A mechanistic stress model of protein evolution accounts for sitespecific evolutionary rates and their relationship with packing density and flexibility
 TsunTsao Huang†^{1, 2},
 María Laura del Valle Marcos†^{3},
 JennKang Hwang^{1, 2} and
 Julian Echave^{3}Email author
DOI: 10.1186/147121481478
© Huang et al.; licensee BioMed Central Ltd. 2014
Received: 16 January 2014
Accepted: 21 March 2014
Published: 9 April 2014
Abstract
Background
Protein sites evolve at different rates due to functional and biophysical constraints. It is usually considered that the main structural determinant of a site’s rate of evolution is its Relative Solvent Accessibility (RSA). However, a recent comparative study has shown that the main structural determinant is the site’s Local Packing Density (LPD). LPD is related with dynamical flexibility, which has also been shown to correlate with sequence variability. Our purpose is to investigate the mechanism that connects a site’s LPD with its rate of evolution.
Results
We consider two models: an empirical Flexibility Model and a mechanistic Stress Model. The Flexibility Model postulates a linear increase of sitespecific rate of evolution with dynamical flexibility. The Stress Model, introduced here, models mutations as random perturbations of the protein’s potential energy landscape, for which we use simple Elastic Network Models (ENMs). To account for natural selection we assume a single active conformation and use basic statistical physics to derive a linear relationship between sitespecific evolutionary rates and the local stress of the mutant’s active conformation.
We compare both models on a large and diverse dataset of enzymes. In a proteinbyprotein study we found that the Stress Model outperforms the Flexibility Model for most proteins. Pooling all proteins together we show that the Stress Model is strongly supported by the total weight of evidence. Moreover, it accounts for the observed nonlinear dependence of sequence variability on flexibility. Finally, when mutational stress is controlled for, there is very little remaining correlation between sequence variability and dynamical flexibility.
Conclusions
We developed a mechanistic Stress Model of evolution according to which the rate of evolution of a site is predicted to depend linearly on the local mutational stress of the active conformation. Such local stress is proportional to LPD, so that this model explains the relationship between LPD and evolutionary rate. Moreover, the model also accounts for the nonlinear dependence between evolutionary rate and dynamical flexibility.
Keywords
Protein evolution Sitespecific substitution rate Local packing density Elastic network model Flexibility Stress Mean square fluctuation Mean local mutational stressBackground
Due to functional and biophysical constraints, different protein sites evolve at different rates of aminoacid substitution [1–6]. The most popular structural correlate of a site’s substitution rate is its Relative Solvent Accessibility (RSA) [7–10]. In a thorough assessment of many structural properties as predictors of sitespecific rates of evolution, Franzosa and Xia showed that the only two with significant independent contributions are RSA and CN, the Contact Number, with RSA performing slightly better [9]. However, in a more recent study, Yeh et al. compared RSA with two Local Packing Density (LPD) measures, CN and the Weighted Contact Number (WCN), and found that both LPD measures correlate better than RSA with evolutionary rates [11]. Moreover, they found that once LPD is controlled for, the independent contribution of RSA is small. Thus, LPD seems to be the main structural determinant of rate of evolution at site level. The purpose of the present work is to study possible mechanisms that connect LPD to evolutionary rates.
