• Can be applied conveniently for the analysis of
initial-rate kinase kinetics.
6
Part 3: Symbolic Model

Mechanism:              k1
E+A           EA
k-1

k2          kp          k3
EA + B         EAB         EPQ         EQ + P
k-2         k-p         k-3

k4
EQ          E+Q
DynaFit data:            k-4

[mechanism]
E + A <==> EA             :    k1      k-1
EA + B <==> EAB           :    k2      k-2
EAB <==> EPQ              :    kp      k-p
EPQ <==> EQ + P           :    k3      k-3
EQ <==> E + Q             :    k4      k-4

7
Part 4: Differential-Equation (DE) Model

Mathematical model internally derived by DynaFit :
d[E]/dt       =      -k1[E][A]+k-1[EA]+k4[EQ]-k-4[E][Q]
d[A]/dt       =      -k1[E][A]+k-1[EA]
d[EA]/dt      =      +k1[E][A]-k-1[EA]-k2[EA][B]+k-2[EAB]
d[B]/dt       =      -k2[EA][B]+k-2[EAB]
d[EAB]/dt     =      +k2[EA][B]-k-2[EAB]-kp[EAB]+k-p[EPQ]
d[EPQ]/dt     =      +kp[EAB]-k-p[EPQ]-k3[EPQ]+k-3[EQ][P]
d[EQ]/dt      =      +k3[EPQ]-k-3[EQ][P]-k4[EQ]+k-4[E][Q]
d[P]/dt       =      +k3[EPQ]-k-3[EQ][P] … overall rate
d[Q]/dt       =      +k4[EQ]-k-4[E][Q]
8
Example 1: Hexokinase Inhibition by EuATP
1. Raw Data

5
2

E=
uT0
P] 0
[ A1
0
2

1
5                            u
E=P]
T0
[ A5
1/v

1
0                            u
E=P
T]
[ A0

5
.0
0
00         .5
0
00         0
.0
01         .5
0
01

/g
[ T
M
1A]
P
9
Morrison & Cleland (1980) Biochemistry 19, 3127-3131. Figure 3.
Example 1: Hexokinase Inhibition by EuATP
2. Mechanism

ka(Mg)
2+
Mg        + ATP                           S
kd(Mg)

ka(Eu)
2+
Eu        + ATP                           I
kd(Eu)

ka(S)                 kr
E+S                       ES                     E+P
kd(S)

ka(I)
E+I                             EI
kd(I)

10
Morrison & Cleland (1980) Biochemistry 19, 3127-3131.
Example 1: Hexokinase Inhibition by EuATP
3. Algebraic Model

b  2c / K m  b 2  4 a c
app
v  Vmax
2(aKm  b  c / K m )
app         app

where           [ I ]   [ S ]  [ I ]  Kd [ Mg 2 ]
a  1 
        1 
                
    Ki        Ki  Ki K Mg. ATP

 [S ]  [ I ]                Kd         [S ]  [ I ] 
b  [ S ] 1 
                [ Mg 2 ]             1            
     Ki                   K Mg. ATP   
     Ki      

c   [ Mg 2 ]
Kd
[ S ]  [ I ]
K Mg. ATP
11
Morrison & Cleland (1980) Biochemistry 19, 3127-3131. Equation (11).
Example 1: Hexokinase Inhibition by EuATP
4. Symbolic Model

DynaFit input data (partial display):

data = velocities

[mechanism]
Mg + ATP <===> S       :    ka(Mg)   kd(Mg)
Eu + ATP <===> I       :    ka(Eu)   kd(Eu)

E + S <===> E.S        :    ka(S)    kd(S)
E.S ----> E + P        :    kr
E + I <===> E.I        :    ka(I)    kd(I)

[constants] ...

12
Example 1: Hexokinase Inhibition by EuATP
5. Differential Equation Model

Generated automatically by DynaFit:
d[Mg]/dt =    -kamg[Mg][ATP]+kdmg[S]
d[ATP]/dt =   -kamg[Mg][ATP]+kdmg[S]-kaeu[ATP][Eu]+kdeu[I]
d[S]/dt =     +kamg[Mg][ATP]-kdmg[S]-kas[S][E]+kds[E.S]
d[Eu]/dt =    -kaeu[ATP][Eu]+kdeu[I]
d[I]/dt =     +kaeu[ATP][Eu]-kdeu[I]-kai[I][E]+kdi[E.I]
d[E]/dt =     -kas[S][E]+kds[E.S]+kr[E.S]-kai[I][E]+kdi[E.I]
d[E.S]/dt =   +kas[S][E]-kds[E.S]-kr[E.S]
d[P]/dt =     +kr[E.S]
d[E.I]/dt =   +kai[I][E]-kdi[E.I]
13
Example 1: Hexokinase Inhibition by EuATP
6. Results of Fit using DynaFit

.
1
05

E=
uT
P]
[ A0

E=
uT0
P]
[ A5
.
1
00

E=
uT0
P] 0
[ A1
v
.
0
05

.
0
00

0         0
0
2          4
0
0           0
0
6

Mg
T
[A]
P
14
Morrison & Cleland (1980) Biochemistry 19, 3127-3131.
Example 1: Hexokinase Inhibition by EuATP
6. Results of Fit (continued)

