Data set

The wild type structure of BsLipA with the highest resolution (PDB ID: 1ISP; resolution = 1.3 Å) was obtained from the Protein Data Bank (PDB; www.pdb.org) [57]. For probing the sensitivity of the CNA results on the conformation of the input structures, five additional crystal structures of wild type BsLipA were analyzed (PDB IDs: 1I6W, 1R4Z, 1R50, 2QXT, 2QXU). We included in our study all mutants from Rao et al. for which T_m values were determined [44–48]. In addition, we included the three most thermostable mutants developed in the last rounds of iterative saturation mutagenesis by Reetz et al. [42]. Models of mutant structures for which crystal structures were not available in the PDB were generated with the SCWRL program [58], using the respective BsLipA structure as a template that is closest in sequence to the mutant. SCWRL constructs mutant models by predicting backbone-dependent side chain conformations with the help of a rotamer library; coordinates of backbone atoms remain unchanged. Conformations of side chains of all residues within 8 Å of a mutated residue were re-predicted in order to allow for a local structural relaxation. For all structures, hydrogen atoms were added using REDUCE [59]; side chains of Asn, Gln, and His were flipped in this stage if necessary to optimize the hydrogen bond network. All water molecules, buffer ions, and crystal solvents were removed from the structures. Finally, all structures were minimized by 5000 steps of conjugate gradient minimization (including an initial steepest descent minimization for 100 steps) or until the root mean-square gradient of the energy was < 1.0·10⁻⁴ kcal mol^-1 Å^-1. The energy minimization was carried out with Amber11 [60] using the Cornell et al. force field [61] with modifications for proteins (ff99SB) [62] and the GB^OBC generalized Born model [63]. All variants of BsLipA used in this study are summarized in Table 1.

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Table 1. Summary of BsLipA variants used in the study.

https://doi.org/10.1371/journal.pone.0130289.t001

Construction of the constraint network and rigidity analysis

As described in the previous section, only the protein part was considered for network construction, i.e., all non-protein molecules including water molecules were discarded. This was done based on previous findings that including water molecules does not significantly change the rigidity analysis results [64, 65]. Proteins were modeled as constraint networks in a body-and-bar representation (see section “Body-and-bar networks” in S1 File) [66, 67] using the CNA software [35] that acts as a front- and back-end to the Floppy Inclusion and Rigid Substructure Topography (FIRST) program [51, 68]. Once the constraint network is built, rigidity analysis is carried out, which identifies (rigid) clusters of atoms with no internal motion and flexible links in between, using the pebble game algorithm [52, 53] as implemented in the FIRST software [51].

Thermal unfolding simulation

By sequentially removing non-covalent constraints from a network, one can simulate a loss of structural rigidity due to a temperature rise. Specifically, hydrogen bonds were removed from the network in increasing order of their strength following the idea that stronger hydrogen bonds break at higher temperatures than weaker ones [69]. As such, only hydrogen bonds with an energy E_HB ≤ E_cut(σ) were included in the network of state σ. A thermal unfolding trajectory of 60 network states was generated for each input network by decreasing E_cut from −0.1 kcal mol⁻¹ to −6.0 kcal mol⁻¹ with a step size of 0.1 kcal mol⁻¹. According to the linear relationship between E_cut and the temperature T introduced by Radestock and Gohlke on 20 pairs of orthologs from mesophilic and thermophilic organisms, respectively (Eq 1) [12, 13], the range of E_cut used in this study is equivalent to increasing the temperature of the system from 302 K to 420 K with a step size of 2 K. Because hydrophobic interactions remain constant or become even stronger as the temperature increases [70, 71], the number of hydrophobic tethers were kept unchanged throughout the thermal unfolding simulation. Rigidity analysis was performed on all such generated network states, and then local and global rigidity characteristics were calculated (see section “Local and global rigidity indices” in S1 File). The setup of the thermal unfolding simulation and the subsequent rigidity analysis were performed using the CNA software [35], which is available from http://cpclab.uni-duesseldorf.de/software. A web service for performing CNA analysis can be accessed via http://cpclab.uni-duesseldorf.de/cna [34].

