As before all calculations are
As before , , all calculations are based on thermodynamically first- and second-order hydropathic (amino acid) scales , , linearly scaled to a common center and a common range for each of the 20 amino acids. These are then converted to a triangular matrix , where is the length of a sliding window centered on each amino filipin site. We have studied the range , which is includes values of W much larger than the small value, fixed at , in most calculations using sliding windows . Just as one focuses a microscope to optimize its image, one scans W to optimize its recognition of allometric regularities of hydropathic hot spots (hydrophobic extrema of ) at special values of .
Results The hydrophobic extrema of neuroglobin form sophisticated patterns that are closely related to the evolution of specific species. For example, mouse and rabbit escape predators in different ways, and these differences are recognizable in their profiles . There are several other examples already of proteins whose hydrophobic extrema form level sets. In Fig. 1 we plot the profiles of Uba1 (E1) for humans and slime mold. The choice levels the human hydrophobic extrema, and simultaneously aligns the slime mold extrema linearly with a small tilt (about 15% of the overall range). Such excellent alignments (to within 1%) are unlikely and not accidental. For instance, the differences between the MZ and KD scales are small (85% correlation ), yet as Fig. 2 shows, the successful pivotal alignment with the MZ scale is lost with the KD scale. Structural data are most complete for Uba1 (E1) yeast, and the human and yeast profiles are compared in Fig. 3. The differences are small, and are mentioned in the Fig. 3 caption. Before we compare the long-range (“allosteric”) correlations of these figures, we show in Fig. 4 the results for fruit fly, which has a lifetime of days, not years. This implies that its Uba1 kinetics are ∼103 faster than human Uba1 kinetics. It is plausible that the two hydrophilic minima discussed in Fig. 4, which are 3–6 lower values of (5%–10% of the full range) than in the human , are good indicators of this kinetics acceleration.
Discussion Before comparing our results with structural studies, we can turn to the Wiki on transition state theory (1935), which explains the reaction rates of elementary chemical reactions in terms of two parameters in one dimension. Structural studies contain information on the ground state, a minimum in configuration space, whereas rates are determined by the properties of transition states, technically also saddle points in configuration space. Both minima and saddle points are also thermodynamic critical points, where long-range attractive and short-range repulsive interactions are equal at the critical temperature, close to body temperature. Our conjecture here is that studying extrema, both the hydrophobic pivots and the hydrophilic hinges , provides one-dimensional insights into extremal sets of two parameters describing transition state properties, which can be compared to structural studies ,  of ground state properties. Because Uba1 is so large, the free energy differences between these two states are very small, and the static and dynamic structural differences are expected to involve interdomain conformational motion. According to , Uba1-E1 consists of four building blocks: first, the adenylation domains composed of two motifs (labeled IAD (1–169) and AAD (404–594), for “inactive” and “active” adenylation domain, respectively), the latter of which binds ATP and Ub; second, the catalytic cysteine half-domains, which contain the E1 active site cysteine (CC (169–268) and CCD (594–860)) inserted into each of the adenylation domains; third, a four-helix bundle 4HB (268–356) that represents a second insertion in the IAD; and fourth, the C-terminal ubiquitin-fold domain (UFD (926–1024)), which recruits specific E2s. How do these structural and functional domains compare with our one-dimensional hydropathic profiles? Fig. 5 shows an excellent match, with the domain boundaries associated either with pivots or one amphiphilic side of a hinge. One could argue here that the separation of 4HB is unnecessary, but this secondary structure includes the deep and wide hydrophilic minimum 330–370, with a sharp edge at 320 of .