Self-Organized Networks with Long-Range Interactions: Tandem Darwinian Evolution of $alpha$ and $beta$ Tubulin

Cytoskeletons are self-organized networks based on polymerized proteins: actin, tubulin, and driven by motor proteins, such as myosin, kinesin and dynein. Their positive Darwinian evolution enables them to approach optimized functionality (self-organized criticality). Our theoretical analysis uses hydropathic waves to identify and contrast the functional differences between the polymerizing $alpha$ and $beta$ tubulin monomers, which are similar in length and secondary structures, as well as having indistinguishable phylogenetic trees. We show how evolution has improved water-driven flexibility especially for $alpha$ tubulin, and thus facilitated heterodimer microtubule assembly, in agreement with recent atomistic simulations and topological models. We conclude that the failure of phylogenetic analysis to identify functionally specific positive Darwinian evolution has been caused by 20th century technical limitations. These are overcome using 21st century quantitative mathematical methods based on thermodynamic scaling and hydropathic modular averaging. Our most surprising result is the identification of large level sets, especially in hydrophobic extrema, with both thermodynamically first- and second-order scaled water waves. Our calculations include explicitly long-range water-protein interactions described by fractals. We also suggest a much-needed corrective for large protein drug development costs.