Download Quantification in Life Sciences Systems Biology: Reaction Kinetics

Quantification in Life Sciences
Systems Biology: Reaction Kinetics
Dr. Nacho Molina
[email protected]
Outline
1. What is Systems Biology?
2. Reactions in biology
3. Production and degradation processes
4. mRNA production and degradation
5. Ligand-receptor process
5. Enzyme kinetics
6. Gene regulation
What is Systems Biology?
§ 
Systems biology is the study of an organism,
viewed as an integrated and interacting network
of genes, proteins and biochemical reactions
which give rise to life (Institute of Systems
Biology)
§ 
Emerging laws: Whole is greater than the
sum of the parts.
§ 
There are new experimental technologies
that permit to measure simultaneously many
molecules (mRNA, proteins, metabolites) and
how they change over time.
§ 
The amount of data is vast. Mathematical
models and computational algorithms are
required to analyze the data and to make
predictions.
§ 
Biology is changing from a descriptive
science to a quantitative science:
mathematics and programming are
becoming fundamental!
Bibliography
Uri Alon’s book:
Rob Philips’ (et al.) book:
Reaction Kinetics
All basic biological process as
metabolism, gene regulation,
cell cycle, etc. can be understood
in terms of chemical reactions.
-  How many molecules there are?
-  How do they change over time?
-  Where do they are?
Reaction Kinetics
All basic biological process as
metabolism, gene regulation,
cell cycle, etc. can be understood
in terms of chemical reactions.
-  How many molecules there are?
-  How do they change over time?
-  Where do they are?
Production process
Zero-order reaction:
k
k
;!
M
Rate equation:
vM
dM
=
=k
dt
Molecules (#/cell)
k: rate constant (molecules/minutes)
700
600
500
400
300
200
100
M0
100
200
300
400
500
Time (minutes)
Time-dependent solution:
M (t) = kt + M0
Molecules (#/cell)
1000
k = 2 · mol./min.
800
600
400
200
0
100
200
300
400
Time (minutes)
500
Degradation process
First-order reaction:
M !;
: rate constant (hours-1)
1/ : expected life (hours)
M
Time-dependent solution:
M (t) = M0 e
M0
800
t1/2 = 2.77 · hours
600
400
200
2
4
6
8
10
Time (hours)
Half-life:
M0 /2 = M0 e
+
log(2)
t1/2 =
1000
t
= 0.25 · hours
1000
t1/2
Molecules (#/cell)
vM
dM
=
=
dt
Molecules (#/cell)
Rate equation:
100
10
1
0.1
2
4
6
Time (hours)
8
10
1
mRNA production-degradation process
Transcription & Translation: fundamental processes in biology
A simple model of mRNA accumulation:
1) combination of production and degradations processes
2) both process are constant over time
k
mRNA production-degradation process
Reactions:
k
G!
G+M !;
k: rate constant (molecules/minutes)
: rate constant (1/minutes)
Rate equation:
dM
=
=k
dt
Molecules (#/cell)
M
Chemical equilibrium:
0=k
Meq ) k = Meq
+
k
Meq =
Meq
2000
1500
1000
= 0.01 · min
t1/2 = 69 · min
500
200
Time-dependent solution:
k
(1
600
800
k = 10 · mol/min
k = 5 · mol/min
1000
e
t
) + M0 e
k = 10 · mol/min
1000
t
1
= 0.005 · min
2000
1500
M (t) =
400
k = 20 · mol/min
1
Time (min)
Molecules (#/cell)
vM
k
500
200
400
600
Time (min)
800
1000
= 0.01 · min
= 0.02 · min
1
1
mRNA production-degradation process
Reactions:
k
G!
G+M !;
k: rate constant (molecules/minutes)
: rate constant (1/minutes)
Rate equation:
dM
=
=k
dt
M
Molecules (#/cell)
vM
k
Chemical equilibrium:
0=k
Meq ) k = Meq
+
k
Meq =
(1
600
k =?
=?
