Post on 13-Jan-2016
Lecture 5: Chemical Reactions
Outline:• basic concepts• Nonlinearities:
• saturation: Michaelis-Menten kinetics• switching: Goldbeter-Koshland
Basics
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A + B r ⏐ → ⏐ CSimple reaction
Basics
€
A + B r ⏐ → ⏐ CSimple reaction
Number of AB pairs in volume v:
€
ρ(A)ρ(B)v 2 = [A][B]v 2
Basics
€
A + B r ⏐ → ⏐ CSimple reaction
Number of AB pairs in volume v:
Reaction equation:
€
ρ(A)ρ(B)v 2 = [A][B]v 2
€
d [C]V( )dt
= −d [A]V( )
dt= −
d [B]V( )dt
= r ⋅[A][B]v 2 ⋅V
v⇒
Basics
€
A + B r ⏐ → ⏐ CSimple reaction
Number of AB pairs in volume v:
Reaction equation:
€
ρ(A)ρ(B)v 2 = [A][B]v 2
€
d [C]V( )dt
= −d [A]V( )
dt= −
d [B]V( )dt
= r ⋅[A][B]v 2 ⋅V
v⇒
d[C]
dt= −
d[A]
dt= −
d[B]
dt= rv[A][B] ≡ k[A][B]
Basics
€
A + B r ⏐ → ⏐ CSimple reaction
Number of AB pairs in volume v:
Reaction equation:
Note: r has units 1/t, k has units volume/t
€
ρ(A)ρ(B)v 2 = [A][B]v 2
€
d [C]V( )dt
= −d [A]V( )
dt= −
d [B]V( )dt
= r ⋅[A][B]v 2 ⋅V
v⇒
d[C]
dt= −
d[A]
dt= −
d[B]
dt= rv[A][B] ≡ k[A][B]
Reversible reactions, stoichiometry
Reaction can go both ways:
€
A + B k,k '← → ⏐ C
Reversible reactions, stoichiometry
Reaction can go both ways:
€
A + B k,k '← → ⏐ C
€
d[A]
dt=
d[B]
dt= −
d[C]
dt= −k[A][B] + ′ k [C]
Reversible reactions, stoichiometry
Reaction can go both ways:
Equilibrium:€
A + B k,k '← → ⏐ C
€
d[A]
dt=
d[B]
dt= −
d[C]
dt= −k[A][B] + ′ k [C]
[A][B]
[C]=
′ k
k
Reversible reactions, stoichiometry
Reaction can go both ways:
Equilibrium:
Stoichiometry:
€
A + B k,k '← → ⏐ C
€
d[A]
dt=
d[B]
dt= −
d[C]
dt= −k[A][B] + ′ k [C]
[A][B]
[C]=
′ k
k
€
mA + nB k,k'← → ⏐ pC
Reversible reactions, stoichiometry
Reaction can go both ways:
Equilibrium:
Stoichiometry:
€
A + B k,k '← → ⏐ C
€
d[A]
dt=
d[B]
dt= −
d[C]
dt= −k[A][B] + ′ k [C]
[A][B]
[C]=
′ k
k
€
mA + nB k,k'← → ⏐ pC
€
1
m
d[A]
dt=
1
n
d[B]
dt= −
1
p
d[C]
dt= −k[A]m[B]n + ′ k [C]p
Reversible reactions, stoichiometry
Reaction can go both ways:
Equilibrium:
Stoichiometry:
€
A + B k,k '← → ⏐ C
€
d[A]
dt=
d[B]
dt= −
d[C]
dt= −k[A][B] + ′ k [C]
[A][B]
[C]=
′ k
k
€
mA + nB k,k'← → ⏐ pC
€
1
m
d[A]
dt=
1
n
d[B]
dt= −
1
p
d[C]
dt= −k[A]m[B]n + ′ k [C]p
[A]m[B]n
[C]p=
′ k
k
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
€
E + S ↔ C → E + Preversible rates a,d
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
€
E + S ↔ C → E + P
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
€
E + S ↔ C → E + Preversible rates a,d
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
€
E + S ↔ C → E + Preversible irreversible rates a,d rate k
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
€
E + S ↔ C → E + Preversible irreversible rates a,d rate k
€
d[S]
dt= −a[E][S] + d[C]Rate equations:
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
€
E + S ↔ C → E + Preversible irreversible rates a,d rate k
€
d[S]
dt= −a[E][S] + d[C]
d[E]
dt= −a[E][S] + d[C] + k[C]
Rate equations:
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
€
E + S ↔ C → E + Preversible irreversible rates a,d rate k