(1)

Ensemble of networks generated by using fuzzy noncovalent constraints

For improving the robustness of rigidity analyses, CNA is generally carried out on an ensemble of structures (e.g., generated by molecular dynamics (MD) simulations), and then results are averaged [11, 64]. The preceding MD simulation compromises the efficiency of the rigidity analysis, however. To overcome this drawback, Pfleger et al. [56] recently introduced an approach that performs rigidity analyses on an ensemble of network topologies (ENT^FNC) generated from a single input structure by using fuzzy noncovalent constraints. Here, the number and distribution of non-covalent constraints (hydrogen bonds and hydrophobic tethers) are modulated by random components within certain ranges as specified in ref. [56], thus simulating thermal fluctuations of a biomacromolecule without actually moving atoms. An ensemble of 2000 network configurations was generated using these definitions of fuzzy noncovalent constraints for all BsLipA variants, respectively. Finally, average local indices were calculated, as were average phase transition temperatures identified by the global index cluster configuration entropy H_type2. The index H_type2 monitors the degree of disorder in the realization of a given network state σ: As long as a network is dominated by a very large rigid cluster, H_type2 tends to be low because there are only a few configurations of a system with a large rigid cluster possible; H_type2 increases when larger rigid clusters break down in smaller clusters (see section “Local and global rigidity indices” in S1 File and ref. [55] for details).

Clustering of unfolding pathways

Recently, we showed that curves of the rigidity order parameter, which characterizes the general percolation behavior of a constraint network during thermal unfolding, for mesophilic proteins and their thermophilic counterparts are almost identical except for a shift of the curve of the thermophilic protein to higher temperatures [12]. This finding supported the hypothesis of corresponding states according to which mesophilic and thermophilic enzymes are in corresponding states of similar rigidity and flexibility at their respective optimal temperature [12]. The percolation index p_i is a local analog to the rigidity order parameter. It monitors for each bond when it segregates from the largest rigid cluster present at the beginning of a thermal unfolding simulation (see section “Local and global rigidity indices” in S1 File and ref. [55] for details). That way, a residue-wise p_i profile of a protein, generated by taking the lower of the p_i values of the two backbone bonds for each residue, expresses the hierarchical break-down of the largest rigid cluster during a thermal unfolding simulation.

We thus reasoned that the (dis)similarity of unfolding pathways of BsLipA variants can be measured by Manhattan distances between their respective p_i profiles. We used this distance measure for clustering the network topologies of all BsLipA variants into 10 clusters using the Partitioning Around Medoids algorithm [72] as implemented in the R program (http://www.r-project.org). This optimal number of clusters was chosen based on monitoring the change in the objective function of the clustering (the mean of the dissimilarities of all objects to their nearest medoids) as a function of the number of clusters (Figure A in S1 File) and visual inspection of cluster medoids for their dissimilarity to other medoids (residue-wise p_i profiles for medoids of the 10 clusters are shown in Fig 1). A clustering in more than 10 clusters essentially created additional clusters that were very similar to other clusters. From this, the cluster distribution (frequencies of network topologies in each of the 10 clusters out of in total 2000 network topologies) for each BsLipA variant was calculated by counting the number of networks that belongs to each of the 10 clusters. A high (low) correlation between cluster distributions for two BsLipA variants then indicates that both variants unfold in a similar (different) manner. Finally, a matrix of all pairwise correlations of cluster distributions of BsLipA variants was generated.

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Fig 1. Residue-wise p_i plots for medoids of the 10 clusters.

Secondary structure elements as computed by the DSSP program [88, 89] are indicated on the top of the plots and are labeled: α-helix (red rectangle), β-strands (green rectangle), loop (black line).

https://doi.org/10.1371/journal.pone.0130289.g001

Data set

BsLipA is a protein of 181 amino acids with a minimal α/β hydrolase fold; in this fold, a central parallel β-sheet of six β-strands is surrounded by six α-helices. Ser77, Asp133, and His156 constitute the catalytic triad (Fig 2). Unlike other lipases, the catalytic site in BsLipA is not covered with a lid. Hence, BsLipA does not show interfacial activation [40]. The data set used in this study contains structures of the wild type BsLipA, thirteen mutants from Rao et al. [44–48], and three mutants from Reetz et al. [42, 49] (Table 1). The mutants differ from the wild type by three to twelve mutations, i.e., the sequence identity is > 93%. Models for the mutants for which X-ray structures were not available were built using the SCWRL program. As the number of mutations in the modeled variants is ≤ 7 with respect to the template structures (< 4% with respect to the sequence length) (Table 1), an overall similar backbone confirmation can be expected as can be an overall reliable modeling of side chain conformations by SCWRL. This was also evident from a very good structural alignment and low root-mean-square deviations (RMSD) between the wild type and those mutants for which crystal structures were available (C_α atom-based RMSD values between the wild type and the mutants < 0.38 Å). The high structural similarity allows a direct comparison of results from rigidity analyses for these structures [11–13].