Initial Fit    Error %
Km =    (kds+kr)/kas
kdeu     1       1.4    0.21   15.0
62.0 mM
kds      10000 6040 300        4.9
kr       200     157    2.3    1.5        Kd =    kdeu/kaeu
0.14 mM
kdi      1000    1880   110    5.7
kamg     10                               Ki =    kdi/kai
kdmg     12                                       18.8 mM

kas      100                              Vmax = [E]  kr
kai      100                                     0.157 mM/sec
[E]      0.001
15
Example 1: Hexokinase Inhibition by EuATP
7. Comparison of Results

DynaFit         Morrison &
Cleland (1980)

Km (mM)                62               63  7

Kd (mM)                0.14             0.16  0.04

Ki (mM)                19               18  2

Vmax (mM/sec)          0.157            0.158  0.006

16
Morrison & Cleland (1980) Biochemistry 19, 3127-3131.
Example 1: Hexokinase Inhibition by EuATP
8. Conclusions

• Algebraic method and the differential-equation
method (used in DynaFit) give the same results.

• Algebraic model is tedious to derive and prone
to error. DynaFit model is derived by the computer.

• Algebraic model cannot be extended to “tight binding”.
DynaFit model is applicable to “tight binding”
without change.

17
Rapid Equilibrium Approximation
Quick Summary

• Applicable when the catalytic step in the mechanism
is relatively slow compared to binding and dissociation.

• DynaFit solves multiple simultaneous equilibria by
using a special numerical (iterative) method :

I & Nancollas (1972) Anal. Chem. 44, 1940-1950.

• This numerical method is equally applicable to
“tight-binding” and to classical enzyme inhibition.

18
Rapid Equilibrium Approximation
Example Script
data   = velocities

[mechanism]
E + S <===> ES     :   Ks     dissoc.
ES ---> E + P      :   kcat
ES + S <===> ES2   :   Ks2    dissoc.
E + I <===> EI     :   Ki     dissoc.
ES + I <===> ESI   :   Kiu    dissoc.

[constants]
Ks = 5000    ?, kcat = 400000 ?
Ks2 = 2000   ?
Ki =    10   ?
Kiu =   10   ?
...                                        19
Example 2: p56lck Tyrosine Kinase Inhibition
1. Raw Data - Peptide as Varied Substrate
Lineweaver-Burk plot
O   O
5
1
NH2
N        N
[I] = 0                             N
0
1
WIN-61651
N
1/v

N
5

[I] = 80 mM
0

0           1              2
/R
[S
R
1R]
C
20
Faltynek et al. (1995) J. Enz. Inhib. 9, 111-122.
Example 2: p56lck Tyrosine Kinase Inhibition
2. Peptide Kinetics: Michaelis-Menten Model

[mechanism]
E + S <===> ES
0
0
1
ES ---> E + P
v

0
5

0

0        0
0
20           4
0
00          0
0
60
RR
R
[S]
C
21
Faltynek et al. (1995) J. Enz. Inhib. 9, 111-122.
Example 2: p56lck Tyrosine Kinase Inhibition
3. Peptide Kinetics: Substrate Inhibition

[mechanism]
E + S <===> ES
0
0
1
ES ---> E + P
ES + S <===> ES2
v

0
5

0

0        0
0
20           4
0
00          0
0
60
RR
R
[S]
C
22
Faltynek et al. (1995) J. Enz. Inhib. 9, 111-122.
Example 2: p56lck Tyrosine Kinase Inhibition
4. Peptide Kinetics: Inhibition Mechanism
Mixed-type noncompetitive inhibition + substrate inhibition

[mechanism]
0
0
1
E + S <===> ES
ES ---> E + P
ES + S <===> ES2
E + I <===> EI
ES + I <===> ESI
rate

0
5

0

0       0
0
20           4
0
00          0
0
60
Rm
R,R
C]
[ SM
23
Faltynek et al. (1995) J. Enz. Inhib. 9, 111-122.
Example 2: p56lck Tyrosine Kinase Inhibition
5. Peptide Kinetics: Published Mechanism
Mixed-type noncompetitive inhibition (Fig. 1B in Faltynek et al.)

[mechanism]
0
0
1
E + S <===> ES
ES ---> E + P

E + I <===> EI
ES + I <===> ESI
rate

0
5

0

0       0
0
20           4
0
00          0
0
60
Rm
R,R
C]
[ SM
24
Faltynek et al. (1995) J. Enz. Inhib. 9, 111-122.
Example 2: p56lck Tyrosine Kinase Inhibition
6. Peptide Kinetics: Comparison of Results

DynaFit            Faltynek
et al. (1995)

Ks (mM)          9100  3700 990  140
Ks2 (mM)         1100  450    —
Ki (mM)          28  2      18  4
Kiu (mM)         14  5      67  18

squares          2.1                19.5

25
Faltynek et al. (1995) J. Enz. Inhib. 9, 111-122.
Example 2: p56lck Tyrosine Kinase Inhibition
7. Peptide Kinetics: Conclusions

DynaFit                            Faltynek
et al. (1995)
• Peptide substrate of             • Peptide substrate of p56lck
p56lck kinase shows                kinase follows pure
substrate inhibition.              Michaelis-Menten kinetics.

