intro chemical kinetics

7/27/2019 Intro Chemical Kinetics

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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

intro chemical kinetics

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