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Fig 2. Cartoon representation of wild type BsLipA with mutated residues indicated by spheres of their C_α atoms (mutations from Rao et al. [44–48]: magenta; Reetz et al. [42, 49]: orange; mutations common in both data sets: cyan).

The catalytic triad (Ser77-Asp133-His156) is shown in stick representation with yellow carbons. The protein is colored according to secondary structure (α-helices: red; β-sheets: yellow; loops: green). The right view (B) differs from the left (A) by an anti-clockwise rotation of ~90° about a horizontal axis. All figures of BsLipA structures were generated with PyMOL (http://www.pymol.org).

https://doi.org/10.1371/journal.pone.0130289.g002

The melting temperature T_m of the wild type is 329.15 K. The T_m values of the mutants of Rao et al. range from 334.35 to 351.35 K (Table 1). For the mutants of Reetz et al. no T_m values are available. Rather, unfolding initiation temperatures T_i were reported, which are lower by 2.5 to 6.2 K than that of the wild type. This suggests that mutants of Reetz et al. are thermodynamically less thermostable than the wild type [49], in contrast to mutants from Rao et al. [44–48]. However, we note that, while T_m reports on the temperature at which 50% of the protein is unfolded and, hence, properly describes the folded-unfolded equilibrium of a protein, T_i only reports on the temperature at which the unfolding transition begins. Therefore, we will only consider relations within mutants of Rao et al. and to the wild type and distinguish those from relations within mutants of Reetz et al. and to the wild type. Finally, the T₅₀^t values of the mutants of Reetz et al. are higher than that of the wild type (Table 1), showing that these mutants more efficiently refold upon cooling after incubation at high temperatures than does the wild type. The location of mutations in all of the mutants investigated in this study is shown in Fig 2; all mutations are located on the protein surface.

Thermal unfolding pathway of BsLipA

From monitoring the loss in rigidity percolation during thermal unfolding simulations, major phase transitions in the protein can be identified that relate to the unfolding pathway [11–13, 54, 73]. Here, we describe the loss of rigidity percolation of the wild type BsLipA (PDB ID 1ISP) as an example. Similarity or dissimilarity, respectively, of the unfolding pathways across all variants is described below. During the thermal unfolding, a giant rigid cluster that exists at low temperature (equivalent to a high E_cut) breaks down in smaller sub-clusters until, finally, the whole protein becomes flexible at a high temperature (Fig 3; see also S1 Video showing the loss of rigidity percolation during the thermal unfolding of the wild type). As such, nearly the entire protein structure constitutes a single giant rigid cluster initially (at 302 K; Fig 3). As the temperature increases, loops segregate first from the giant rigid cluster. Then, at 314 K, α-helix D (αD) and αE segregate to form individual small rigid clusters (Fig 3), as do αA and αF at 318 K. The giant rigid cluster at this temperature is formed by the central β-sheet region and the two helices αB and αC (Fig 3). Next, the β-sheet region becomes sequentially flexible, beginning with β4 and β8 at 320 K (Fig 3). Then, the remaining β-strands become flexible in the order β3, β7, and β5−β6, leading to a completely flexible β-sheet region at 332 K (Fig 3 The immediate next step at which αB and αC become two separate rigid clusters is identified as a phase transition point: Now most of the structure has become flexible. This transition is most prominent with respect to going from a structurally stable wild type BsLipA to an unfolded one (Figure B in S1 File). After this phase transition point, the remaining rigidity is sequentially lost, and the structure finally becomes completely flexible at 374 K (Fig 3).

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Fig 3. Average loss of structural rigidity of the wild type BsLipA during a thermal unfolding simulation.

