Basic Principles of KineticsBasic Principles of Kinetics

and Thermodynamicsand Thermodynamics

First Order ReactionsFirst Order Reactions

First order reactions involve the conversion of a single reactant to one or more products, the kinetics of which follow exponential kinetics.

First order processes are common and involve many phenomena such as protein denaturation and radioactive decay.

0 10 20 30 40 50 60

Time (min)

0

20

40

60

80

100

Am

ou

nt

A Pk

Rate (vt) = k[At]

A

P

vo

vt

v∞

vt = k [At]

Integrate from t=0 → t=∞:

At = Ao e-kt

Integrated first-order rate equation

Since Ao = At + Pt

(the total A we started with equals the A remaining plus product formed)

Then At = Ao – Pt

(Remaining A equals total A minus product formed)

Pt = Ao(1 - e-kt)Integrated first-order rate equation expressed as product formed

Background: You are studying the phosphorylation of a regulatory protein within a cultured cell line. To follow phosphorylation of the protein, the cells are incubated with radioactive inorganic phosphate (32Pi) which is taken up by the cells, rapidly incorporated into ATP from which it is used by a specific protein kinase to transfer 32P to your protein. After several hours you prepare an extract from the cells and isolate your protein by immunoprecipitation (the jargon is to “IP” the protein) using an antibody specific for the protein. Radioactivity in the immunoprecipitate is too small to measure directly since most proteins are present at exceedingly small levels within cells so you resolve the sample by Sodium Dodecyl Sulfate-PolyAcrylamide Gel Electrophoresis (SDS-PAGE) to separate by relative molecular weight. The gel is dried and placed against x-ray film to produce an autoradiogram.

Problem: You originally did this experiment on July 23 but find while writing the paper that you need to compare this result to a later experiment conducted under different conditions. This requires you to run a new SDS-PAGE containing an aliquot of your earlier sample and a new sample from the more recent experiment conducted with fresh 32Pi. You recall that the half life for 32P is 14 days. How would you directly compare the two samples by autoradiography?

Application

Bla

nk

GS

T

GS

T-E

2 ep

f

Bla

nk

GS

T

GS

T-E

2 ep

f

Bla

nk

GS

T

GS

T-E

2 ep

f

Bla

nk

GS

T

GS

T-E

2 ep

f

1:10 1:20 1:40 1:100Sample dilution

CP1

GST-CP1

Effect of Sample Load on Non-Specific Cross Reaction

Mem

bra

ne

Cyt

oso

l

Decreasing exposure time

Mem

bra

ne

Cyt

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Mem

bra

ne

Cyt

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Mem

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Cyt

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Mem

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Exposure Time and the “Limit of Detection”

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Time (min)

0

20

40

60

80

100

Am

ou

nt

Estimating k from the half life

50%

t½

At/Ao = e-kt½

0.5 = e-kt½

ln 0.5 = -kt½

0.693 = kt½

ln 2 = kt½

The reaction must be first order

Remember:

ln 10 = 2.3

Natural or hyperbolic logarithmNatural or hyperbolic logarithm

Base 10 logarithmBase 10 logarithm

log 10 = 1

ConversionConversion

ln x = 2.3 log x

ln e = 1

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Time (min)

0

20

40

60

80

100A

mou

nt

ΔP

Δt

vo = ΔP/Δt

Estimating k from the initial velocity The rate must be linear over the entire measurement period

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Time (min)

0

20

40

60

80

100

Am

ou

nt

Estimation of k from a single time point

At/Ao = 0.25e-k•10

One must know the process is first order

Estimation of k from the entire time course

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Time (min)

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Am

ou

nt

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Time (min)

0.1

1

10

100

% R

em

ain

ing

(A t/A

o)

ln At/Ao = -kt

Slope = -k

100100

1010

11

0.10.1

100100

1010

11

Two-cycle semi-log paperTwo-cycle semi-log paper Three cycle semi-log paperThree cycle semi-log paper

100100

1010

1100

11

0.70.7

1.71.7

22

Plotting Data With Semi-log PaperPlotting Data With Semi-log Paper

On semi-log paper, the y-axis is On semi-log paper, the y-axis is scaled linearly as the log (ln) of scaled linearly as the log (ln) of the number but plotted as the the number but plotted as the actual number. This avoids actual number. This avoids having to determine the log of having to determine the log of each value beforehand. each value beforehand.

time At/Ao x 100

0125102050

1009387715025

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