A possible link could be dynamical flexibility. A site’s flexibility, quantified by its Mean Squared Fluctuation (MSF), is approximately proportional to 1/LPD [12]. A flexibilitybased explanation assumes that a site’s rate of evolution increases with its dynamical flexibility. Within this framework 1/LPD would be just a “proxy” of a site’s flexibility, which would be the actual determinant of its evolutionary rate. Such interpretation would seem to be supported by empirical correlation studies of sequence variability vs. MSF [13] and variability vs. 1/LPD [14, 15], and by a recent study based on a different dynamical flexibility measure [16]. Such a flexibilitybased explanation not only makes some intuitive sense, but it is attractive because it is in line with the increasing acknowledgement of the role of dynamics for protein function [17, 18]. Therefore, we postulate as our null model an explicit empirical Flexibility Model according to which a site’s rate of evolution depends linearly on its MSF.
The main drawback of the previous flexibilitybased interpretations, and the empirical Flexibility Model we set up to make their underlying assumptions explicit, is that no mechanism is proposed. To this end, here we propose a mechanistic alternative model. We model mutations as random perturbations of the parameters of the protein’s potential energy landscape and natural selection as a function of the probability that a mutant adopts a specific active conformation. Using basic statistical physics and certain simplifying assumptions, we derive that according to this model a site’s evolutionary rate will depend on the local stress introduced in the active structure by mutating it. Therefore, we shall call it the Stress Model.
We will show that the Stress Model explains both the dependence of sitespecific rates of evolution on packing density and on dynamical flexibility in terms of the local stress introduced by mutations on the protein’s active structure.
Methods
Elastic network models
Let the conformation of an Nsites protein be represented by the column vector of the 3 N Cartesian coordinates of its N C_{ α } atoms: r = (x_{1} y_{1} z_{1} x_{2} y_{2} z_{2} … x_{ N } y_{ N } z_{ N } )^{ T }. r_{ i } = (x_{ i } y_{ i } z_{ i })^{ T } is the position vector of the ith C_{ α }. The vector joining sites i and j is d_{ ij } = r_{ j }  r_{ i } with length d_{ ij } = d_{ ij }. We use r^{0} for the protein’s equilibrium conformation in which the ith site is at ${\mathit{r}}_{\mathit{i}}^{0}$.
where ${\mathit{d}}_{\mathit{ij}}^{0}$ and k_{ ij } are, respectively, the equilibrium length and force constant of spring ij. As far as we know, all models proposed so far assume that ${\mathit{d}}_{\mathit{ij}}^{0}={\mathit{d}}_{\mathit{ij}}\left({\mathit{r}}^{0}\right)={\mathit{r}}_{\mathit{j}}^{0}{\mathit{r}}_{\mathit{i}}^{0}$, i.e. that at the equilibrium conformation r^{0}, all springs are relaxed.
Fluctuations and flexibility
where ∫ … dτ stands for integration over the whole of conformational space.
An empirical flexibility model
where $\tilde{{\mathit{\omega}}_{\mathit{i}}}\phantom{\rule{0.25em}{0ex}}$ is the relative rate of substitution of the ith site. In general, for sitespecific scalar properties we will use relative values obtained by zscore normalization. For any given sitespecific property x_{ i }, we the zscore normalized values are $\tilde{{\mathit{x}}_{\mathit{i}}}=\raisebox{1ex}{$\left({\mathit{x}}_{\mathit{i}}\u3008\mathit{x}\u3009\right)$}\!\left/ \!\raisebox{1ex}{$\sqrt{\u3008{\mathit{x}}^{2}\u3009\u3008{\mathit{x}}^{2}\u3009}$}\right.$, where the averages are calculated over all sites of the same protein. The subscript P is used to note that a priori the coefficients may depend on the protein considered. We emphasize that the Flexibility Model is empirical: rather than derived from first principles, it is postulated, based on the intuitive notion that flexible sites should accommodate mutations more easily.
A mechanistic stress model
where 〈 … 〉_{mut @ i} stands for averaging over random mutations at the ith site.
This equation specifies the stress model.
