Self-organized criticality (SOC) is a popular concept that has been the subject of more than 3000 articles in the last 25 years. The characteristic signature of SOC is the appearance of self-similarity (power-law scaling) in observable properties. A characteristic observable protein property that describes protein-water interactions is the water-accessible (hydropathic) interfacial area of compacted globular protein networks. Here we show that hydropathic power-law (size- or length-scale-dependent) exponents derived from SOC enable theory to connect standard Web-based (BLAST) short-range amino acid (aa) sequence similarities to long-range aa sequence hydropathic roughening form factors that hierarchically describe evolutionary trends in water - membrane protein interactions. Our method utilizes hydropathic aa exponents that define a non-Euclidean metric realistically rooted in the atomic coordinates of 5526 protein segments. These hydropathic aa exponents thereby encapsulate universal (but previously only implicit) non-Euclidean long-range differential geometrical features of the Protein Data Bank. These hydropathic aa exponents easily organize small mutated aa sequence differences between human and proximate species proteins. For rhodopsin, the most studied transmembrane signaling protein associated with night vision, analysis shows that this approach separates Euclidean short- and non-Euclidean long-range aa sequence properties, and shows that they correlate with 96% success for humans, monkeys, cats, mice and rabbits. Proper application of SOC using hydropathic aa exponents promises unprecedented simplifications of exponentially complex protein sequence-structure-function problems, both conceptual and practical.