€
d[S]
dt= −a[E][S] + d[C]
d[E]
dt= −a[E][S] + d[C] + k[C]
d[C]
dt= a[E][S] − d[C] − k[C]
Rate equations:
Michaelis-Menten
Enzyme + substrate <-> complex -> enzyme + product
€
E + S ↔ C → E + Preversible irreversible rates a,d rate k
€
d[S]
dt= −a[E][S] + d[C]
d[E]
dt= −a[E][S] + d[C] + k[C]
d[C]
dt= a[E][S] − d[C] − k[C]
d[P]
dt= k[C]
Rate equations:
Reduction:
€
[E](t) + [C](t) = const = E0 ⇒ [E] = E0 −[C]Eliminate E:
Reduction:
€
[E](t) + [C](t) = const = E0 ⇒ [E] = E0 −[C]
d[S]
dt= −a E0 −[C]( )[S] + d[C]
d[C]
dt= a E0 −[C]( )[S] − (d + k)[C]
Eliminate E:
To solve:
Reduction:
€
[E](t) + [C](t) = const = E0 ⇒ [E] = E0 −[C]
d[S]
dt= −a E0 −[C]( )[S] + d[C]
d[C]
dt= a E0 −[C]( )[S] − (d + k)[C]
S(0) = S0
C(0) = 0
Eliminate E:
To solve:
Initial conditions:
Initial regime:Lots of S, [S] hardly changes from S0
Initial regime:Lots of S, [S] hardly changes from S0
€
d[C]
dt= a E0 −[C]( )[S] − (d + k)[C]
Initial regime:Lots of S, [S] hardly changes from S0
€
d[C]
dt= a E0 −[C]( )[S] − (d + k)[C]
≈ aE0S0 − aS0 + d + k( )[C]
Initial regime:Lots of S, [S] hardly changes from S0
€
d[C]
dt= a E0 −[C]( )[S] − (d + k)[C]
≈ aE0S0 − aS0 + d + k( )[C]
[C] ≈aE0S0
aS0 + d + k( )1− exp − aS0 + d + k( )t( )[ ]
Initial regime:Lots of S, [S] hardly changes from S0
€
d[C]
dt= a E0 −[C]( )[S] − (d + k)[C]
≈ aE0S0 − aS0 + d + k( )[C]
[C] ≈aE0S0
aS0 + d + k( )1− exp − aS0 + d + k( )t( )[ ]
=E0S0
S0 + KM
1− exp t /τ fast( )[ ]
Initial regime:Lots of S, [S] hardly changes from S0
€
d[C]
dt= a E0 −[C]( )[S] − (d + k)[C]
≈ aE0S0 − aS0 + d + k( )[C]
[C] ≈aE0S0
aS0 + d + k( )1− exp − aS0 + d + k( )t( )[ ]
=E0S0
S0 + KM
1− exp t /τ fast( )[ ]
KM =d + k
a,
1
τ fast
= a(S0 + KM )
Initial regime:Lots of S, [S] hardly changes from S0
€
d[C]
dt= a E0 −[C]( )[S] − (d + k)[C]
≈ aE0S0 − aS0 + d + k( )[C]
[C] ≈aE0S0
aS0 + d + k( )1− exp − aS0 + d + k( )t( )[ ]
=E0S0
S0 + KM
1− exp t /τ fast( )[ ]
KM =d + k
a,
1
τ fast
= a(S0 + KM )
Michaelis constant
Slow dynamics
If [S] changes slowly compared with τfast,
€
[C] ≈E0[S]
[S] + KM
Slow dynamics
If [S] changes slowly compared with τfast,
€
[C] ≈E0[S]
[S] + KM
⇒d[S]
dt= −a E0 −
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟[S] + d
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟
Slow dynamics
If [S] changes slowly compared with τfast,
€
[C] ≈E0[S]
[S] + KM
⇒d[S]
dt= −a E0 −
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟[S] + d
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟
=−aKM + d
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟E0[S]
Slow dynamics
If [S] changes slowly compared with τfast,
€
[C] ≈E0[S]
[S] + KM
⇒d[S]
dt= −a E0 −
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟[S] + d
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟
=−aKM + d
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟E0[S] =
−ad + k
a
⎛
⎝ ⎜
⎞
⎠ ⎟+ d
[S] + KM
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
E0[S]
Slow dynamics
If [S] changes slowly compared with τfast,
€
[C] ≈E0[S]
[S] + KM
