Miap

Casp6

Casp3

N

NGF-P75

Modeling Apoptosis

p75

NGF

NGF

f(NGF)

01( ) (1 tanh( ))

2 10

NGF NGFf NGF

MiapCasp6

Casp3

N

NGF-P75

aakai cccc 6363

3 6 3 6ka i a ac c c c

3 6 3 62 2ki a a a ajc c c c c

26 675 75ki aNGF p c NGF p c

26 6

ka a a iMIAP c MIAP c

LAW OF MASS ACTION

Calculus Review

( )X t For us, the concentration vs. time

dX

dt

< 0 then X is decreasingdX

dt

> 0 then X is increasing

Law of Mass Action• Rate of any given chemical reaction is proportional to the

product of the concentrations of the reactants.

• Example: with reaction ratekA B C k

[ ][ ][ ]

d Ak A B

dt

[ ][ ][ ]

d Bk A B

dt

[ ][ ][ ]

d Ck A B

dt

•Rate of change of concentration of B:

•Rate of change of concentration of C:

•[A], [B], concentration of respective chemicals, t is time.

•Rate of change of concentration of A:

+

A B C

Law of Mass Action (Cont.)

• Many reactions are reversible: e.g.

, forward rate constant, , reverse rate constant.

• After applying the law of mass action:

1

k

kA B C

k 1k

1

[ ][ ] [ ][ ]

d Ak C k A B

dt

1

[ ][ ] [ ][ ]

d Bk C k A B

dt

1

[ ][ ] [ ][ ]

d Ck C k A B

dt

dwkwD

dtdD

kwDdt

2kD w w

Miap

Casp6

Casp3

f(NGF)

636

3 6

dck c c

dt 6ikMIAPc6( )uf NGF c

636

3 6

dck c c

dt 6ikMIAPc6( )uf NGF c

636

3 6( ( ) )i

dck c uf NGF kMIAP c

dt

66( )

dca c

dt When a is positive, c6 will

increase forever!6 6( )n c

63 6 66

3 6( ( ) ) ( )i

dck c uf NGF kMIAP c n c

dt

3

6 3 3 363 MIAP ( )in

dck c k c n c

dt

MiapCasp6

Casp3

N

f(NGF)

m(e-f(NGF))(MIAP)(n -MIAP)dMIAP

dt

3 6( )N fcn c c

3336633 MIAP' CnCkCkC in

Now what does it do?

MIAP663 inkCka

3333 )(' CnCaC

332

33 )(' CnCaC

Simplification

663MIAP Ckkin

33 Cn

0a 33

233 ' CnCC

332

33 )(' CnCaC

ExcitationInhibition

MIAP663 inkCka

Casp6

Casp3

Miap

3336633 MIAP' CnCkCkC in So what happens?

33 Cn

'3C

3C

0a 332

33 ' CnCC

ExcitationInhibition

Casp6

Casp3

N

The Cell May Die

663MIAP Ckkin

33 Cn

0a

332

33 ' CnCC

332

33 )(' CnCaC

ExcitationInhibition

MIAP663 inkCka

3336633 MIAP' CnCkCkC in So what happens?

33 Cn

0a 332

33 ' CnCC

ExcitationInhibition

The Cell May Live'3C

3C

Casp6

Casp3

N

6663366 MIAP)(' CnCkCkuNGFfC i

MIAP)( 336 ikCkuNGFfa

662

66 )(' CnCaC

336-MIAP)( CkkuNGFf i

The Cell May Die The Cell May Live0a 0a

The Casp6 Equation

336-MIAP)( CkkuNGFf i

The Miap Equation

'Miap

Miap

f(NGF)-ek tscl1MIAP' e-MIAP MIAP-Nm

Casp6

Casp3

N

Exactly when does the cell die?

XNCell Lives When

X1-C+Ctanh+12

1=N 63

112tanhC+C 163 X

ConstantC+C 63

Casp6

Casp3

N

3336633 MIAP' CnCkCkC in

6663366 MIAP)(' CnCkCkuNGFfC i

When does the cell die?

6,0 n

0,3n3C

6C 63,nn

Cell Will Probably Die

Cell Will Probably Live

0,0

ConstantC+C 63 Cell Lives When

63,CC