Maximum entropy: A complement to Tikhonov regularization for determination of pair distance distributions by pulsed ESR

Tikhonov regularization (TIKR) has been demonstrated as a powerful and valuable method for the determination of distance distributions of spin-pairs in bi-labeled biomolecules directly from pulsed ESR signals. TIKR is a direct method, which requires no iteration, and, therefore, provides a rapid and unique solution. However, the distribution obtained tends to exhibit oscillatory excursions with negative portions in the presence of finite noise, especially in the peripheral regions of the distribution. The Shannon-Jaynes entropy of a probability distribution provides an intrinsic non-negativity constraint on the probability distribution and an unbiased way of obtaining information from incomplete data. We describe how the maximum entropy regularization method (MEM) may be applied to solve the ill-posed nature of the dipolar signal in pulsed ESR. We make use of it to suppress the negative excursions of the distance distribution and to increase the tolerance to noise in the dipolar signal. Model studies and experimental data are investigated, and they show that, with the initial or “seed” probability distribution that is required for MEM taken as the TIKR result, then MEM is able to provide a regularized solution, subject to the non-negativity constraint, and it is effective in dealing with noise that is problematic for TIKR. In addition we have incorporated into our MEM method the ability to extract the intermolecular dipolar component, which is embedded in the raw experimental data. We find that MEM minimization, which is implemented iteratively, is greatly accelerated using the TIKR result as the seed, and it converges more successfully. Thus we regard the MEM method as a complement to TIKR by securing a positive pair distance distribution and enhancing the accuracy of TIKR.