⇒d[S]
dt= −a E0 −
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟[S] + d
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟
=−aKM + d
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟E0[S] =
−ad + k
a
⎛
⎝ ⎜
⎞
⎠ ⎟+ d
[S] + KM
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
E0[S]
= −kE0[S]
[S] + KM
= −[S]
τ slow
Slow dynamics
If [S] changes slowly compared with τfast,
€
[C] ≈E0[S]
[S] + KM
⇒d[S]
dt= −a E0 −
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟[S] + d
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟
=−aKM + d
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟E0[S] =
−ad + k
a
⎛
⎝ ⎜
⎞
⎠ ⎟+ d
[S] + KM
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
E0[S]
= −kE0[S]
[S] + KM
= −[S]
τ slow
τ fast
τ slow
=kE0
a(S + KM )2<<1
Slow dynamics
If [S] changes slowly compared with τfast,
€
[C] ≈E0[S]
[S] + KM
⇒d[S]
dt= −a E0 −
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟[S] + d
E0[S]
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟
=−aKM + d
[S] + KM
⎛
⎝ ⎜
⎞
⎠ ⎟E0[S] =
−ad + k
a
⎛
⎝ ⎜
⎞
⎠ ⎟+ d
[S] + KM
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
E0[S]
= −kE0[S]
[S] + KM
= −[S]
τ slow
τ fast
τ slow
=kE0
a(S + KM )2<<1
consistent
Result:
€
d[P]
dt= −
d[S]
dt=
kE0[S]
KM + [S]< kE0
Reaction rate
Result:
€
d[P]
dt= −
d[S]
dt=
kE0[S]
KM + [S]< kE0
Reaction rate
saturation
Cooperative binding
€
E + nS ↔ C → E + nP
Cooperative binding
€
E + nS ↔ C → E + nP
€
d[S]
dt= −a E0 −[C]( )[S]n + d[C]
d[C]
dt= a E0 −[C]( )[S]n − (d + k)[C]
Equations ->
Cooperative binding
€
E + nS ↔ C → E + nP
€
d[S]
dt= −a E0 −[C]( )[S]n + d[C]
d[C]
dt= a E0 −[C]( )[S]n − (d + k)[C]
[C] =E0[S]n
KM + [S]nAfter fast transient:
Equations ->
Cooperative binding
€
E + nS ↔ C → E + nP
€
d[S]
dt= −a E0 −[C]( )[S]n + d[C]
d[C]
dt= a E0 −[C]( )[S]n − (d + k)[C]
[C] =E0[S]n
KM + [S]n
Hill coefficient n
After fast transient:
Equations ->
Goldbeter-Koshland switching
2 MM reactions, 1 in each direction:
€
S1 + E1 ↔ C1 → S2 + E1
S2 + E 2 ↔ C2 → S1 + E 2
Goldbeter-Koshland switching
2 MM reactions, 1 in each direction:
€
S1 + E1 ↔ C1 →S2 + E1
S2 + E 2 ↔ C2 →S1 + E 2
dS1
dt= −a1S1E1 + d1C1 + k2C2
Goldbeter-Koshland switching
2 MM reactions, 1 in each direction:
€
S1 + E1 ↔ C1 →S2 + E1
S2 + E 2 ↔ C2 →S1 + E 2
dS1
dt= −a1S1E1 + d1C1 + k2C2
dC1
dt= a1S1E1 − (d1 + k1)C1
Goldbeter-Koshland switching
2 MM reactions, 1 in each direction:
€
S1 + E1 ↔ C1 →S2 + E1
S2 + E 2 ↔ C2 →S1 + E 2
dS1
dt= −a1S1E1 + d1C1 + k2C2
dC1
dt= a1S1E1 − (d1 + k1)C1
dS2
dt= −a2S2E2 + d2C2 + k1C1
dC2
dt= a2S2E2 − (d2 + k2)C2
Goldbeter-Koshland switching
2 MM reactions, 1 in each direction:
€
S1 + E1 ↔ C1 →S2 + E1
S2 + E 2 ↔ C2 →S1 + E 2
dS1
dt= −a1S1E1 + d1C1 + k2C2
dC1
dt= a1S1E1 − (d1 + k1)C1
dS2
dt= −a2S2E2 + d2C2 + k1C1
dC2
dt= a2S2E2 − (d2 + k2)C2
€
Stot = S1 + S2 + C1 + C2 = const
Goldbeter-Koshland switching
2 MM reactions, 1 in each direction:
€
S1 + E1 ↔ C1 →S2 + E1
S2 + E 2 ↔ C2 →S1 + E 2
dS1
dt= −a1S1E1 + d1C1 + k2C2
dC1
dt= a1S1E1 − (d1 + k1)C1
dS2
