The QHA bases poorly align with directions that indicate high-energy states. Ubiquitin is universally expressed in eukaryotes and plays a fundamental role in the proteosomal degradation pathway by labeling specific proteins. The protein’s three-dimensional structure is highly conserved over evolution. Further, it is known to bind a large number of proteins with high specificity implying that its intrinsic mechanism of binding is finely tuned to respond to its diverse set of targets. Recently, it was proposed that the solution structure of ligand-free ubiquitin exhibits all of its conformational diversity required to bind diverse targets. These studies imply that ligand-free ubiquitin might Orbifloxacin occasionally visit conformations that resemble the ligand-bound structure. Hence, it is of interest to quantify from an ensemble, how many of these conformations exhibit the required diversity to resemble ligand-bound conformations. We can reduce the fourth order dependencies by minimizing the sum of the Catharanthine sulfate cross-cumulant terms, which is equivalent to diagonalizing the tensor K. However, no closed form solution exists for diagonalizing a tensor, but an approximate solution can be found using efficient algebraic techniques such as Jacobi rotations. We next examine if these conformational wells exhibit any similarity in terms of their internal energies, defined as the sum of van der Waals and electrostatic energy over all interactions in the protein and computed using the program NAMDEnergy. We plot the scaled internal energy values on the data in Figure 4 and illustrate it in Figure 5. Scaled internal energy refers to the sum of non-bonded interaction energies between all residues in the protein that have been normalized. While cluster I shows considerable diversity in its internal energies, clusters II, III and IV are homogeneous. The homogeneity in the internal energy distributions are quantified further in Figure S3 and supporting text S1. Clusters I and III are separated by 
highenergy structures possibly indicating a transition state between the two wells. The largest conformational well is highly diverse with respect to its internal energy distributions and positional deviations. Thus, we can examine the conformational diversity in this cluster by iteratively performing QAA only for this subset of conformations to see if a subsequent decomposition might homogenize this landscape. This corresponds to Level 2 in the conformational hierarchy. Figure 5 reveals that cluster I separates into 3 sub-states having unique structural and energetic properties. The separation between the high- and low-energy conformations from each cluster, as identified by QAA, provides a unique opportunity to examine the biophysical relevance of the relative populations and its impact on ubiquitin binding. Note that at any given level of the conformational hierarchy, the presence of a minor population of conformations sharing either high- or lowinternal energy. These minor populations deviate from the largest heterogenous cluster in exhibiting motions along functionally relevant regions. As one descends the conformational hierarchy, it becomes clear that the flexible regions of the protein do not change; only the amplitude of the actual conformational change changes. These changes in both motions and energetics allow ubiquitin to sample conformations that may in fact exceed the observed diversity in all of its bound conformations. Observe that the top 3 anharmonic modes of motion covers all of the conformational heterogeneity exhibited by the bound X-ray ensemble. The hierarchy of motions in ubiquitin allow the protein to sample conformations that involve modulating the pincer regions to varying degrees. This subtle interplay between global conformational fluctuations as well as its ability to modulate local motions can thus enhance ubiquitin’s ability to target multiple substrates. Overall, QAA allows the identification of energetically homogenous sub-states as well as a multi-level hierarchy of internal motions for ubiquitin.