Rigid clusters are depicted as uniformly colored bodies, with the largest rigid cluster shown in blue and smaller rigid clusters in the order of the colors green, magenta, cyan, orange, and violet. Temperatures are indicated for each depiction of a rigid cluster decomposition. At the beginning of the thermal unfolding simulation (302 K), almost the complete structure is part of the giant rigid cluster; in contrast, the structure becomes completely flexible at temperatures ≥374 K. The right views differ from the left ones by an anti-clockwise rotation of ~90° about a horizontal axis. Important secondary structure elements are labeled. Note that the unfolding pathway shown here represents an average loss of rigidity percolation calculated from a stability map (see section “Local and global rigidity indices” in S1 File) averaged over all unfolding trajectories obtained for the ensemble of 2000 network topologies. Hence, the temperature at the phase transition point identified that way (Figure B in S1 File) cannot be compared to the average phase transition temperature, which is obtained from 2000 individual T_p values and used for predicting the thermodynamic thermostability of BsLipA variants (see section “Prediction of thermodynamic thermostability of BsLipA variants”)

https://doi.org/10.1371/journal.pone.0130289.g003

During the thermal unfolding of BsLipA, helices segregate from the giant rigid cluster as independent small rigid clusters. This is due to two reasons: First, in the body-and-bar network representation, a helix with a minimum of seven amino acids is already rigid by itself due to constraints arising from covalent and backbone hydrogen bonds [66]. Second, with the current energy function E_HB [69], all backbone hydrogen bonds are assigned a very similar strength, irrespective of their location along a helix. Thus, a helix will persist as an independent rigid cluster during the thermal unfolding simulation until all backbone hydrogen bonds break almost simultaneously at a high temperature, which most likely represents an overstabilization of a helix [74]. Considering this behavior, the unfolding pathway identified for the wild type BsLipA is in good agreement with respect to the early segregation of α-helices with experimental findings on the unfolding of proteins with an α/β hydrolase fold [75, 76]. This indicates that side chain-mediated interactions between amino acids are well represented by the applied definitions of non-covalent constraints in the network. This is important as we want to detect effects of changes in such interactions due to mutations.

Prediction of thermodynamic thermostability of BsLipA variants based on the global index H_type2

From the thermal unfolding simulations, the temperature of the phase transition point T_p was identified as described in the section “Local and global rigidity indices” in S1 File. Note that T_p values determined that way should be considered relative values only, as stated in previous studies [12, 34, 35]. Initially, we calculated phase transition points using single network topologies generated from the input structures of wild type BsLipA and mutants of Rao et al.; however, this resulted in a very poor prediction of thermodynamic thermostability with a coefficient of determination (R²) for a linear fit between experimental T_m and predicted T_p of 0.22 (Figure C in S1 File). We anticipated that this result reflects the high sensitivity of CNA on the conformation of the input structures as also found previously [11, 56, 64, 65]. We thus resorted to averaging T_p values over an ensemble of BsLipA, applying the recently developed ENT^FNC approach. This approach generates an ensemble of network topologies from a single input structure and has been shown to yield results of rigidity analyses both at the local and global level that agree almost perfectly with those obtained from MD simulations-generated ensembles of structures [56]. However, this yielded a significant (p = 0.002) correlation between T_p and T_m with R² = 0.58 only if the two structures with the lowest (wild type) and highest (mutant 6B) T_m were considered outliers (Fig 4A; see below for an explanation regarding the outliers; note that removing the two outliers in the case of using single network topologies only marginally improved R² from 0.22 to 0.29). The mutants IX, X and XI of Reetz et al. were predicted to be slightly less thermostable than the wild type (Fig 4A). This is in line with experimental findings by Reetz et al. that suggest that these mutants are thermodynamically less stable than the wild type [49]. In summary, these results suggest that CNA coupled with the ENT^FNC approach can sense effects on the thermodynamic thermostability that arise from only a few sequence variations (pairwise sequence identity > 93%; pairwise RMSD < 0.38 Å). However, the false predictions for wild type BsLipA and mutant 6B are dissatisfying.

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Fig 4. Correlation between predicted and experimental thermostabilities (T_m values) of BsLipA variants; for the predictions, the ENT^FNC approach was used.

A: Correlation between T_p derived from the global index H_type2 and T_m values for thermodynamically thermostabilized mutants from Rao et al. Data points colored red were considered outliers (see main text for explanation) and excluded when calculating R² values and the correlation lines. B: Correlation between and T_m values for thermodynamically thermostabilized mutants from Rao et al. Data points shown as empty squares represent values for five additional wild type crystal structures (see main text for details; two of the squares closely overlap; mean over all six data points for wild type structures is shown as a small horizontal line: 315.9 ± 0.6 K). A and B: Error bars represent the standard error in the mean. T_p and values for kinetically thermostabilized mutants from Reetz et al. are marked by arrows on the corresponding ordinates.

