# A Mathematical Modelling of Signal Transduction System via Insulin

Document Sample

```					Identification for Insulin Signal Kinetics in
HEK293 Cells via Mathematical Modeling

Department of Mathematics. POSTECH       Kwang Ik Kim
Department of Life Science, POSTECH      Sung Ho Ryu

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Introduction

   Insulin signal transduction is a signaling path process
from external stimulus to a cellular response.
   The fundamental motif in signaling network is the
phosphorylation and dephosphorylation which have a
dynamic profile.

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Introduction

   To identify the dynamics of insulin signal transduction
system, a mathematical model, which governs the signal
transduction from an extracellular stimulation to the
activation of intracellular signal molecules is constructed.
   In insulin signal transduction, each signal protein has its
own kinetic profile in such a way that IR, IRS , Akt and
Erk are phosphorylated transiently.

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Introduction
   These kinetic profiles are determined by their kinases
and phosphatases appropriately for their physical roles
in insulin signal transduction.
   Through this system, it is possible to predict each
signaling proteins quantitatively, once the concentration
of treated insulin is given, which is very important to
regulate the pharmaceutical control of insulin
concentration

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Kinetic scheme of insulin-induced
Insulin

PTP1B     IRS      IRS*     PI3K     PI3K*    PDK       PDK*    Akt      Akt*
Grb2/Sos

RasGDP      RasGTP Raf                                  PP2A
Raf*

PP1
MEK        MEKP      MEKPP

PP2A
Insulin-bound insulin receptor initiates important
signal transductions, IRS-PI3K-PDK-Akt and IRS-                                   ERK     ERKP    ERKPP
Ras-Raf-MEK-ERK pathways:            , mass action:

MKP3
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Simplified kinetic model of insulin
signaling
Insulin
k1
IR                        IR*
E1
k3            k2
IR*-E1    k-2                              k5
E1                                        k4
IRS              IR*-IRS            IRS*
k-5
k7                   E2
k6
IRS*-E2
E2                         k-6

k9                                     k
Akt             IRS*-Akt           AKt*   ERK           IRS*-ERK K13         ERK*
k8                 k-9                  k12                   -13

k11                 E3                    k15                E4
k10    k-10                             k14       k-14
Akt* -E3                               ERK* -E4
E3                                        E4

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Basic module in signal transduction

k1          k2
E1 + S           C         P + E1
k-1
E2
k4           k3
E2P        k-3

E2

Michelis-Menten forward and backward kinetics

dp/dt = k2[E1][S] / (KM+[S]) – 4[E2][[P] / (KM`+[P] ) ,

where KM=(k-1+k2) / k1, KM`=(k-3+k4)/k3

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Kinetic equation in insulin signal
transduction

d[ IR*]
 k1[ I ][ IR]  k 3{[ IR *0 ]  [ IR*]}  k 3[ IR]
dt
d [ IRS*]
 k 5{[ IR *0 ]  [ IR* ]}  k 7{[ IRS *0 ]  [ IRS*]}  k 7[ IRS ]
dt

d[ ERK *]
 k13{[ IRS *0 ]  [ IRS*]}  k15{[ ERK *0 ]  [ ERK *]}  k15[ ERK ]
dt
d [ Akt*]
 k 9{[ IRS *0 ]  [ IRS*]}  k 11{[ Akt *0 ]  [ Akt*]}  k11[ Akt ]
dt

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Kinetic equations modified from the
insulin signal transduction kinetics

d[IR*] / dt = k1[I][IR] – k3[E10][IR*] / (K2+[IR*])

d[IRS*] / dt = k5[IR0*][IRS] / (K3+[IRS]) – k7[E20][IRS*] / (K4+[IRS*])

d[Akt*] / dt = k9[IRS0*][Akt] / (K5+[Akt]) – k11[E30][Akt*] / (K6+[Akt*])

d[ERK*] / dt = k13[IRS0*][ERK] / (K7+[ERK]) – k15[E40][ERK*] / (K8+[ERK*])

Where
K2 = (k-2+k3) / k2, K3 = (k-4+k5) / k4, K4 = (k-6+k7) / k6,
K5 = (k-8+k9) / k8, K6 = (k-10+k11) / k10,
K7 = (k-12+k13) / k12, K8 = (k-14+k15) / k14

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Experimental materials and methods

1. Cell preparation
HEK 293 cells were subcultured in 6cm tissue dishes with Dulbecco’s Modified
Eagle Medium (DMEM) containing 10 % fetal bovine serum.

