Smoothing protein energy landscapes by integrating folding models with structure prediction

Abstract

Decades of work has investigated the energy landscapes of simple protein models, but what do the landscapes of real, large, atomically detailed proteins look like? We explore an approach to this problem that systematically extracts simple funnel models of actual proteins using ensembles of structure predictions and physics-based atomic force fields and sampling. Central to our effort are calculations of a quantity called the relative entropy, which quantifies the extent to which a given set of structure decoys and a putative native structure can be projected onto a theoretical funnel description. We examine 86 structure prediction targets and one coupled folding-binding system, and find that in a majority of cases the relative entropy robustly signals which structures are nearest to native (i.e., which appear to lie closest to a funnel bottom). Importantly, the landscape model improves substantially upon purely energetic measures in scoring decoys. Our results suggest that physics-based models-including both folding theories and all-atom force fields-may be successfully integrated with structure prediction efforts. Conversely, detailed predictions of structures and the relative entropy approach enable one to extract coarse topographic features of protein landscapes that may enhance the development and application of simpler folding models.

Figure 1

Schematic of the landscape smoothing approach. Structures from an initial collection of webserver predictions are first energy-minimized using an all-atom physiochemical force field. Short (40 ps) molecular dynamics runs then serve to explore the immediate metabasin surrounding each. The average energies and interstructure distances are projected onto a simple, analytical funnel-shaped model that is tuned so as to minimize the relative entropy. Finally, structures are scored by the value of the relative entropy when the funnel minimum is coincident with them.

Figure 2

Landscape smoothing for the αβ target T0471. Minimized (top) and average MD energies (middle) do not show a strong correlation with near-nativeness. On the other hand, the relative entropy (bottom) does and successfully picks out the top structure, without any knowledge of the native one. For convenience, all S_rel values are shifted such that the minimum is zero.

Figure 3

Best predictions for T0471 as selected by minimum average energies and minimum relative entropy. The latter is in much better agreement with the native structure at 3.5/2.5 Å vs. 15.0/9.0 Å overall backbone RMSD/dRMSD. Interestingly, the parts of the top S_rel structure in worst agreement with the native tend to lie in loop regions that display increased flexibility across multiple NMR models.

Figure 4

Landscape smoothing for five selected CASP9 targets. (Left panels) Average MD energies. (Right panels) Corresponding relative entropy values. A dramatic improvement in correlation with near-nativeness is found for the relative entropy, across proteins of a variety of secondary structure content, size, and quality of predictions. (Dotted red lines) Value of the relative entropy for the true native structure; it is close to zero, as expected. The native structures depicted do not fall on top of any data points.

Figure 5

Docking predictions and selection for hirudin-thrombin binding. (Left) All 100 predictions of hirudin-thrombin complex obtained using the RosettaDock server. Ten hirudin structures are initially generated by replica exchange MD before docking. (Right) The lowest S_rel structure (teal) finds the appropriate binding pocket and is close to the native (green) at 5.4 Å RMSD.

Figure 6

Landscape smoothing for hirudin-thrombin binding. Similar to what was found for purely folding cases, this coupled folding-binding problem shows that minimized (top) and average MD energies (middle) do not signal proximity to the true binding conformation, whereas the relative entropy strategy (bottom) does. The large numbers of structures with S_rel near a value of 13 are cases in which the energy landscape was found to have a maximum rather than a funnel structure (α < 0), and the largest positive-α relative entropy was instead assigned. These structures turn out to cluster near the same region of the thrombin surface (Fig. S10).