• WIN-61651 has                    • WIN-61651 has greater
greater affinity for               affinity for peptide site.
ATP site than for
peptide site.

• These conclusions are
incorrect.
26
Faltynek et al. (1995) J. Enz. Inhib. 9, 111-122.
Beyond Kinases
Analysis of highly complex mechanisms

• At least in principle, mechanisms shown so far can
be described by algebraic models. (Exception:
tight-binding inhibition).

• However, many biochemical mechanisms cannot
be described by algebraic models. A single rate
equation can never be derived.

• In the latter case, tools such as DynaFit become
a necessity, not just a convenience.

27
Example 3: Tissue Factor Pathway to Thrombin
1. Symbolic Definition of Mechanism
[mechanism]
IX + TF.VIIa     <==>    IX.TF.VIIa               :   k6    k16
IX.TF.VIIa       -->     TF.VIIa + IXa            :   k11
X + TF.VIIa      <==>    X.TF.VIIa                :   k6    k17
X.TF.VIIa        -->     TF.VIIa + Xa             :   k12
X + VIIIa.IXa    <==>    X.VIIIa.IXa              :   k6    k18
X.VIIIa.IXa      -->     VIIIa.IXa + Xa           :   k13
IX + Xa          -->     Xa + IXa                 :   k15
V + Xa           -->     Va + Xa                  :   k1
VIII + Xa        -->     VIIIa + Xa               :   k3
V + IIa          -->     IIa + Va                 :   k2
VIII + IIa       -->     VIIIa + IIa              :   k4
II + Va.Xa       <==>    II.Va.Xa                 :   k6    k19
II.Va.Xa         -->     Va.Xa + mIIa             :   k14
mIIa + Va.Xa     -->     Va.Xa + IIa              :   k5
VIIIa + IXa      <==>    VIIIa.IXa                :   k7    k9
Va + Xa          <==>    Va.Xa                    :   k8    k10
28
Jones & Mann (1994) J. Biol. Chem. 269, 23367.
Example 3: Tissue Factor Pathway to Thrombin
2. Results of DynaFit Simulation

.
1
00

a
V
conetraion, mM
I
VI
I
I
VI
Ia
X
I
.
0
05
X
a
X

.
0
00

0       0
0
1          2
0
0          0
0
3
i ()
m
tee
sc
29
Jones & Mann (1994) J. Biol. Chem. 269, 23367.
Example 3: Tissue Factor Pathway to Thrombin
2. Results of DynaFit Simulation (contd.)

.
0
1

I m
aI
I+a
I
conetraion, mM

.
5
0

.
0
0

0       0
0
1          2
0
0          0
0
3
i ()
m
tee
sc
30
Jones & Mann (1994) J. Biol. Chem. 269, 23367.
Example 4: Fatty-Acid Biosynthesis Assay
1. Symbolic Definition of Mechanism
[mechanism]

; Malonyl transfer (fabD)
MalCoA + fabD <==> MalCoA.fabD                :   k1    k-1
MalCoA.fabD + AcACP <==> MalCoA.fabD.AcACP    :   k2    k-2
MalCoA.fabD.AcACP --> MalAcACP + CoA + fabD   :   k3

; Condensation (fabF)
MalAcACP + fabF <==> MalAcACP.fabF            :   k4    k-4
MalAcACP.fabF <==> KeBuACP.fabF               :   k5    k-5
KeBuACP.fabF --> KeBuACP + fabF               :   k6

; Reduction (fabG)
KeBuACP + fabG <==> KeBuACP.fabG             :    k7    k-7

; Coupled reduction (FMN oxidoreductase)
31
Example 4: Fatty-Acid Biosynthesis Assay
2. Results of DynaFit Simulation

.
0
1

M
N
F

0
.
5               it c nn )
i
i l na (
noti sr
a eoM
ct    m
a 0
b .1
0
fD 0
conetraion,M

a 0
b .1
fF0 0
a 0
b .1
fG00
c
A .1
C0
P0
A 0
E0.1
0
0
a
lA
C 0
o
M 1
A
D 0
P
H
N 1
M1
N 0
F2
H

.
0
0

0     0
0
10       2
0
00       0
0
30
i (
m s
e
te)c
32
Symbolic Analysis of Initial Rate Kinetics
Summary and Conclusions

• Program DynaFit can be used to analyze initial
reaction velocities observed in enzyme assays.

• The two main advantages are (a) convenience and (b)
general applicability to an arbitrarily complex mechanism.

• REMAINING PROBLEM : A major limitation on analysis
of complex mechanisms is that sufficient information
(i.e., rate constants) must be available for individual steps.

• POSSIBLE SOLUTION : Customize DynaFit to simplify
kinetic analysis where insufficient information is available
“differential-algebraic” systems).                         33

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