2. Fasting
Dishes to be processed on the same day were plated with equal number of
cells. The cells were incubated for 24h in DMEM.

24h

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Experimental materials and methods

3. Insulin Stimulation
At various times, insulin was added to each plate at the final concentration
indicated and incubated for the time interval specified. At the end point of
the experiment, each plate was washed twice with ice-cold Dulbecco’s
phosphate buffered saline and lysed in 150nM of ice-cold buffer containing
40mM HEPES.

4. Sonication
Each lysate transferred to Eppendorf tube after scapping was sonicated and
contrifuged at 4 °C for 15 min to acquire supernatant. The protein concentration
of each lysate was measured by Bradford assay.

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Experimental materials and methods

5. Centrifugation
To quantify the phosphorylation of signal proteins, cell lysate samples
containing equal amounts of proteins were resolved by SDS-PAGE and
electrophoretically transferred to nitrocellulose membrane.

-   -     -    -

-

-   -     -    -

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Experimental materials and methods

6. Electrophoresis
NC

Zel               -                   +

Zel

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Experimental Materials and Methods
7. Antibody
After blocking with 5 % skimmed milk in TTBS (10 mM Tris/HCl, pH7.5, 150 mM
NaCl and 0.5 %(w/v) tween 20), the membranes were incubated with the antibodies
(anti-phospho-IRS, anti-phospho-IR, anti-phospho-Akt, anti-phospho-ERK and
anti-actin). Washed with TTBS, the membranes were incubated with peroxidase-
conjugated goat anti-rabbit IgG (KPL) and peroxidase-conjugated goat anti-mouse
IgA+IgG+IgM (H+L) (KPL).

8. Quantitative Analysis
To visualize the phosphorylated proteins, the enhanced chemillominescence system
(ECL system from Amersham Corp.) was used and proteins bands were quantified
using densidomiter (Fuji-Film Corp.)

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Phosphorylation patterns of signal proteins
with respect to insulin stimulation time

A          Insulin 10 nM                                      B          Insulin 100 nM
Time                                                          Time
0.25

0.25

0.5
0.5

20
20

10
10
(min):                                                        (min):

0

2
5
1
WB
0

2

5
1

WB
p-IR                                                          p-IR
(pY1158)                                                      (pY1158)

p-IRS                                                         p-IRS
(pY989)                                                       (pY989)

p-Akt                                                         p-Akt
(pS473)                                                       (pS473)

p-ERK                                                         p-ERK
(pT202                                                        (pT202
/Y204)                                                        /Y204)
Actin                                                         Actin

HEK 293 cells are deprived of serum for 24h before treatment and stimulated
with 10 nM and 100 nM of insulin for indicated time and lysed.The lysates are
subjected to SDS-PAGE and immunoblotted.
A: HEK 293 cells are stimulated with 10 nM of insulin.
B: HEK 293 cells are stimulated with 100 nM of insulin.

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Regresstion with in vivo data via least
squares method for p-IR
(B)
(A)                                     b x
y
xa

a=2.78201
10 nM
b=0.68833
b x
y       c              a=1.39433
xa      100 nM
b=0.54915

Graphs from in vivo experimental data and in silico analysis
(A) Based on the in vivo data, kinetic graphs for insulin signal proteins
were drawn.
(B) After regression with in vivo data, in silico graph were obtained.

