Calculating ensemble averaged descriptions of protein rigidity without sampling

Abstract

Previous works have demonstrated that protein rigidity is related to thermodynamic stability, especially under conditions that favor formation of native structure. Mechanical network rigidity properties of a single conformation are efficiently calculated using the integer body-bar Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations to accurately account for fluctuations in network topology. We have developed a mean field Virtual Pebble Game (VPG) that represents the ensemble of networks by a single effective network. That is, all possible number of distance constraints (or bars) that can form between a pair of rigid bodies is replaced by the average number. The resulting effective network is viewed as having weighted edges, where the weight of an edge quantifies its capacity to absorb degrees of freedom. The VPG is interpreted as a flow problem on this effective network, which eliminates the need to sample. Across a nonredundant dataset of 272 protein structures, we apply the VPG to proteins for the first time. Our results show numerically and visually that the rigidity characterizations of the VPG accurately reflect the ensemble averaged [Formula: see text] properties. This result positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability.

Figure 1. The respective network descriptions are compared.

Equilibrium rigidity properties (designated as ) are calculated by averaging across an ensemble of binary networks where H-bonds are either present or not. Conversely, the VPG describes the network with H-bond probabilities.

Figure 2. Two different rigid cluster decomposition examples are compared.

In the first example, (a), there are 1.4 free pebbles available (located on vertices and ), whereas the capacities of edges , , and is, respectively, 0.6, 5.0, 5.0. If a hypothetical edge is added between any pair of vertices, there is always going to be possible to find DOF, therefore the three vertices result in single bodies (highlighted by color differences). Conversely, in the second example, (b), only the six trivial DOF can be found (on vertex ). That is, no free pebbles remain in the network (they have all be consumed by the edges). As such, the five vertices belong to a single rigid cluster.

Figure 3. Quantifying and VPG similarity.

(a) The Rand Measure (RM) is plotted versus for four exemplar proteins that span a range of sizes (from 64 to 315 residues) and topological architectures. All proteins across the full dataset have the same characteristic shape where the minima in RM is related to the protein structure's rigidity transition. The value corresponding to the worst RM is defined as . (b) Histogram detailing values for each protein within our dataset. Encouragingly, an overwhelming majority of cases have RMs greater than . (c) Histogram detailing the agreement measure for each backbone torsion within our dataset at each protein's respective value. (d) Histogram detailing the Pearson correlation coefficient comparing the and VPG mechanical coupling maps across the dataset at each protein's respective value.

Figure 4. Rigid Cluster visualizations for four example proteins.

The first column highlights the rigid clusters identified within the PG realization that corresponds to the median RM value, designated . The middle column corresponds to the rigid clusters identified by the VPG. Finally, differences between the two algorithms are highlighted in the third column.

Figure 5. Agreement measure (AM) results.

AM histograms for the (a) methyltransferase, (b) FLAP endonuclease and (c) disulfide oxidoreductase at their respective values. (d) Differences between the and VPG are mapped to the ribosylglycohydrolase structure from M. jannaschii, which is presented as a typical case. Red coloring indicates that the VPG overestimates rigidity relative to , whereas blue indicates an underestimate. Across our dataset, as shown in this example, differences occur most frequently in loop regions.

Figure 6. Rigid cluster maps (RCM) of chemotaxis receptor methyltransferase CheR structure is plotted at two different values.

Red coloring identifies residue pairs that are co-rigid. results are presented in the upper triangle, whereas the VPG is presented in the lower. At , the protein is mostly flexible due to a lack of crosslinking H-bonds. However, the structure becomes increasingly rigid as H-bonds are added to the network. At , the VPG slightly under-predicts the extent of rigidity. For this protein .

Figure 7. Rigid Cluster Maps (RCM) for four different example proteins near their respective values.

results are presented in the upper triangle, whereas the VPG is presented in the lower. Note that the presented proteins are the same from Fig. 3a.

Figure 8. Mechanical Coupling Maps (MCM) provide a more nuanced description of co-rigidity.

Specifically, the continuous scale provides a normalized description of how many free pebbles (DOF) are shared between each residue pair (0 = 0 pebbles, whereas 1 = 6 pebbles). Again, each MCM is plotted near their respective values for the same four proteins presented Figs. 3a and 7.

Figure 9. Boxplots describing the ensemble of Pearson correlations coefficients comparing each PG realization to the behavior.

The red line represents the correlation between the and the VPG. In all cases, the to VPG similarity is greater than the 75th percentile of the intrinsic fluctuations within the PG ensemble.

Figure 10. Rigid Cluster Susceptibility (RCS) is plotted versus for 12 typical protein examples (  =  solid line and VPG  =  dashed line).

Note that the proteins presented in the first column are the same from Fig. 3a.

Figure 11. Rigidity transition effects.

(a) The rigidity transition ( ) is compared across the and VPG algorithms. (b) Similarly, the average cluster size (ACS) at their respective values are compared across the two algorithms. The value of with the worst RM (called ) is compared to calculated using the (c) VPG and (d) . The linear relationships occur because the mean field approximation is maximally inaccurate in this range. Note, a few proteins do not have completed peaks in their rigid cluster susceptibility curves because the protein never crosses the rigidity transition, which have been excluded from panels (c) and (d).