We review the development of thermodynamic protein hydropathic scaling theory, starting from backgrounds in mathematics and statistical mechanics, and leading to biomedical applications. Darwinian evolution has organized each protein family in different ways, but dynamical hydropathic scaling theory is both simple and effective in providing readily transferable dynamical insights for many proteins represented in the uncounted amino acid sequences, as well as the 90 thousand static structures contained in the online Protein Data Base. Critical point theory is general, and recently it has proved to be the most effective way of describing protein networks that have evolved towards nearly perfect functionality in given environments, self-organized criticality. Darwinian evolutionary patterns are governed by common dynamical hydropathic scaling principles, which can be quantified using scales that have been developed bioinformatically by studying thousands of static PDB structures. The most effective dynamical scales involve hydropathic globular sculpting interactions averaged over length scales centered on domain dimensions. A central feature of dynamical hydropathic scaling theory is the characteristic domain length associated with a given protein functionality. Evolution has functioned in such a way that the minimal critical length scale established so far is about nine amino acids, but in some cases it is much larger. Some ingenuity is needed to find this primary length scale, as shown by the examples discussed here. Often a survey of the Darwinian evolution of a protein sequence suggests a means of determining the critical length scale. The evolution of Coronavirus is an interesting application; it identifies critical mutations.
How Life Works: Darwinian Evolution of Proteins