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Regresstion with in vivo data via least squares
method for p-IRS

b x
y                             y
b x
xa                          xa

a=0.83907
10 nM
b=1.32975

a=0.25139
100 nM
b=0.91993

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Regresstion with in vivo data via least squares
method for p-Akt

1.4

1.2

2 ymax
y                ymax
1

 ax
0.8                                         1 e
0.6

0.4            10nM Insulin
100nM Insulin

0.2

0
0   5   10        15         20

ymax=0.85000
10 nM
a=2.25335
ymax=1.06250
100 nM
a=4.44860

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Regresstion with in vivo data via least
squares method for p-ERK

a=0.35000                                           a=0.86600
b=0.17241                                           b=0.02858

ax1.9  bx cx  d gx2   c=0.57564
ax1.5  bx cx d gx2   c=0.35690
y                  e                                y                 e
10nM
0.3  x cx  f         d=0.17306         100nM
0.3  x cx f         d=0.78620

f=- 0.71380                                         f=- 0.71380

g=- 0.00992                                         g=- 0.01272

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Kinetic graphs for p-IR in vivo and in silico
least squares fitted data

p-IR In vivo experimental data          p-IR In silico fitted data

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Kinetic graphs for p-IRS in vivo and in silico
least squares fitted data

p-IRS In vivo data              p-IRS least squares fitted data

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Kinetic graphs for p-Akt in vivo and in
silico least squares fitted data

p-Akt In vivo data          p-Akt least squares fitted data

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Kinetic graphs for p-ERK in vivo and in
silico least squares fitted data

p-ERK In vivo data          p-ERK least squares fitted data

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Relative kinetic graphs for phosphorylation
of IR

Phosphorylation of IR for 10nM      Phosphorylation of IR for 100nM

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Relative kinetic graphs for phosphorylation
of IRS

Phosphorylation of 10nM IRS         Phosphorylation of 100nM IRS

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Relative kinetic graphs for phosphorylation
of Akt

Phosphorylation of 10nM Akt            Phosphorylation of 100nM Akt

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Relative kinetic graphs for phosphorylation
of ERK

Phohphorylation of 10nM ERK     Phohphorylation of 10nM ERK

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Linearlized System for Insulin Signaling
Kinetics

B(t  h)  hAX  B(t )  O(h),where
 [ IR*](t  h)                           [ IR*](t ) 
                                                       
B (t  h)   [ IRS *](t  h)               B (t )  
[ IRS *](t ) 
 [ Akt*](t  h)                          [ Akt*](t ) 
 [ ERK *](t  h) 
                                        
 [ ERK *](t ) 
                                                       

X   k1[ IR], k5 , k9 , k13  k3 , k7 , k11 , k15  k3[ IR], k7 [ IRS ], k11[ Akt ], k15 [ ERK ]
T

[I ]        0                  0                    0           [ IR0 ]  [ IR* ]
*
0                   0                      0              1000
                                                                                                                                                            
0 [ IR0 ]  [ IR ]
*         *
0                    0                              [ IRS0 ]  [ IRS ]
*          *
0100
A
0                                       0                      0

 0          0         [ IRS0 ]  [ IRS * ]
*
0                   0                   0          [ Akt0 ]  [ Akt * ]
*
0              0010

 0                                                                                                                                                          
[ IRS0 ]  [ IRS * ]                                                              [ ERK 0 ]  [ ERK * ] 0 0 0 1 
*                                                                                   *
            0                  0                                        0                   0                   0                                           

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Reaction coefficients Identified via Pseudo-
Inverse with Householder transformation

0.045                                          16
Insulin
0.04                                          14
10nM Insulin
10nM Insulin
10nM Insulin
0.035                                          12             100nM Insulin
k1                                               100nM Insulin

IR                             IR*          0.03
10
E1                    0.025
8
0.02
k3                                    0.015
6
k2
4
IR*-E1                           0.01
k-2                                                              2
E1                                        IRS   0.005
0                                           0

-0.005                                         -2
0   5      10        15          20     0   5   10            15     20

k1[IR]                                  k3

[ IR* ](t  h)  {k1[ I ][ IR](t )  k3 ([ IR* ]  [ IR0 ])(t )  k3[ IR](t )}h  [ IR* ](t )  O(h)
*

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Identified reaction coefficients and p-IR
signal proteins
0.3                                                     0.3

10nM IR*
0.25                              k [IR]                 0.25                          100nM IR*
1                                                   k [IR]
k                                                     1
3                                                   k
3
0.2                                                     0.2