Relationship of flexibility and stress with packing density
The purpose of this work is to investigate why LPD correlates with rate of evolution at site level. The previous models relate rates of evolution with MSF (Eq. 6) and MLmS (Eq. 12). Here we derive the relationship between these properties and LPD measures.
Note that Eq. (14) is an approximation while Eq. (15) is an identity.
where R_{ cut } is typically between 10 Å and 18 Å.
Note that while MSF is approximately equal to 1/LPD, MLmS is exactly equal to LPD (for relative znormalized values).
Calculation details
We used the dataset of 213 monomeric enzymes of Yeh et al. [11]. The dataset includes proteins of diverse sizes, functional, and structural classes (Additional file 1: Table S1).
We used the evolutionary rates of [11]. They were inferred from multiple alignments of homologous sequences using Rate4Site, which builds the phylogenetic tree using a neighbourjoining algorithm and estimates rates with an empirical Bayesian approach and the JTT model of sequence evolution [27, 28]. To keep in mind that we are not dealing with the (unknown) “true rates”, but with Rate4Siteinferred rates, we use the notation ${\tilde{\mathit{\omega}}}_{\mathit{i}}^{\mathit{R}4\mathit{S}}$.
From the pdb equilibrium structure of each protein we calculated the spring constants of pfANM (Eq. 16) and ANM (Eq. 17), for which we used a cutoff distance of 13 Å [11]. Given a protein and ENM model, we calculated the Hessian matrix, inverted it to obtain the variancecovariance matrix, and calculated the sitespecific flexibility values $\tilde{\mathit{MS}{\mathit{F}}_{\mathit{i}}}$ using Eq. (5) and znormalizing. Regarding stress, we obtained the relative sitespecific values $\tilde{\mathit{MLm}{\mathit{S}}_{\mathit{i}}}$ using Eq. (15) and znormalizing.
Since we always use znormalized relative values, for the sake of notational simplicity, we shall use ω^{R4S}, MSF, and MLmS to refer to znormalized values from now on.
We performed two analyses. In a proteinbyprotein analysis, we performed linear fits of ω^{R4S} with either MSF (Flexibility Model) or MLmS (Stress Model) using the lm() function of the base package of R for each protein. In a global analysis we pooled together all sites of all proteins and performed similar global fits.
To assess the goodnessoffit of a model to the data, we used the Akaike Information Criterion AIC = 2k  2 ln L, where k is the number of parameters and L is the model’s likelihood given the data. When comparing models, the AIC weight of evidence for each model is given by $\mathit{w}\left(\mathit{AIC}\right)\propto {\mathit{e}}^{\mathit{\u2012}\frac{1}{2}\mathrm{\Delta}\left(\mathit{AIC}\right)}$, where Δ(AIC) = AIC ‒ min(AIC) [29, 30].
We also calculated Pearson’s correlation coefficients between evolutionary rates and the independent variable that defines each model. When comparing two models, we calculated partial correlation coefficients of evolutionary rates with the independent variable of each model controlling that of the other.
Results and discussion
We aim to elucidate whether a site’s rate of evolution depends on flexibility or mutational stress as measured by MSF and MLmS, respectively. To address this issue, for each site of each of the 213 proteins of a dataset of monomeric enzymes, we used the Rate4Site program to estimate its relative evolutionary rate ω^{R4S}, we calculated its MSF using both the pfANM model and the ANM model using Eq. (5), and we calculated its MLmS for the pfANM and ANM models using Eq. (15). We also considered as a measure of flexibility the Bfactors of the pdb files. As described in Methods, all relative sitespecific values were zscore normalized for each protein. All these values for the 77141 sites of the 213 proteins can be found in Additional file 2: Table S2.
Stress vs. flexibility: proteinbyprotein analysis
Model comparison: proteinbyprotein analysis
Potential  Model  y  x  AIC  <w( AIC)>  N _{ prot }  <R>  <pR> 