400
200
400
600
800
1000
Time (min)
Transcriptional pulse
1000
e
t
) + M0 e
t
Molecules (#/cell)
M (t) =
Meq
800
200
Time-dependent solution:
k
1000
800
600
400
500 · min
M (t = 500 · min)
200
200
400
600
Time (min)
800
1000
Bimolecular reaction / Ligand-receptor process
Second order reaction:
A+B
kf
C
kb
kf : forward rate constant (mol-1min-1)
kb : backward rate constant (min-1)
Reaction rates:
!
forward reaction:
rf = kf AB
backward reaction:
rb = k b C
(mol/min)
(mol/min)
Rate equation:
dA
= kf AB + kb C
dt
dB
vB =
= kf AB + kb C
dt
dC
vC =
= kf AB kb C
dt
vA =
Molecules (#/cell)
2000
A0
1500
1000
Aeq
B0
Beq
500
Ceq
C0
1
2
3
Time (hours)
4
5
Chemical Equilibrium / Steady State
At equilibrium the concentration
does not change:
vA = 0
+
kf Aeq Beq + kb Ceq = 0
+
Law of conservation of mass:
molecular mass
2000
Molecules (#/cell)
kb
Aeq Beq
=
kf
Ceq
A0
1500
1000
Aeq
B0
Beq
500
Ceq
C0
1
2
3
Time (hours)
mA A + mB B + mC C = mA A0 + mB B0 + mC C0
4
5
Enzyme kinetics and Michaelis-Menten equation
Reactions:
E+S
kf
k
c
C!
E+P
kr
kf : forward rate constant (mol-1min-1)
kr : reverse rate constant (min-1)
kc : catalysis rate constant (min-1)
Rate equations:
dE
=
dt
dS
vS =
=
dt
vE =
kf ES + kr C + kc C
kf ES + kr C
vC =
dC
= kf ES
dt
vP =
dP
= kc C
dt
kr C
kc C
[C]
Enzyme kinetics and Michaelis-Menten equation
Reactions:
E+S
kf
k
c
C!
E+P
kr
kf : forward rate constant (mol-1min-1)
kr : reverse rate constant (min-1)
kc : catalysis rate constant (min-1)
Michaelis-Menten approximation:
kf ES + kr C
kf ES + kr C = 0
E + C = E0
+
E0 = 2000 · mol
(mol/min)
dS
=
dt
+
150
E0 = 1500 · mol
100
E0 = 1000 · mol
vP
vS =
2000
E0 S
C=
Kd + S
4000
6000
8000 10 000
S (molecules)
+
k c E0 S
v P = kc C =
Kd + S
kc = 0.1 · min 1
Kd = 1000 · mol
50
Dissociation or Michaelis constant:
Kd =
kr
(mol)
kf
Gene regulation
Transcription is a highly regulated process
A simple model of transcription: when a
transcription factor is bound to the gene
there is mRNA production with a rate
constant k
M
k
kb
C
ku
G+T
Gene regulation
Reactions:
A+G
kf
M
kc
C !A+G+M
kr
kb : binding rate constant (mol-1min-1)
ku: unbinding rate constant (min-1)
k : production rate constant (mol/min)
k
kb
Effective production rate:
ku
C
Activation coefficient:
A+G
K = ku /kb (mol)
effective rate (mol/min)
ke↵
A
=k
A+K
Chemical equilibrium:
A k
Meq =
A+K
Time-dependent solution:
A k
M (t) =
(1 e t ) + M0 e
A+K
k = 10 · mol/min
10
8
6
K = 100 · mol
K = 500 · mol
4
2
1000
2000
3000
4000
A (molecules)
t
5000
Gene regulation with cooperativity
Reactions:
nA + G
kf
M
kc
C ! nA + G + M
kr
kb : binding rate constant (mol-nmin-1)
ku: unbinding rate constant (min-1)
k : production rate constant (mol/min)
n : hill coefficient (number)
k
kb
Effective production rate:
K n = ku /kb (moln)
Chemical equilibrium:
An
k
Meq = n
A + Kn
Time-dependent solution:
k
An
M (t) =
(1 e
An + K n
ku
C
Activation coefficient:
effective rate (mol/min)
ke↵
An
=k n
A + Kn
A+G
K = 2000 · mol
k = 10 · mol/min
n=8
n=2
n=1
10
8
6
4
2
1000
2000
3000
4000
A (molecules)
t
) + M0 e
t
5000