0.15                                                     0.15

0.1                                                     0.1

0.05                                                     0.05

0                                                       0

-0.05                                                   -0.05
0        5       10      15          20                 0      5       10     15           20

p-IR with K1 and k3[IR]                                 p-IR with K1 and k3[IR]
for 10 nM insulin                                       for 100 nM insulin

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Reaction coefficients Identified via Pseudo-
Inverse with Householder transformation
0.04                                          0.2

IR*                                        0.03                                         0.15
10nM Insulin
10nM Insulin
100nM Insulin
100nM Insulin
0.02
0.1
k5           0.01
k4                       IRS*
IRS               IR*-IRS                                                               0.05
k-5            0

k7                    E2                                                            0
-0.01
k6
IRS*-E2                                                                -0.05
E2                          k-6            -0.02

-0.03                                        -0.1

-0.04                                        -0.15
ERK           0   5     10       15           20           0   5   10     15            20
Akt                                                k5                                        k7

[ IRS * ](t  h)  {k5 ([ IR0 ]  [ IR* ])(t )  k7 ([ IRS * ]  [ IRS0 ])(t )  k7[ IRS ](t )}h  [ IRS * ](t )  O(h)
*                                         *

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Identified reaction coefficients versus
p-IRS signal proteins

0.8                                                  1.2

0.7
10nM IRS*               1
100nM IRS*
k
0.6                            5                                              k
5
k
7                     0.8                      k
7
0.5

0.4                                                  0.6

0.3                                                  0.4
0.2
0.2
0.1
0
0

-0.1                                                 -0.2
0        5      10         15      20             0       5      10     15          20

p-IRS with K5 and k7                              p-IRS with K5 and k7
for 10 nM insulin                                 for 100 nM insulin

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5   ([ ERK * ]  [ ERK0 ])(t )  k15[ ERK ](t)}h  [ ERK * ](t)  O(h)
*

Reaction coefficients Identified via Pseudo-
Inverse with Householder transformation

0                                       0.35
IRS*
-0.1                                     0.3
10nM Insulin
k9                                                                                 100nM Insulin

Akt             IRS*-Akt               AKt*     -0.2                                     0.25
k8                     k-9                                 10nM Insulin
100nM Insulin
E3                    -0.3                                     0.2
k11                 k10
-0.4                                     0.15
Akt*-E3           k-10
E3                                             -0.5                                     0.1

-0.6                                     0.05

-0.7                                      0
0   5   10         15        20          0   5       10         15          20

k9                                          k11

[ Akt * ](t  h)  {k9 ([ IRS0 ]  [ IRS * ])(t )  k11 ([ Akt * ]  [ Akt0 ])(t )  k11[ Akt ](t )}h  [ Akt * ](t )  O(h)
*                                            *

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Identified reaction coefficients versus
p-Akt signal proteins

1                                                    1.2

1
0.8
0.8
0.6                                                                             100nM Akt*
10nM Akt*               0.6                       k
9
k
9                                                k
0.4                           k                                                  11
11                     0.4

0.2                                                   0.2

0
0
-0.2
-0.2
-0.4

-0.4                                                  -0.6
0       5      10       15         20              0       5      10       15         20

p-Akt with K9 and k11                              p-Akt with K9 and k11
for 10 nM insulin                                  for 100 nM insulin

Combinatorial and Computational Mathematics Center
Reaction coefficients Identified via Pseudo-
Inverse with Householder transformation
0.2                                       0.5

IRS*                                        0.1                                       0.4
10nM Insulin
100nM Insulin

k13            0                                        0.3

ERK            IRS*-ERK               ERK*
k12                    k-13          -0.1                                      0.2
E4                                       10nM Insulin
k15                                     -0.2                 100nM Insulin        0.1
k14
ERK*-E4         k-14           -0.3                                       0
E4
-0.4                                      -0.1

-0.5                                      -0.2
0   5   10         15         20          0   5   10      15            20

k15
k13

[ ERK * ](t  h)  {k13 ([ IRS0 ]  [ IRS * ])(t )  k15 ([ ERK * ]  [ ERK0 ])(t )  k15[ ERK ](t)}h  [ ERK * ](t)  O(h)
*                                            *

Combinatorial and Computational Mathematics Center
Identified reaction coefficients and p-
ERK signal proteins
2.5                                                  3

2                                                  2.5                          100nM ERK*
10nM ERK*                                           k