https://doi.org/10.1371/journal.pone.0130289.g004

Difference in unfolding pathways explains outliers

Next, we investigated why the thermostabilities of the wild type and the mutant 6B were predicted falsely. Since the precision of the computations shown in Fig 4A is high (the standard error in the mean is < 0.38 K in all cases), we reasoned that the false prediction must arise from a systematic difference between the wild type and 6B versus all other mutants of Rao et al. Thus, we mutually compared all unfolding pathways of the systems as described in “Materials and Methods”. After partitioning unfolding pathways of BsLipA variants characterized on a residue basis by the percolation index p_i into 10 clusters (see Fig 1 for the p_i profiles of the 10 cluster medoids), we calculated correlation coefficients from the resulting cluster distributions for all pairs of variants (Fig 5; Tables A and B in S1 File).

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Fig 5. Pairwise correlations of cluster distributions (using 10 clusters) of unfolding pathways of wild type BsLipA and mutants from Rao et al.

The upper triangle shows pairwise correlation coefficients as dial plots where a filled portion of a pie indicates the magnitude of the correlation (r) and blue (red) color indicates a positive (negative) correlation. The lower triangle shows 68% data ellipses (depicting the bivariate mean ± 1 standard deviation) [90] and scatterplots of the respective cluster distributions (frequencies of network topologies in each of the 10 clusters) of the two BsLipA variants as red lines smoothed by locally-weighted polynomial regression [91]. Data points and axes for the plots in the lower triangle are omitted for clarity. The blow-up in the right bottom corner depicts exemplarily for the TM versus WT case axes, axes labels, data points, the smoothing line obtained by locally-weighted polynomial regression, and the 68% data ellipsis. The figure was generated using the “corrgram” package [92] of the R program (http://www.r-project.org).

https://doi.org/10.1371/journal.pone.0130289.g005

These results revealed that the wild type enzyme shows an unfolding pathway distribution very distinct from other BsLipA variants from Rao et al. with correlation coefficients r ranging from −0.69 to 0.54 (Fig 5, Table A in S1 File). The average r value for the wild type against all other variants from Rao et al. is −0.06 ± 0.14 (mean ± SEM), which is lower than that of the other variants (≥°0.16 except for the outlier 6B) (Table A in S1 File). The second outlier, mutant 6B, has an average r value of 0.12 ± 0.16 when comparing its unfolding pathway distribution to those of other variants from Rao et al. This average r value is lower than the corresponding average r values of all other mutants from Rao et al. (Table A in S1 File). The thermal unfolding pathway of 6B is shown in Figure D in S1 File. While the overall unfolding pathway of 6B is comparable with that of the wild type BsLipA in that the helices segregate from the giant rigid cluster as individual rigid clusters in the early phase of unfolding, they do so in a different order (αD, αA−αF, αE, αB−αC; Figure D in S1 File). A probability density function (PDF) of r values of unfolding pathway distributions of the two outliers wild type and mutant 6B with all other variants shows a bimodal distribution and is shifted towards lower r values compared to the PDF of the r values of other mutants from Rao et al. Furthermore, about half of this distribution is related to negative r values (Figure E in S1 File). In all, this suggests that the two outliers have unfolding pathways different from all other mutants from Rao et al. for which the prediction of thermodynamic thermostability was successful. Finally, we note that the unfolding pathway distributions of the wild type and the three mutants from Reetz et al. are highly similar to each other (r > 0.79; p ≤ 0.001; Table B in S1 File).

These findings have important implications: First, the results strongly suggest that the misprediction of the thermostabilities of the wild type and mutant 6B arises from them showing different unfolding pathways from all of the remaining mutants from Rao et al.. Apparently, the present approach of identifying phase transition points by monitoring the global index H_type2 (see section “Local and global rigidity indices” in S1 File) is too sensitive with respect to the details of such pathways. Consequently, alternative methods should be explored (see section “Median stability of rigid contacts between residue neighbors as a new measure for predicting thermodynamic thermostability”). Second, the results suggest that the history of the generation of the BsLipA structures may play a role for the observed differences in the unfolding pathways: generally, the most similar unfolding pathways (Tables A and B in S1 File) (and then the most coherent T_p predictions) are found for those variants that originate from a common structural “ancestor” (Table 1). Third, the results propose to apply the similarity/dissimilarity of unfolding pathway distributions as a measure to judge the reliability of thermostability predictions in future studies: the lower the similarity for two variants, the less confident should one be that relative thermostability predictions are correct. Finally, we cannot exclude at the present stage that thermostabilizing mutations lead to an unfolding pathway that is different from the one of the wild type. Considering that intrinsic and extrinsic modifications in other systems that led to thermostabilization have been shown to influence not just the folded state but the entire (un)folding free energy landscape [77, 78], this possibility also exists for BsLipA mutants [45, 47].

