Cooperative ligand binding is a fundamental property of many biological macromolecules,
notably transport proteins, hormone receptors, and enzymes. Positive homotropic cooperativity, the form
of cooperativity that has greatest physiological relevance, causes the ligand affinity to increase as ligation
proceeds, thus increasing the steepness of the ligand-binding isotherm. The measurement of the extent
of cooperativity has proven difficult, and the most commonly employed marker of cooperativity, the
Hill coefficient, originates from a structural hypothesis that has long been disproved. However, a wealth
of relevant biochemical data has been interpreted using the Hill coefficient and is being used in studies
on evolution and comparative physiology. Even a cursory analysis of the pertinent literature shows that
several authors tried to derive more sound biochemical information from the Hill coefficient, often unaware
of each other. As a result, a perplexing array of equations interpreting the Hill coefficient is available
in the literature, each responding to specific simplifications or assumptions. In this work, we summarize
and try to order these attempts, and demonstrate that the Hill coefficient (i) provides a minimum
estimate of the free energy of interaction, the other parameter used to measure cooperativity, and (ii)
bears a robust statistical correlation to the population of incompletely saturated ligation intermediates.
Our aim is to critically evaluate the different analyses that have been advanced to provide a physical
meaning to the Hill coefficient, and possibly to select the most reliable ones to be used in comparative
studies that may make use of the extensive but elusive information available in the literature.
The minimization of matrix logarithms: On a fundamental property of the unitary polar factor
