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Introduction to Chemical Kinetics
and Computational Modeling
Hana El-Samad
Byers Hall (QB3), Rm 403D
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Why mathematics in biology?
k
B E F k
A F B
k
A D E
k
A B C input
output
Given some initial concentrations, what is output give some prescribed input?
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Chemical reactions are collisions of molecules
Molecule
B
Molecule
A
k
A B C
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A
Re
actantA
time
MoleculeA
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A1
A2
t1
Rea
ctantA
time
t2
Change in Concentration: DA= A2-A1
Change in time: Dt=t2-t1
A
t
D
D
Average rate of change in concentration during time Dt=
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A1
A2
t1
Rea
ctantA
t2
Change in Concentration: DA= A2-A1
dA
dtinstantaneous rate of change in concentration during time dt=
Derivative of A with
Respect to time
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A
.dA Adt
Rate of change of A
Degradation constant
Concentration of A
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k
D A
. .dA k D Adt
Rate of change of A
Degradation constant
Concentration of AConcentration of D
Production constant
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k
A B C
. .dA
k A Bdt
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k
nA mB C
. .n mdA
k A Bdt
k
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2k
A B C
1k
D A
1 2 2. . . .dA
k D A k A B k C dt
2k
C A B
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Reaching steady-state (equilibrium)
0dAdt
k
D A
. .dA
k D Adt
Steady-state: No more change in A
.K DA
. . 0k D A
Production of A balances degradation
A is constant
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Chemical Reactions inside the cellCentral Dogma of Molecular Biology
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start
promoter
end
terminator
gene
Transcription
factor
DNA
Transcription
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mRNA Translation
Proteins
ribosomes
mRNA
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Modeling activator binding and production of mRNA
start
promoter
end
terminator
gene
2
:
: :k
A DNA A DNA
A DNA mRNA A DNA
Fast
slow
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:A DNA A DNA Fast
1 2
[ : ]. . .[ : ]
d A DNAk A DNA k A DNA
dt 0
[ : ]total
DNA DNA A DNA 1
2
. .k A DNADNA
k
1
2
.(1 )
k ADNA
k
1 2. . .[ : ]k A DNA k A DNA1
2
. .[ : ] k A DNAA DNAk
but
1
2
.(1 )
totalDNA
DNAk A
k
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1
2
.(1 )
totalDNA
DNAk A
k
1
2
. .[ : ]
k A DNAA DNA
k
1
2
1
2
[ : ]1
total
kA
k
A DNA DNAkA
k
2
1
d
kk
k
Dissociation constant
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[ : ]
1
d
total
d
A
kA DNA DNA
A
k
totalDNA
[ : ]A DNA
Hill Function
A
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2
: :k
A DNA mRNA A DNA
3 1.[ : ]dmRNA
k A DNA mRNAdt
[ : ]
1
d
total
d
A
kA DNA DNA
A
k
3 1.
1
d
total
d
A
kdmRNAk DNA mRNA
Adt
k
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3
3 13
( )
.1 ( )
d
total
d
A
kdmRNA
k DNA mRNAAdt
k
Cooperativity
start
promoter
end
terminator
gene
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3 1
( )
.1 ( )
n
d
total
n
d
A
kdmRNA
k DNA mRNAAdt
k
Cooperativity
start
promoter
end
terminator
gene
n molecules
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start
promoter
end
terminator
gene
3 1
1.
1 ( )total
n
d
dmRNAk DNA mRNA
Xdt
k
Repressor
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3
1.
1 ( )total
n
d
k DNAX
k
X
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