Median stability of rigid contacts between residue neighbors as a new measure for predicting thermodynamic thermostability

The above findings called for predicting the thermodynamic thermostability in a way that is less sensitive to the details of the unfolding pathway than the present approach relying on the global index H_type2. The sensitivity arises here from the need to accurately identify the phase transition point from the percolation behavior of the constraint network as the most pronounced jump in H_type2 during the unfolding (Figure B in S1 File). As shown previously, however, the percolation behavior of networks from protein structures is complex [13] (in contrast to that of network glasses [54, 79]), reflecting that a protein structure is hierarchical and composed of modules. As a consequence, often more than one pronounced jump in H_type2 is observed, which then makes it difficult to assign a phase transition point (Figure B in S1 File).

As an alternative, we set out to characterize thermodynamic thermostability at the local level [55], i.e., by monitoring residue pair-wise descriptors of local stability within a protein structure as a function of the temperature. The most comprehensive information in that direction is provided by stability maps rc_ij [12], which depict when a rigid contact rc between two residues i and j ceases to exist along a thermal unfolding trajectory. As such, rc_ij contains information cumulated over all states σ of a network along the trajectory as to which parts of the network are (locally) mechanically stable at a given state σ, and which are not [12, 55]. Of note, this stability information is not only available in a qualitative manner (i.e., in terms of local rigidity and flexibility) but also quantitatively in that each rc_ij has associated with it the energy E_cut at which this rigid contact is lost. Thus, ∑_{i,j > i} rc_ij represents the chemical potential energy due to non-covalent bonding, obtained from the coarse-grained, residue-wise network representation of the underlying protein structure. With respect to a reference state where no non-covalent interactions are present anymore (i.e., an unfolded state), ∑_{i,j > i} rc_ij can be considered an unfolding energy then. Three modifications were applied to ∑_{i,j > i} rc_ij here for technical reasons. I) In order to stress the locality of interactions within a protein, which will later aid in understanding how structural differences relate to thermostability differences (see section “Influence of mutations on local structural rigidity”), we focused on the stability of rigid contacts rc_ij,neighbor between structurally close residues only (i.e., those residues where at least one pair of respective atoms is within 5 Å distance). II) To suppress the influence of extreme values in the double summation on the outcome of the unfolding energy, we used the median stability of rigid contacts instead. Such extreme values can occur in regions that are highly stabilized by interactions to hydrophobic atoms [56]. III) Applying the ENT^FNC approach, were averaged over ensembles of 2000 constraint networks, which has been shown to significantly improve the robustness of rigidity analyses [56]. The values are given in Table 1. In addition, Table 1 and Fig 4B show these values after converting them to a temperature scale via Eq 1

A significant and fair linear correlation of with T_m values of the thermodynamically stable mutants from Rao et al. is obtained (R² = 0.46, p = 0.004; Fig 4B). No outlier is observed now, indicating that our definition of an average local stability correctly reflects differences in the thermodynamic thermostability. This finding substantiates our above interpretation of as an approximation to the unfolding energy, because under the condition of a temperature-independent heat capacity the unfolding energy is linearly correlated to the melting temperature, with the heat capacity as the scaling factor [80]. The slope of the correlation line (0.26) in Fig 4B deviates from unity. This indicates that the linear relationship in Eq 1 used for converting to a temperature scale, which was derived for H_type2-based thermostability prediction [12, 13], may need to be reparameterized for application with . In this case, as the heat capacity has been shown to scale linearly with the number of residues for small globular proteins [80, 81], a normalization with respect to protein size needs to be applied. When only considering the six X-ray structures in the dataset of Rao et al., a good correlation of with T_m values of R² = 0.87 (p = 0.007) is found (Table 1). In contrast, a weaker correlation (R² = 0.33, p = 0.07) is obtained for the eight variants that were modeled using SCWRL (). This suggests an influence of the quality of the input structures on the prediction of thermodynamic thermostability. Finally, as before, the mutants from Reetz et al. are found to have a lower thermodynamic thermostability than the wild type, in very good agreement with experimental findings (see above and Table 1) [49].

