18 RADIOISOTOPES IN BIOLOGY AND AGRICULTURE 



protein and measures 1000 counts/min to give a specific activity of 1000 

 counts/min/mg. If 100 mg of normal A is added to another 10-g sample 

 of protein and the isolation again carried out to yield 1 mg of the amino 

 acid, which now measures only 250 counts/min, it can be calculated that 



M* = jKry^ K^ "=" 33.3 mg of amino acid contained in 10 g of protein. 



A useful modification has been reported by Keston et al. (62), who pre- 

 pared labeled derivatives of the unknown substance quantitatively and 

 then added amounts of the unlabeled derivative to carry through the 

 inverse isotope-dilution procedure. Double-labeling techniciues to allow 

 correction for secondary losses and combinations with such powerful sep- 

 aration techniques as paper chromatography and countercurrent separa- 

 tions have been employed (63). Isotope dilution has also been useful in 

 studies involving the separation and isolation of racemic mixtures (64). 



The Kinetic Approach. It is of value to consider the kinetic aspects of 

 the various physical and biological processes, in addition to tracer prob- 

 lems concerned with the metabolic route and the identification of constit- 

 uents. A mathematical description of the process usually leads to a 

 better vniderstanding of the mechanics, to increased generalizations from 

 specific data, and to a useful basis for intercomparison of information. 

 In particular, kinetic analyses permit (a) estimation of the rate of transfer 

 of a substance from one phase to another, (6) determination of the 

 amounts present in each phase at a given time, and (c) identification of 

 the processes that may be involved. As always, for optimum usefulness 

 there must be a balance between maximum generality and rigorous treat- 

 ment, on the one hand, and oversimplification, on the other. Emphasis 

 will be given to those treatments which have been found of practical use 

 in the simpler systems. Mathematical analyses of this type have been 

 described many times (10, 65 to 67). 



In the discussion the numerical values for typical cases will often be 

 presented to allow the reader to plot and analyze the curves and to use 

 the equations. Unless otherwise noted, the equations are based on the 

 amount of labeled isotope. Provided that one is consistent and takes 

 into consideration such factors as volume changes, it is also appropriate 

 to use concentration or specific-activity values. 



A great many processes involve the so-called first-order reaction . These 

 reactions proceed at a rate that is proportional to the amount of sub- 

 stance present and may be described by the equation 



^ = -kA (,.,0) 



where A.4 = change in amount of substance A in unit time A^ 

 k = rate constant for process 



