Guinier Derivation ... by Robert P. Rambo, Ph.D.
The radius-of-gyration, Rg, describes the mass distribution of the macromolecule around its center of gravity. It is potentially a very powerful parameter to measure in a experiment especially if the macromolecule is undergoing conformational changes during substrate binding. The Rg is sometimes called the second moment of inertia, please see this wiki for more information.
The Guinier analysis is a linearization of the small angle scattering data in the region closest to the zero scattering angle. Starting with the classic description of scattering, the following is a derivation of the Guinier plot and analysis.
The above sine function can be approximated by a Taylor series expansion. However, the equation will evaluate the experimental data closest to the zero angle, therefore, we can set a = 0 to produce the McLaurin series expansion of sine.
The McLaurin representation of sine simplifies the expression thereby re-writing the classic scattering equation as a polynomial in terms of q & r (see following derivation). Now, if we consider the fact that the function will be evaluated when q is very small, say less than 0.003, then the higher order terms will essentially make very little contribution to the function (q4 = 0.000000000081).
Thus, we can consider the first two terms in the expansion as significant and ignore the higher order terms, q4, q6, q8, ...This greatly simplifies the polynomial expansion of the scattering function resulting in:
Now, we need to define I(0) and Rg as follows and substitute.
The equation is not a linear equation; however, Guinier's approach recognized the boxed region below to be a Taylor series expansion of ex. This substitution is an approximation to the function and still requires q to be small.
Finally, we can linearize the equation by taking the natural logarithm of both sides thereby producing the recognizable form of y = m x + b.