pfANM  Stress  ω  MLmS  190508  0.97  206  0.54  0.33 
Flexibility  ω  MSF  198662  0.03  7  0.45  0.06  
ANM  Stress  ω  MLmS  194154  0.98  209  0.52  0.39 
Flexibility  ω  MSF  207258  0.02  4  0.35  0.04 
We think that it is most meaningful to compare between MLmS and MSF calculated using the same potential energy landscape (pfANM or ANM). However, the znormalized MSF values can also be obtained from the Bfactors available from the pdb files. We compared the Stress Model, both pfANMbased and ANMbased with a Bfactorbased flexibility model and the conclusions are the same (results not shown). In general Bfactor based Flexibility Models are the worst (see Additional file 3: Table S3 and Additional file 4: Table S4). This is not surprising because Bfactors usually depend very strongly on several factors including experimental conditions, method used to estimate them, crystal disorder, etc. (see [31] and references therein).
To summarize, whether using the pfANM or the ANM potentials, stress (MLmS) predicts evolutionary rates better than dynamical flexibility (MSF) for almost all proteins of the dataset and the independent contribution of MSF is very small once MLmS is controlled for.
Stress vs. flexibility: global analysis
Model comparison: global analysis
Potential  Model  y  x  AIC  w(AIC)  R  pR 

