Scatchards Equation for Macro Ions

Scatchard s equation (1946) is derived to explain Ai for proteins. The change in dimensions (e.g., mean-square end-to-end distance) is not of any concern in Scatchard s derivation. No model was assumed and no statistical mechanics was used. Scatchard successfully correlated the osmotic pressure with the distribution of diffusible solutes across a semipermeable membrane by manipulating the terms of activities of the components (such as protein, salt, and water) with changing composition of the solutions. The mathematical detail is simple but messy. According to Scatchard, the interactions involved in protein solutions are not limited to the exclusion of volume between the segments of macromolecules but also includes the Donnan effect and the binding of small ions to macro ions in a given system. For simplicity, let us consider a three-component system, and let 1 represent the solvent (or buffer), 2 the macromolecule (such as protein), and 3 a salt (e.g., NaCl). Scatchard derived an equation of the second vitial coefficient 

64) consists of three terms. The first describes the Donnan effect, which states that in osmosis the small ions are not distributed equally on the two sides of the membrane. The Donnan effect always causes an increase in osmotic pressure. To minimize this effect, the protein solution should have high ionic strength or should be near the isoelectric pH. The second term represents the excluded volume effect and the interaction between charges on different macro ions. It is basically similar to the parameter Xi in the properties of synthetic polymer solutions but is a little more complicated because of the charges involved. The third term involves the interaction between macro inons and the actual binding of small ions. This term is particularly important to the study of the behavior of proteins in dilute salt solutions. 

Experimental values for all of the three parameters (P22. p23 P33) molecular weight M2 can be determined from osmotic pressure experiments as follows. The plot of n /c2 versus C2 is expected to be a straight line from which the intercept A and the slope A2 are obtained. From the intercept we obtain M2 using the relation 

This is obtained from a separate and independent experiment (not included in the osmotic pressure measurement). The parameter P23 is calculated using 

In the above equations is original concentration of the salt in the protein solution, m3 is the concentration of the salt in the solution after diffusion, and tmj is the concentration of salt diffused out through the membrane. The quantities tMj, m3, and m3 are all measured in the same osmotic pressure experiment. After aU of the above parameters are calculated, the parameter P22 be determined directly from the slope A2 using Eq. (9.64). 

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