k                                                    13
13                                                 k
k                                                    15
15                     2
1.5
1.5
1
1
0.5
0.5

0                                                   0

-0.5                                                -0.5
0       5       10      15        20                0        5      10       15        20

p-ERK with K13 and k15                              p-ERK with K13 and k15
for 10 nM insulin                                   For 100 nM insulin

Combinatorial and Computational Mathematics Center
Interpolation with identified parameters
for 30nM insulin concentration

0.3                                                              0.9

0.8
0.25
0.7

0.2                                                              0.6

0.5
0.15
0.4

0.1                                                              0.3

0.2
0.05
0.1

0                                                                0
0   2   4   6   8   10   12   14   16   18   20                  0   2   4   6   8   10   12   14   16   18   20

Predicted p-IR protein signal for 30 nM insulin                   Predicted p-IRS protein signal for 30 nM insulin

Combinatorial and Computational Mathematics Center
Interpolation with identified parameters
for 30nM insulin concentration

1.2                                                            2.5

1
2

0.8
1.5

0.6

1
0.4

0.5
0.2

0                                                              0
0   2   4   6   8   10   12   14   16   18   20                0   2   4   6   8   10   12   14   16   18   20

Predicted p-Akt protein signal for 30 nM insulin              Predicted p-ERK protein signal for 30 nM insulin

Combinatorial and Computational Mathematics Center
Phosphorylation pattern of signal
proteins for 30nM insulin stimulation
Insulin 30 nM

Time
WB

0.25
(min):

0.5

20
10
0

1

5
2
p-IR
(pY1158)

p-IRS
(pY989)

p-Akt
(pS473)

p-ERK
(pT202
/Y204)

Actin

HEK 293 cells are deprived of serum for 24h before treatment
and stimulated with 30 nM insulin for indicated time. HEK 293
cells are stimulated with 30 nM of insulin.

Combinatorial and Computational Mathematics Center
Regresstion with in vivo data via least
squares method for protein signals
Regression parameters for 30 nM insulin concentration by
least squares method

a=1.87940
p-IR               b x
y
xa                       b=0.58406

b x                    a=0.76379
p-IRS             y
xa                    b=1.33801

2 ymax                    ymax=0.9000
p-Akt          y       ax
 ymax
1 e                        a=3.03422
a=0.33628
b=0.00669
ax1.95  bx cx d gx2        c=0.57565
p-ERK         y           cx f
e
0.3  x                     d=0.22306
f=- 1.72694
g=- 0.00634

Combinatorial and Computational Mathematics Center
Regression with 30nM invivo data via
least squares method

0.9

0.3
0.8

0.25                                                             0.7

0.6

0.2
0.5

0.15                                                             0.4

0.3

0.1
0.2

0.05                                                             0.1

0

0                                                                        0   5      10   15    20

0   2   4         6   8   10   12   14   16   18   20

p-IR                                                  p-IRS

1.2                      2.5

1
2

Combinatorial and Computational Mathematics Center
0.8
1.5

0.6

1
Regression with 30nM invivo data via
least squares method
1.2                                          2.5

1

2

0.8

1.5

0.6

1

0.4

0.5

0.2

0                                            0

0   5      10   15   20                      0   5       10   15    20

p-Akt                                        p-ERK

Combinatorial and Computational Mathematics Center
Comparison with predicted and least
squares fitted data

0.3                                                                       0.3
predicted value                                                           predicted value
least squares method                                                      least squares method
0.25                                                                      0.25

0.2                                                                       0.2

0.15                                                                      0.15

0.1                                                                       0.1

0.05                                                                      0.05

0                                                                         0
0   2   4   6   8   10   12     14    16     18      20                   0   2   4   6   8   10   12     14    16     18      20

p-IR                                                                      p-IRS

Combinatorial and Computational Mathematics Center
Comparison with predicted and least
squares fitted data

1.2                                                                       2.5
predicted value
least squares method
1
predicted value                       2
least squares method

0.8
1.5

0.6

1
0.4

0.5
0.2

0                                                                         0
0   2   4   6    8   10   12     14    16     18      20                  0   2   4   6   8   10   12     14    16     18      20

p-Akt                                                                 p-ERK

Combinatorial and Computational Mathematics Center
Conclusion

 Kinetics for Insulin transduction is identified.

 It is possible to predict [IR*], [IRS*], [Akt*], and [ERK*]
without actural experiment

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Future Study

More invivo data for different Insulin medication cases
are necessary to verify the effectiveness of our results.

Combinatorial and Computational Mathematics Center

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