Visualization of distance distribution from pulsed double electron-electron resonance data

Double electron-electron resonance (DEER), also known as pulsed electron-electron double resonance (PELDOR), is a time-domain electron paramagnetic resonance method that can measure the weak dipole-dipole interactions between impaired electrons. DEER has been applied to discrete pairs of free radicals in biological macromolecules and to clusters containing small numbers of free radicals in polymers and irradiated materials. The goal of such work is to determine the distance or distribution of distances between radicals, which is an underdetermined problem. That is, the spectrum of dipolar interactions can be readily calculated for any distribution of free radicals, but there are many, quite different distributions of radicals that could produce the same experimental dipolar spectrum. This paper describes two methods that are useful for approximating the distance distributions for the large subset of cases in which the mutual orientations of the free radicals are uncorrelated and the width of the distribution is more than a few percent of its mean. The first method relies on a coordinate transformation and is parameter-free, while the second is based on iterative least-squares with Tikhonov regularization. Both methods are useful in DEER studies of spin-labeled biomolecules containing more than two labels.