The -based measure is apparently less sensitive to differences in the unfolding pathway because the wild type and mutant 6B are now much better ranked. However, comparing the prediction of thermostabilities by and H_type2, the latter yields a better correlation with T_m for mutants with similar unfolding pathways. From an application point of view, we thus recommend using H_type2-derived T_p values for comparing thermostabilities of variants of a protein unless the underlying unfolding pathways are dissimilar; in that case, we recommend using .

When applied to hen egg white lysozyme the ENT^FNC approach has been shown to significantly improve the robustness of rigidity analyses with respect to the conformation of the input structures [56]. To probe if this also holds for BsLipA investigated here, we computed using the ENT^FNC approach for five additional crystal structures of wild type BsLipA (see section “Materials and methods”). The standard error of the mean in over all six wild type BsLipA structures is 0.57 K (Fig 4B) including PDB ID 1ISP discussed so far. This error is likely within the experimental uncertainty, confirming our previous results of robust rigidity analyses with ENT^FNC [56]. Still, if the average over all six crystal structures (315.9 K; see horizontal line in Fig 4B;Table 1) is considered for the versus T_m correlation, the quality of the correlation improves considerably to R² = 0.55 (p = 0.001) compared to if only of PDB ID 1ISP is used (see above). This indicates that the use of multiple input structures in connection with the ENT^FNC approach further increases the accuracy of thermostability predictions.

Influence of mutations on local structural rigidity

Considering that the average local stability defined above correctly reflects differences in the (macroscopic) thermodynamic thermostability, we analyzed on a residue basis how changes in thermostability relate to changes in local structural stability (rigidity). First, we compared stability maps of variants from Rao et al. with distinct thermostabilities to analyze the effect of mutations on the local rigidity. In particular, we compared the wild type to a more thermostable variant 1-14F5 and the most thermostable variant 6B. We averaged stability maps of the six wild type structures (see above and Fig 4B) and used this average for comparison against the thermostable variants of BsLipA. Difference stability maps for 1-14F5/wild type (Fig 6A) and 6B/wild type (Fig 6B) pairs demonstrate that mutations in general improve the strength of rigid contacts to and in between neighboring residues of the mutations (lower triangles in Fig 6A and 6B) but also in between residue pairs not in contact distance (upper triangles in Fig 6A and 6B). This effect is more pronounced for 6B/wild type than 1-14F5/wild type.

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Fig 6. Differences in the stability of rigid contacts between wild type and mutants of BsLipA.

Maps depict differences between stability maps of the respective mutants and an average stability map of the six wild type structures (see the main text for explanation) for A: mutant 1-14F5, B: mutant 6B, and C: mutant X. A red (blue) color indicates that a rigid contact in the mutant is more (less) stable than in the wild type (see color scale at the bottom). The upper triangles show differences in the stability values for all residue pairs; the lower triangles show differences in the stability values only for residue pairs that are within 5 Å of each other, with values for all other residue pairs colored gray. Secondary structure elements as computed by the DSSP program [88, 89] are indicated on both abscissa and ordinate and are labeled: α-helix (red rectangle), β-strands (green rectangle), loop (black line). Arrows represent the mutation positions with respect to the wild type sequence: Common mutations in 1-14F5 (A) and 6B (B) are shown in magenta, unique mutations in 6B (B) are shown in green, and mutations in X (C) are shown in orange. The differences in the stability of rigid contacts for residue neighbors is also displayed on the structures of the mutants by sticks connecting C_α atoms of residue pairs colored according to the color scale of the maps for D: 1-14F5, E: 6B, and F: X. Only those contacts that are stabilized by ≥ 4 K or destabilized by ≥ 3 K are shown for clarity; for the same reason, contacts between two residues of the same secondary structure element are not shown. Mutated residues are shown as sticks and a sphere at their C_α atoms (D, E, and F) in the same color used for arrows (A, B, and C).

https://doi.org/10.1371/journal.pone.0130289.g006

In more detail, the four mutations (indicated by arrows in Fig 6A and shown in Fig 6D) on 1-14F5 stabilize contacts of αD with its neighboring helix αC and contacts of αA with αF (Fig 6A and 6D). More importantly, the contacts of helices αA and αF with their neighboring β-strands in the central β-sheet region are stabilized, which delays the early loss of these helices observed during the thermal unfolding of the wild type (Fig 3). Similarly, the contacts between αB and the central β-sheet region also become stronger, which delays the decay of structural stability of the β-sheet during thermal unfolding. On average, contacts between all residue neighbors are ~−0.1 kcal mol⁻¹ or ~2 K more stable in 1-14F5 than in the wild type.

