Global measures of the chain structure

The mean-squared end-to-end distance depends on an average over all the conformations of a chain molecule and gives a single-number measure of the global extent of the macromolecule. Another important quantity that depends on an average over the conformations of the whole chain is the mean-squared radius of gyration ) Each macromolecule is characterized by the location of its center of mass, r. In the frame of reference of the center of mass, the location of each atom can be expressed as a vector, s, that is equal to the vector difference between the location of the atom and the center of mass  

The mean-squared radius of gyration is the mass-weighted average squared distance of an atom in the macromolecule from its center of mass  

For a homopolymer, it is convenient to calculate the mean-squared radius of gyration by dividing the chain into the same m subunits discussed in Section 2.6 for the random coil chain. Because all of the subunits have the same weight, the mass weighting ceases to be necessary. There is also a theorem due to Lagrange that allows to be expressed in terms of the intersubxmit vectors, R f. 

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