  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.