pfANM  Stress  ω  MLmS  191424  1.00  0.55  0.32 
Flexibility  ω  MSF  199645  0.00  0.47  0.04  
ANM  Stress  ω  MLmS  194589  1.00  0.52  0.40 
Flexibility  ω  MSF  207993  0.00  0.36  0.02 
Evolutionary rates vs. flexibility and stress
Even though previous sequenceflexibility studies used Pearson correlations, which, rigorously, make sense only for linear relationships, they already found nonlinear sequenceflexibility plots similar to those of Figure 1 (left panels) [14, 32, 13]. In spite of this, they either dismissed the nonlinear part [14]or interpreted it in terms of different selection regimes [13]. From Figure 1 (left panels) it is clear that the nonlinearity follows naturally from the proposed Stress Model, suggesting that evolutionary rates depend nonlinearly on MSF because they depend (approximately) linearly on MLmS, and MSF ≈ 1/MLmS, which can be derived from Eqs. (19) and (20).
To conclude this subsection, we must observe that inferred rates are larger than stressbased predictions for the slowest sites and smaller for the fastest. A reason could be that inference methods overestimate small rates and underestimate large ones [33]. However, close inspection of the rate vs. stress curves (right panels of Figure 1) indicates that despite the very good fit of the linear Stress Model, there still seems to be some remaining nonlinearity of the ω^{R4S} vs. MLmS plots. A possible reason is the weakselection approximation used to linearize the exponential in Eq. (11), however, resolving this issue is beyond the scope of the present report.
pfANM vs. ANM
To finish this section, we compare ANM with pfANM. Figure 1 shows that both pfANM and ANM result in similar qualitative dependence of rate vs. flexibility (left panels) and rate vs. stress (right panels). However, the pfANM potential (top panels) results in better fits to the inferred rates than the ANM potential (bottom panels). Accordingly, the AIC values (Table 1 and Table 2) show that the pfANMbased stress model is better than the one based on ANM. This is in agreement with the finding that WCN correlates better than CN with evolutionary rates [11].
Conclusion
We introduced a mechanistic Stress Model of protein sequence evolution. Mutations are modelled as random perturbations of the protein’s potential energy landscape, represented using Elastic Network Models. To model natural selection, we used basic statistical physics to derive the expected probability that a mutant samples a specific functional structure. From this, we deduced a linear relationship between a site’s mean evolutionary rate and the mean local mutational stress (MLmS) of the functional conformation. We compared this model with an empirical Flexibility Model that postulates that a site’s evolutionary rate is linearly dependent on its flexibility (measured by its MSF). We compared both models and found strong support for the Stress Model. Moreover, the independent contribution of flexibility is negligible once stress is controlled for.
The MLmS is proportional to Local Packing Density and, therefore, the Stress Model provides a mechanism for the connection between a site’s LPD and its evolutionary rate. Regarding the sequenceflexibility relationship, previous empirical correlation studies had already found that the sequenceflexibility relationship is nonlinear and either dismissed the nonlinear parts or attempted an interpretation in terms of different selection regimes [14, 32, 13]. We found the nonlinearity follows naturally from the Stress Model: evolutionary rates depend nonlinearly on MSF because they depend (approximately) linearly on MLmS, and MSF ≈ 1/MLmS. To summarize, the Stress Model accounts for the observed sitedependency of evolutionary rates and its relationship with packing density and flexibility.
A note of caution is in order here. For the Stress Model mutational stress was not postulated to be the determinant factor a priori but, rather, it was derived from the assumptions of the model that are essentially two (1) there is a single active conformation and (2) mutants are flexible and therefore can sample the active conformation so that they are at least partly functional. Therefore even though Stress Model was chosen to designate this mechanistic model, it should be kept in mind that it demonstrates the importance of protein flexibility.
It is worthwhile to mention some of the possible caveats and further developments of the Stress Model. First, we assume a single active conformation. In principle, it would be reasonable to assume that only changes of the activesite conformation should affect fitness. However, we note that if protein sites are strongly coupled, which is often the case, any conformational change will affect the active site conformation. For a strongly coupled elastic network forcing the active site to adopt a given conformation makes the rest of the protein move accordingly. Therefore, assuming that the whole protein conformation must be in the “active conformation” for the protein to function is not necessarily an important limitation. However, for cases where the coupling is not very strong, if the active site is known, this could be easily tackled using a modified version of the selection function that integrates away all coordinates except for those of the active site (i.e. uses marginal conformational distributions rather than the full ones in the definition of selection function).
Second, in Eq. (11) we performed a linear approximation of the exponential function. This is reasonable a priori only for weak selection, and a posteriori by the good fit of the resulting model to the data. We should note, however, this approximation can be easily removed, and the actual mean of the exponential can be calculated via simulation. Further work is needed to explore this possibility.
Third, we note that the znormalized MLmS values, on which the Stress Model is based, are identical to the znormalized LPD measures WCN (for the pfANM potential) and CN (for the ANM potential). For other potentials this need not be the case and it is for that reason that we chose to keep the notation MLmS in the present tables and figures, to make them comparable with further research based on estimating MLmS using different, perhaps better, potential energy functions.
To close, we note that the mutational part of the Stress Model accounts for observed patterns of evolutionary divergence of protein structure and dynamics [21–23]. Regarding structural divergence, unselected random mutations reproduce very well the evolutionary conservation of a “structural core” and account for the observation that structures diverge mainly within the space spanned by a few lowenergy collective normal modes [21, 22]. Regarding protein motions, unselected random mutations explain the higher conservation of the lowenergy normal modes in terms of their mutational robustness [31, 23]. In general, those studies could found no evidence of natural selection at the levels of structural or dynamical divergence. Clearly, without natural selection, all sites would evolve at the same rate, which is not the case. The Stress Model proposed here accounts rather well for the variation of rates of evolution among sites. It would be interesting to study the effect of the selection function introduced here on structural and dynamical divergence and compare the observed patterns with those that result from unselected mutations. This could advance our understanding of the effect of selection at the levels of structure and dynamics. In general, we think the Stress Model provides a possible unifying framework to study evolutionary protein divergence at the levels of sequence, structure, and dynamics.
Notes
Abbreviations
 RSA:

Relative solvent accessibility
 LPD:

Local packing density
 WCN:

Weighted contact number
 CN:

Contact number
 ENM:

Elastic network model
 ANM:

Anisotropic network model
 pfANM:

parameterfree anisotropic network model
 MSF:

Mean square fluctuation
 MLmS:

Mean local mutational stress
 AIC:

Akaike information criterion.
Declarations
Acknowledgements
This research was supported in part by Academic Summit Program of National Science Council with grant number NSC1022745B009001 and the “Center for Bioinformatics Research of Aiming for the Top University Program” of the National Chiao Tung University and Ministry of Education, Taiwan, R.O.C. JE is a researcher of CONICET.
Authors’ Affiliations
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