Computing energy landscape maps and structural excursions of proteins

Abstract

Background: Structural excursions of a protein at equilibrium are key to biomolecular recognition and function modulation. Protein modeling research is driven by the need to aid wet laboratories in characterizing equilibrium protein dynamics. In principle, structural excursions of a protein can be directly observed via simulation of its dynamics, but the disparate temporal scales involved in such excursions make this approach computationally impractical. On the other hand, an informative representation of the structure space available to a protein at equilibrium can be obtained efficiently via stochastic optimization, but this approach does not directly yield information on equilibrium dynamics.

Methods: We present here a novel methodology that first builds a multi-dimensional map of the energy landscape that underlies the structure space of a given protein and then queries the computed map for energetically-feasible excursions between structures of interest. An evolutionary algorithm builds such maps with a practical computational budget. Graphical techniques analyze a computed multi-dimensional map and expose interesting features of an energy landscape, such as basins and barriers. A path searching algorithm then queries a nearest-neighbor graph representation of a computed map for energetically-feasible basin-to-basin excursions.

Results: Evaluation is conducted on intrinsically-dynamic proteins of importance in human biology and disease. Visual statistical analysis of the maps of energy landscapes computed by the proposed methodology reveals features already captured in the wet laboratory, as well as new features indicative of interesting, unknown thermodynamically-stable and semi-stable regions of the equilibrium structure space. Comparison of maps and structural excursions computed by the proposed methodology on sequence variants of a protein sheds light on the role of equilibrium structure and dynamics in the sequence-function relationship.

Conclusions: Applications show that the proposed methodology is effective at locating basins in complex energy landscapes and computing basin-basin excursions of a protein with a practical computational budget. While the actual temporal scales spanned by a structural excursion cannot be directly obtained due to the foregoing of simulation of dynamics, hypotheses can be formulated regarding the impact of sequence mutations on protein function. These hypotheses are valuable in instigating further research in wet laboratories.

Keywords: Energy landscape map; Evolutionary algorithm; Low-cost paths; Mechanical work; Multi-basin energy landscape; Multi-state protein; Nearest-neighbor graph; Protein equilibrium dynamics; Sample-based representation; Structural excursion.

Fig. 1

Run time profiling. The break-down of the run time along each of the three main components of the methodology is shown for SOD1, CaM, and H-Ras WT. The path query time refers to the proportion of running time spent on computing the lowest-cost path

Fig. 2

Crossover evaluation. The top panel shows the average fitness/energy among individuals in the hall of fame list (map), as the map is updated over generations. The bottom panel shows the average diversity among individuals in the hall of fame (measured via Euclidean distance) as the map is updated over generations

Fig. 3

Visualization in 2D of map and a lowest-cost path computed for SOD1. The computed map for SOD1 is projected onto 2D and projections are color-coded by Rosetta score12 energy values. Black dots show projections of experimentally-known structures. A lowest-cost path connecting the two visible basins is additionally drawn. Its cost (in REUs) is also listed

Fig. 4

Visualization in 2D of map and a lowest-cost path computed for caM. The computed map for CaM is projected onto 2D and projections are color-coded by Rosetta score12 energy values. Black dots show projections of experimentally-known structures. A lowest-cost path connecting two experimentally-known structures is also drawn; its cost (in REUs) is listed. The PDB ids of the structures the path connects, as well as other experimentally-known structures of interest, are additionally shown where these structures project onto the top two PCs

Fig. 5

Visualization via conditioning plots of map computed for H-Ras WT. The map computed for the H-Ras WT is visualized via conditioning plots, which plot projections of the hall of fame on PC1 and PC2 conditional on projections on PC3 and PC4. Sixteeen plots are generated, considering combinations of all quartiles of PC3 with all quartiles of PC4. The 4D space in each subplot is discretized via hexagons, plotting for each hexagon only the projection of the lowest-energy individual in the hexagon. The blue-to-red color-coding scheme follows the low to blue energy range. Dots in yellow show PC1-PC2 projections of experimentally-known structures of H-Ras WT

Fig. 6

Visualization in 2D of map and paths computed for H-Ras WT. The computed map for H-Ras WT is projected onto 2D and projections are color-coded by Rosetta score12 energy values. Low-cost paths (costs in REUs are listed) modeling the On →Off structural excursion are also drawn. The projections of experimentally-known structures on the top two PCs are related by showing whether the structures are captured in the wet laboratory for the WT or variants

Fig. 7

Visualization in 2D of map and paths computed for H-Ras G12C. The computed map for the H-Ras G12C variant is projected onto 2D and projections are color-coded by Rosetta score12 energy values. Low-cost paths (costs in REUs are listed) modeling the On →Off structural excursion are also drawn. The projections of experimentally-known structures on the top two PCs are related by showing whether the structures are captured in the wet laboratory for the WT or variants

Fig. 8

Visualization in 2D of map and paths computed for H-Ras Q61L. The computed map for the H-Ras Q61L variant is projected onto 2D and projections are color-coded by Rosetta score12 energy values. Low-cost paths (costs in REUs are listed) modeling the On →Off structural excursion are also drawn. The projections of experimentally-known structures on the top two PCs are related by showing whether the structures are captured in the wet laboratory for the WT or variants

Fig. 9

Visualization in 2D of map and paths computed for H-Ras Y32CC118S. The computed map for the H-Ras Y32CC118S variant is projected onto 2D and projections are color-coded by Rosetta score12 energy values. Low-cost paths (costs in REUs are listed) modeling the On →Off structural excursion are also drawn. The projections of experimentally-known structures on the top two PCs are related by showing whether the structures are captured in the wet laboratory for the WT or variants

Fig. 10

Visualization in 3D of map computed for H-Ras WT. The lowest-energy structures in the map computed for H-Ras WT are shown projected onto the top 3 PCs. Projections of the experimentally-known structures are also drawn, as red spheres of a larger radius. The PDB ids of some of these structures are also shown. The four basins that emerge on the WT and the various variants are also delineated and named per the convention described in the main text