Residues mutated in 6B (indicated by arrows in Fig 6B and shown in Fig 6E) include the mutations already found in 1-14F5. This explains a strengthening of inter-helical contacts and of the contacts between α helices and the central β-sheet region as discussed already for 1-14F5 (Fig 6D and 6E). However, the additional mutations in 6B stabilize contacts between other α-helices (αD and αE) and the central β-sheet region and further reinforce those between αA or αF and the β-sheet. On average, contacts between all residue neighbors are ~−0.4 kcal mol⁻¹ or ~8 K more stable in 6B than in the wild type (Fig 6E).

Taken together, contacts between peripheral helices and the central β-sheet region are stronger in 6B than in 1-14F5. This delays the loss of α-helices during thermal unfolding (Fig 3) to a larger extent in 6B than in 1-14F5, explaining at a structural level why 6B is more stable than 1-14F5. Remarkably, many of these stabilizations must arise from the long-range aspect of rigidity percolation [52, 64, 82, 83], because almost all mutations in 6B are on the surface, i.e., far from the central β-sheet region. In contrast, inter-helical contacts of the αB/αC helix pair become weaker in the mutants than in wild type (Fig 6D and 6E) indicating that the strengthened stability between these helices and the central β-sheet region is sufficient to keep the structure folded. At last, for all other thermodynamically more thermostable mutants, a similar profile of changes in contact stability between various secondary structure elements was observed (Figure F in S1 File). Not unexpected, the increase in contact stability compared to wild type (Figure F in S1 File) was generally the more pronounced the higher the thermodynamic thermostability is of the mutant (Table 1).

Second, we compared the mutants from Reetz et al. to the wild type. Regarding mutant X, seven residues have been mutated (indicated by arrows in Fig 6C and shown in Fig 6G). In strict contrast to what was observed for the thermodynamically thermostabilized mutants, this mutant showed a destabilization of rigid contacts both locally and globally (Fig 6C and 6F; see also Figure G in S1 File, where a similar finding is depicted for mutants IX and XI). For mutant X, the average decrease in stability over all residue neighbors is ~0.06 kcal mol^-1 or ~1.2 K. The destabilization found on the local scale agrees with results of a lower T_p found when analyzing the mutants globally. Furthermore, the results are in line with experimental findings which suggest that the mutants are thermodynamically less stable than the wild type (Table 1) [49]. Our findings are also in good agreement with results obtained by comparative crystal structure analysis of wild type and variant X [49]: Loop region 14–21, for which lower B-factors in X than in the wild type structure were observed, shows increased contact stabilities with its neighboring residues in X (Fig 6C and 6F; Figure H in the S1 File). Likewise, regions 129–153 and 177–181, for which higher B-factors in X than in the wild type structure were observed, show decreased contact stabilities with their neighboring residues in X (Fig 6C and 6F; Figure H in the S1 File). However, region 60–70 shows increased contact stabilities in X (Fig 6C and 6F and Figure H in the S1 File) despite higher B-factors observed in the comparative crystal structure analysis. The latter may reflect increased motions of a stabilized region as a whole, taking into consideration that B-factors can report on rigid body motions of a structurally stable part [84].

Finally, it would be very satisfying from both the biochemical and structural biology point of view, if the effects of the three to twelve mutations on increased or decreased local rigidity could be immediately related to the changes in specific interactions with neighboring residues. Our above observation for the mutants from Rao et al. that many of the stability changes must arise from the long-range aspect of rigidity percolation [52, 64, 82, 83] speaks against such an endeavor. Nevertheless, we analyzed differences in the per-residue number of hydrogen bonds and hydrophobic tethers of the variants 1_14F5, 6B, and X with respect to the wild type (Figure I in S1 File). While these analyses reveal differences in the number of interactions > 1 indeed only for a small set of residues, between ~10 and ~35% of all residues show differences of 1. This finding is remarkable given the small number of mutations in the variants and the high structural similarity between crystal structures of the wild type and mutants found for backbone atoms above; apparently, these differences arise from subtle changes in the conformations of the side chains due to the mutations. This finding also suggests that an interpretation of the relationship between increased or decreased local rigidity due to changes in specific interactions of a mutated residue and a change in thermostability of a mutant may fall short of the actual complexity underlying this relationship.