Basic Principles of Kinetics and Thermodynamics

First Order Reactions First order reactions involve the conversion of a single reactant to one or more products and follow exponential kinetics. First order processes are common and involve many phenomena such as protein denaturation and radioactive decay.

A P k Rate (vt) = k[At] v∞ P vt vo A

At = Ao e-kt Pt = Ao(1 - e-kt) vt = k [At] Integrate from t=0 → t=∞: 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

Application 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?

Effect of Sample Load on Non-Specific Cross Reaction Blank GST GST-E2epf 1:10 1:20 1:40 1:100 Sample dilution aCP1 GST-aCP1

Exposure Time and the “Limit of Detection” Membrane Cytosol Membrane Cytosol Membrane Cytosol Membrane Cytosol Membrane Cytosol Decreasing exposure time

Estimating k from the half life The reaction must be first order At/Ao = e-kt½ 0.5 = e-kt½ ln 0.5 = -kt½ 50% t½ ln 2 = kt½ 0.693 = kt½

Remember: ln e = 1 ln 10 = 2.3 log 10 = 1 ln x = 2.3 log x Natural or hyperbolic logarithm ln e = 1 ln 10 = 2.3 Base 10 logarithm log 10 = 1 Conversion ln x = 2.3 log x

Estimating k from the initial velocity The rate must be linear over the entire measurement period ΔP vo = ΔP/Δt Δt

Estimation of k from a single time point One must know the process is first order At/Ao = 0.25e-k•10

Estimation of k from the entire time course Slope = -k ln At/Ao = -kt

Two-cycle semi-log paper Three cycle semi-log paper 100 100 10 10 1 1 0.1

Plotting Data With Semi-log Paper 1 0.7 1.7 2 100 10 1 On semi-log paper, the y-axis is scaled linearly as the log (ln) of the number but plotted as the actual number. This avoids having to determine the log of each value beforehand. time At/Ao x 100 1 2 5 10 20 50 100 93 87 71 25 3 10 20 30 40 50