Fast computational mutation-response scanning of proteins

Abstract

Studying the effect of perturbations on protein structure is a basic approach in protein research. Important problems, such as predicting pathological mutations and understanding patterns of structural evolution, have been addressed by computational simulations that model mutations using forces and predict the resulting deformations. In single mutation-response scanning simulations, a sensitivity matrix is obtained by averaging deformations over point mutations. In double mutation-response scanning simulations, a compensation matrix is obtained by minimizing deformations over pairs of mutations. These very useful simulation-based methods may be too slow to deal with large proteins, protein complexes, or large protein databases. To address this issue, I derived analytical closed formulas to calculate the sensitivity and compensation matrices directly, without simulations. Here, I present these derivations and show that the resulting analytical methods are much faster than their simulation counterparts.

Keywords: Compensatory mutations; Mutational response; Protein.

Figure 1. Comparison between sMRS and aMRS sensitivity matrices. Results shown for Phospholipase A2 (d1jiaa).

The sensitivity matrix S has elements S_ij that measure the structural shift of site i averaged over mutations at site j. sMRS is a simulation-based Mutation Response Scanning method that calculates S by averaging over simulated point mutations. aMRS is an analytical method that calculates S using a closed formula. (A) sMRS response matrix obtained by averaging over 200 mutations (simulation) compared with the aMRS matrix (analytical). (B) Scatterplot of the sMRS vs. aMRS matrix elements of A. (C) Convergence of sMRS with increasing number of mutations per site. In C the d1jiaa case is shown with black lines and points, and the other 9 proteins studied are shown with grey lines. Matrix elements Si j are normalised so that their average is 1. Logarithmic scale is used in A and B and R is the Pearson correlation coefficient between the log-transformed sMRS and aMRS matrices.

Figure 2. Comparison of sMRS and aMRS marginal profiles. Results shown for Phospholipase A2 (d1jiaa).

The influence profile is the average of the sensitivity matrix over rows; element S_j measures the average influence of mutations at site j. The sensitivity profile is the average of the response matrix over columns; element S_i measures the average sensitivity of site i. (A) Sj profiles obtained with sMRS using 200 mutations per site (simulation) and aMRS (analytical); (B) scatter plot of the sMRS vs. aMRS S_j values of A; (C) convergence of the sMRS S_j profile towards the aMRS profile. (D) Si profiles obtained with sMRS using 200 mutations per site (simulation) and aMRS (analytical); (E) scatter plot of the sMRS vs. aMRS S_i values of D; (F) convergence of the sMRS S_i profiles towards the aMRS profile. In C and F, the d1jiaa case is shown with black lines and points, and the other 9 proteins studied are shown using grey lines. Profiles were calculated using the normalised matrix (matrix average is 1). Profile elements are shown in logarithmic scale and R is the Pearson correlation coefficient between log-transformed sMRS and aMRS profiles.

Figure 3. The analytical mutation-response scanning method (aMRS) is much faster than the simulation method (sMRS).

(A) CPU time vs. protein size the for sMRS with 200 mutations per site (simulation) and for aMRS (analytical). Time is shown in logarithmic scale. From the slope of the linear fits it follows that both times scale with N^1.5 (N is the number of sites, each point is one protein). (B) The CPU time of the simulation method increases linearly with the CPU time of the analytical method, with a speedup of 126: t_sMRS = 126×t_aMRS. (C) The speedup, t_sMRS/t_aMRS obtained as shown in B, increases linearly with the number of mutations per site. Calculations were performed on the proteins of Table 1 using the methods implemented in R, with base LAPACK and the optimised AtlasBLAS libraries for matrix operations, on an early-2018 MacBook Pro notebook (processor i7-8850H).

Figure 4. Comparison of sDMRS and aDMRS compensation matrices. Results shown for Phospholipase A2 (d1jiaa).

The compensation matrix D has elements D_ij that measure the maximum compensation of the structural deformation due to a mutation at site i afforded by a second mutation at j. sDMRS is a simulation-based Double Mutation Response Scanning method that calculates D by maximizing the structural compensation over pairs of simulated mutations. aDMRS is an analytical method that calculates D using a closed formula. (A) sDMRS compensation matrix obtained using 200 mutations per site (simulation) compared with the aDMRS matrix (analytical). (B) Scatterplot of the sDMRS vs. aDMRS matrix elements of A. (C) Convergence of the sDMRS matrix towards the aDMRS matrix with increasing number of mutations per site. In C the d1jiaa case is shown with black lines and points, and the other 9 proteins studied are shown with grey lines. D_ij are normalised so that their average is 1, logarithmic scales are used in A and B, and R is Pearson’s correlation coefficient between log-transformed sDMRS and aDMRS matrix elements.

Figure 5. Comparison of sDMRS and aDMRS marginal profiles. Results shown for Phospholipase A2 (d1jiaa). Two marginal profiles are considered.

The D_j profile is the average of the compensation matrix over rows; element D_j measures the ability of j to compensate mutations at other sites. The D_i profile is the average of the compensation matrix over columns; element D_i measures the degree to which a mutation at i can be compensated by mutations elsewhere. (A) sDMRS D_j profile obtained using 200 mutations per site (simulation) and aDMRS D_j profile (analytical); (B) scatter plot of the sDMRS vs. aDMRS D_j values of A; (C) convergence of the sDMRS D_j profile towards the aDMRS profile. (D) sDMRS D_i profile obtained using 200 mutations per site (simulation) and aDMRS D_i profile (analytical); (E) scatter plot of the sDMRS vs. aDMRS D_i values of D; (F) convergence of the sDMRS D_i profile towards the aDMRS profile. In C and F, the d1jiaa case is shown with black lines and points, and the other 9 proteins studied are shown with grey lines. Profiles were calculated with normalised matrices (matrix average is 1), they are in logarithmic scale, and R is the Pearson correlation coefficient between the log-transformed sDMRS and aDMRS profiles.

Figure 6. The analytical double mutation-response scanning method (aDMRS) is much faster than the simulation method (aDMRS).

(A) CPU time vs. protein size for sDMRS with 200 mutations per site (simulation) and for aDMRS (analytical). Time is shown in logarithmic scale. From the slope of the linear fits it follows that both CPU times scale with N³ (N is the number of sites, each point is one protein). (B) The CPU time of the simulation method increases linearly with the CPU time of the analytical method, with a speedup of 137: t_sDMRS = 137×t_aDMRS. (C) The speedup, t_sDMRS/t_aDMRS, increases non-linearly with the number of mutations per site M, tending towards O(M²) for large M. Calculations were performed on the proteins of Table 1 using the methods implemented in R, with base LAPACK and the optimised AtlasBLAS libraries for matrix operations, on an early-2018 MacBook Pro notebook (processor i7-8850H).