Download Introduction to Chemical Kinetics

Introduction to Chemical Kinetics
and Computational Modeling
Hana El-Samad
Byers Hall (QB3), Rm 403D
Hana El-Samad
UCSB
Why mathematics in biology?
input
k
A  B C
k
A DE
k
B EF
k
A F B
output
Given
some
Hana
El-Samad
UCSB
initial concentrations, what is output give some prescribed input?
Chemical reactions are collisions of molecules
Molecule
A
Molecule
B
k
A  B C
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UCSB
Reactant A
A
time
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UCSB
Molecule
A
Change in Concentration: DA= A2-A1
Reactant A
A1
A2
t1
t2
time
Change in time: Dt=t2-t1
Average rate of change in concentration during time Dt=
Hana El-Samad
UCSB
DA
Dt
Reactant A
A1
Change in Concentration: DA= A2-A1
A2
t1 t2
Derivative of A with
Respect to time
instantaneous rate of change in concentration during time dt=
Hana El-Samad
UCSB
dA
dt

A
dA
  . A
dt
Rate of change of A
Degradation constant
Hana El-Samad
UCSB
Concentration of A
k

D  A
dA
 k .D   . A
dt
Rate of change of A
Concentration of D
Concentration of A
Production constant
Degradation constant
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UCSB
k
A  B C
dA
 k . A.B
dt
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UCSB
k
nA  mB  C
dA
n
m
 k . A .B
dt
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UCSB
k1

D  Ak
2
A  B C
k2
C A B
dA
 k1.D   . A  k2 . A.B  k2C
dt
Hana El-Samad
UCSB
Reaching steady-state (equilibrium)

k
D  A
dA
 k .D   . A
dt
Steady-state: No more change in A
dA
0
dt
Production of A balances degradation
A is constant
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UCSB
k .D   . A  0
A
K .D

Chemical Reactions inside the cell
Central Dogma of Molecular Biology
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UCSB
Transcription
Transcription
factor
start
DNA
end
gene
promoter
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UCSB
terminator
mRNA Translation
Proteins
mRNA
ribosomes
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UCSB
Modeling activator binding and production of mRNA
start
end
gene
promoter
A  DNA  A : DNA
k2
A : DNA  mRNA  A : DNA
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UCSB
terminator
Fast
slow
A  DNA  A : DNA
Fast
d [ A : DNA]
 k1. A.DNA  k2 .[ A : DNA]  0
dt
k1. A.DNA  k2 .[ A : DNA]
k1. A.DNA
[ A : DNA] 
k2
but
DNAtotal
Hana El-Samad
UCSB
k1. A.DNA  (1  k1. A ) DNA
 DNA  [ A : DNA]  DNA 
k2
k2
DNAtotal
DNA 
k1. A
(1 
)
k2
DNAtotal
DNA 
k1. A
(1 
)
k2
k1. A.DNA
[ A : DNA] 
k2
k1
A
k2
[ A : DNA] 
DNAtotal
k1
1 A
k2
k2
 kd
k1
Dissociation constant
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UCSB
Hill Function
A
kd
[ A : DNA] 
DNAtotal
A
1
kd
[ A : DNA]
Hana El-Samad
UCSB
DNAtotal
A
k2
A : DNA  mRNA  A : DNA
dmRNA
 k3 .[ A : DNA]   1mRNA
dt
A
kd
[ A : DNA] 
DNAtotal
A
1
kd
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UCSB
A
kd
dmRNA
 k3 .
DNAtotal   1mRNA
A
dt
1
kd
Cooperativity
start
end
gene
promoter
A 3
( )
kd
dmRNA
 k3 .
DNAtotal   1mRNA
A 3
dt
1 ( )
kd
Hana El-Samad
UCSB
terminator
Cooperativity
n molecules
start
end
gene
promoter
A n
( )
kd
dmRNA
 k3 .
DNAtotal   1mRNA
A n
dt
1 ( )
kd
Hana El-Samad
UCSB
terminator
Repressor
start
end
gene
promoter
dmRNA
1
 k3 .
DNAtotal   1mRNA
X n
dt
1 ( )
kd
Hana El-Samad
UCSB
terminator
1
k3 .
DNAtotal
X n
1 ( )
kd
Hana El-Samad
UCSB
X