Extraction of Affinity Spectra

Besides the application of specific models, an important amount of work has been devoted to extract affinity distributions directly from experimental data, attempting to invert Equations 11.31, 11.37, and/or 11.40 or similar. The extraction of affinity distribution from experimental data is an ill-posed problem (Provencher 1982a Cernik, Borkovec, and Westall 1995 Borkovec et al. 1996), because with experimental data one actually has, in the case of Equation 11.31, 

Semianalytical approaches, which involve introducing approximations to invert Equation 11.31 analytically 

Regularization methods, to invert Equation 11.31 numerically by a modified least-squares scheme 

The maximum entropy (MaxEnt) approach, a different numerical method which searches for the most probable constant distribution 

In the following subsections, these approaches are reviewed. It is important to remark here that the deduction of affinity spectra from binding (titration) curves is a hard numerical problem, where the accuracy of the resulting spectrum depends critically on the quality of the input data it is well known that exact distributions can only be obtained with accordingly exact data even with 1% random deviations, variations in the results are found. Besides, titration curves are sensitive only to the main moments of the distribution (peak positions and width) thus in the absence of additional information, further details cannot be obtained (Merz 1980 Borkovec et al. 1996 Orsetti, Andrade, and Molina 2009). 

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The MaxEnt formalism seeks to find the most probable distribution compatible with all known constraints of a given problan it was introdnced by Jaynes (1957, 1982) and since then applied to many different statistical problems (Christakos 1998 Cheeseman and Stntz 2005 Komnitsas and Modis 2006 Vakarin and Badiali 2006 Orton and Lark 2007). Garces, Mas, and Puy (1998) proposed its application to the extraction of affinity spectra as an alternative to regularization methods as follows. The MaxEnt formalism attempts to find the most probable distribution /J of sites, compatible with the set of the M experimental data points obtained, where... 

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