dt= −a2S2E2 + d2C2 + k1C1
dC2
dt= a2S2E2 − (d2 + k2)C2
€
Stot = S1 + S2 + C1 + C2 = const
≈ S1 + S2
Goldbeter-Koshland switching
2 MM reactions, 1 in each direction:
€
S1 + E1 ↔ C1 →S2 + E1
S2 + E 2 ↔ C2 →S1 + E 2
dS1
dt= −a1S1E1 + d1C1 + k2C2
dC1
dt= a1S1E1 − (d1 + k1)C1
dS2
dt= −a2S2E2 + d2C2 + k1C1
dC2
dt= a2S2E2 − (d2 + k2)C2
€
Stot = S1 + S2 + C1 + C2 = const
≈ S1 + S2
E0(1) = E1 + C1 = const
E0(2) = E2 + C2 = const
Goldbeter-Koshland switching
2 MM reactions, 1 in each direction:
€
S1 + E1 ↔ C1 →S2 + E1
S2 + E 2 ↔ C2 →S1 + E 2
dS1
dt= −a1S1E1 + d1C1 + k2C2
dC1
dt= a1S1E1 − (d1 + k1)C1
dS2
dt= −a2S2E2 + d2C2 + k1C1
dC2
dt= a2S2E2 − (d2 + k2)C2
€
Stot = S1 + S2 + C1 + C2 = const
≈ S1 + S2
E0(1) = E1 + C1 = const
E0(2) = E2 + C2 = const
Steady state (add 1st 2 or2nd 2 eqns):
€
k1C1 = k2C2
Steady state:
€
k1C1 = k2C2
Steady state:
€
k1C1 = k2C2
dC1
dt= 0 ⇒ a1S1 E0
(1) − C1( ) = (d1 + k1)C1
Steady state:
€
k1C1 = k2C2
dC1
dt= 0 ⇒ a1S1 E0
(1) − C1( ) = (d1 + k1)C1
⇒ C1 =E0
(1)S1
KM(1) + S1
, KM(1) =
d1 + k1
a1
Steady state:
€
k1C1 = k2C2
dC1
dt= 0 ⇒ a1S1 E0
(1) − C1( ) = (d1 + k1)C1
⇒ C1 =E0
(1)S1
KM(1) + S1
, KM(1) =
d1 + k1
a1
dC2
dt= 0 ⇒ C2 =
E0(2)S2
KM(2) + S2
, KM(2) =
d2 + k2
a2
Steady state:
€
k1C1 = k2C2
dC1
dt= 0 ⇒ a1S1 E0
(1) − C1( ) = (d1 + k1)C1
⇒ C1 =E0
(1)S1
KM(1) + S1
, KM(1) =
d1 + k1
a1
dC2
dt= 0 ⇒ C2 =
E0(2)S2
KM(2) + S2
, KM(2) =
d2 + k2
a2
⇒k1E0
(1)S1
KM(1) + S1
=k2E0
(2) Stot − S1( )KM
(2) + Stot − S1
Steady state:
€
k1C1 = k2C2
dC1
dt= 0 ⇒ a1S1 E0
(1) − C1( ) = (d1 + k1)C1
⇒ C1 =E0
(1)S1
KM(1) + S1
, KM(1) =
d1 + k1
a1
dC2
dt= 0 ⇒ C2 =
E0(2)S2
KM(2) + S2
, KM(2) =
d2 + k2
a2
⇒k1E0
(1)S1
KM(1) + S1
=k2E0
(2) Stot − S1( )KM
(2) + Stot − S1
Quadratic equation for S1
Solution:
€
S1 =
v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟−κ 2
κ1
κ 2
+v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟+
v1
v2
−1−κ 2
κ1
κ 2
+v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
2
+ 4κ 2
v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟
2v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟
v1 = k1E0(1), v2 = k2E0
(2), κ1 = KM(1)
Stot, κ 2 = KM
(2)
Stotwhere
Solution:
€
S1 =
v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟−κ 2
κ1
κ 2
+v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟+
v1
v2
−1−κ 2
κ1
κ 2
+v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
2
+ 4κ 2
v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟
2v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟
v1 = k1E0(1), v2 = k2E0
(2), κ1 = KM(1)
Stot, κ 2 = KM
(2)
Stotwhere
S1,1-S1
Solution:
€
S1 =
v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟−κ 2
κ1
κ 2
+v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟+
v1
v2
−1−κ 2
κ1
κ 2
+v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
2
+ 4κ 2
v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟v1
v2
⎛
⎝ ⎜
⎞
⎠ ⎟
2v1
v2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟
v1 = k1E0(1), v2 = k2E0
(2), κ1 = KM(1)
Stot, κ 2 = KM
(2)
Stotwhere
S1,1-S1
Sharp switching forsmall κ