A hierarchy of coupling free energies underlie the thermodynamic and functional architecture of protein structures

Abstract

Protein sequences and structures evolve by satisfying varied physical and biochemical constraints. This multi-level selection is enabled not just by the patterning of amino acids on the sequence, but also via coupling between residues in the native structure. Here, we employ an energetically detailed statistical mechanical model with millions of microstates to extract such long-range structural correlations, i.e. thermodynamic coupling free energies, from a diverse family of protein structures. We find that despite the intricate and anisotropic distribution of coupling patterns, the majority of residues (>70%) are only marginally coupled contributing to functional motions and catalysis. Physical origins of 'sectors', determinants of native ensemble heterogeneity in extant, ancient and designed proteins, and the basis for allostery emerge naturally from coupling free energies. The statistical framework highlights how evolutionary selection and optimization occur at the level of global interaction network for a given protein fold impacting folding, function, and allosteric outputs.

Keywords: Allostery; Energetic coupling; Energy landscape; Function; Stability; Thermodynamics.

Graphical abstract

Fig. 1

Steps involved in the calculation of thermodynamic coupling free energies from the WSME model. The native protein structure is considered as either residues or blocks (panel A) from which the ensemble is constructed (panel B) with 0 and 1 representing unfolded and folded residue (block) status, respectively. Since the WSME model is structure-centric, the energetics can be directly derived from the interactions present in the native state (vdW interactions, electrostatics, solvation free energy) while the entropic penalties are invoked based on the secondary-structure type identified in the PDB, following which the statistical weights and probabilities of every microstate are calculated. The resulting ensemble is partitioned into four sub-ensembles depending on the conditional folded status of residue j with respect to residue i (panel C). Positive, negative and effective thermodynamic coupling free energies are estimated (panel D) that can either be represented as a coupling matrix (panel E) or as vectors as a function of protein sequence index (panel F). In addition to this, the effective coupling free energies are mapped on to the structure, and those residues exhibiting strong coupling (Z-score > 1) are shown as dots (panel G). The arrow in panel G points to the direction in which the ligand binds.

Fig. 2

Representative examples of coupling free energy calculations on Ank4, Villin and Barstar. Shown are the one-dimensional free energy profiles (panels A, D, G), effective coupling free energy matrices (panels B, E, H) and the structural maps (panels C, F, I). The parameters n, f_c and σ indicate the total number of microstates, the fraction of strongly coupled residues and the standard deviation in effective coupling free energy, respectively. The arrow in panel B signals the weak coupling between 2^nd and 4^th repeat residues in Ank4 while the white block in panel E shows near-zero coupling between helices 1–2 and 3 in Villin. The green arrow in panel I represents the binding surface on Barstar. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 3

Functional connect of marginally coupled residues. Free energy profiles of Cnu, RNase H, T4 lysozyme and PDZ (panels A, D, G, J), the effective coupling free energies with the strongly coupled residues in red and functional residues in yellow (panels B, E, H, K), and the structural map of the effective coupling free energies (panels C, F, I, L). The green arrows in panels C, F and I represent the binding surface. The black arrow in panel K signals the weak coupling of the C-terminal hairpin in PDZ to the rest of the structure, while the spheres in panel L represent ‘sector’ residues. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 4

Larger proteins display a similar anisotropic distribution of coupling free energies. The color code is maintained the same as Fig. 3 for the proteins glucocorticoid receptor (panels A, B, C), adenylate kinase (panels D, E, F) and beta-2 adrenergic receptor (panels G, H, I). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 5

Insights into structural partitioning of coupling free energies in designed proteins and enzymes. (A, B) Mapping of coupling free energies onto the structure (panel A) for S6 and the designed protein Top7 together with their free energy profiles (panel B). The red arrow in panel B indicates the large difference in barrier heights between the two proteins. (C, D) Structural mapping of the coupling free energies of thioredoxins from E. coli and the resurrected LBCA ancestor (panel C), and their corresponding free energy profiles (panel D). The red arrow signals the more ‘disordered’ native ensemble for the LBCA ancestor. (E) Free energy profiles of the designed Kemp Eliminases HG3 and HG3.17. The red arrow signals the large difference in barrier heights between the two and the overall tilting of the HG3.17 towards the folded state (black). (F) The difference in coupling free energies between the two variants HG3 and HG3.17. The positions of additional mutations in HG3.17 are shown in green circles while the red arrows displays the non-intuitive changes in coupling free energies at positions far from the mutated sites. (G) Mapping of the coupling free energy differences onto the structure. The transition state analog is shown in cyan. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 6

Coupling of experimentally known sites to active site residues in PDZ (Y92; panels A, B), CypA (V29; panels C, D) and CheY (D56; panels E, F). In panels A, C, and E, the star and the shaded block highlight the position in consideration, i.e. the residue whose coupling with other residues is considered (a residue is infinitely coupled with itself and hence no coupling free energy value is shown). The yellow circles are the active site residues in panels A and C, while they represent the known allosteric quartet in CheY. Coupling free energies mapped on to the respective structures are shown in panels B, D, and F with the spheres representing the corresponding active site residues in panel B and D, and the allosteric quartet in panel F. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)