Correlated mutation analyses in multiple sequence alignments have long been considered as useful tools to identify sites involved in protein-protein contacts 19, or to obtain distance information for structural prediction and fold recognition 2021. However, Horowitz et al. 22 have observed that coordinated changes in amino acids at two different positions (i.e. compensatory mutations) in an evolutionarily sampled sequence could predict non-additivity in the corresponding double mutant cycle, which is indicative for allosteric coupling. Lockless and Ranganathan 23 later expanded on this idea and suggested that covariance information in a multiple sequence alignment (MSA) could also be used to identify long-range energetic couplings (i.e. allosteric linkage) in proteins. The reasoning was as follows: if proteins have evolved in such a way that a perturbation at one site can be allosterically transmitted to a distant site, then an evolutionary perturbation imparted by a random mutation at one of these sites should affect the evolution of the other. This should result in a dependence of amino acid distributions at the relevant sites, which could be uncovered by covariance analyses in a well sampled MSA. Although this is equivalent to saying that distant sites in a protein may have been co-evolved under common functional or structural constraints, the subtle interpretation that such associations might indicate allosteric linkage between the sites has been proved to be useful 24. However, the free energy-like measure that has originally been suggested by Lockless and Ranganathan for this statistical coupling has potentially serious disadvantages: It is based on a virtual perturbation experiment that yields an asymmetric measure of coupling, and it is extremely biased by the conservation of the sites. The bias is due to the properties of binomial probabilities that play a central role in the measure (see supplemental information in the Additional file 1 for detailed discussion). In general, the bias is so extensive that the measure gauges conservation rather than statistical coupling. Other disadvantages have also been discussed by others 25. In the present analysis, we therefore used a modified measure of dependence as explained below.

We assessed the dependence of amino acid distributions by means of χ² value associated with relevant contingency tables. The tables were constructed as usual. The magnitude of χ² gauges the degree of departure from independence assumption. However, the absolute magnitude of χ² depends on the actual amino acid content of the sites, which can obviously be different for every pair of sites in a protein sequence, and therefore requires normalization in order to be comparable across different pairs of sites. We used the following normalization:

N χ i j 2 = 2 χ i j 2 ( χ i i 2 + χ j j 2 ) { i = j ⇒ N χ i j 2 = 1 i ≠ j ⇒ 0 ≤ N χ i j 2 < 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6ACF@

The normalizing factor is the average "self-coupling" of each site, which gives the upper limit of χ² for that site. For all possible pair of sites in the MSA, Nχij2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaaqaaiabd6eaobaacqaHhpWydaqhaaWcbaGaemyAaKMaemOAaOgabaGaeGOmaidaaaaa@3295@ forms a correlation-like symmetric matrix with diagonal elements equal to 1. As a further refinement we made the following correction: we calculated average Nχij2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaaqaaiabd6eaobaacqaHhpWydaqhaaWcbaGaemyAaKMaemOAaOgabaGaeGOmaidaaaaa@3295@ for each pair of sites by randomly permuting the amino acids at each site in the pair (this procedure affects only the joint distribution without changing the marginals). We considered this value as the "background" Nχij2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaaqaaiabd6eaobaacqaHhpWydaqhaaWcbaGaemyAaKMaemOAaOgabaGaeGOmaidaaaaa@3295@ for the sites i and j. For all possible pairs in the sequence, these values form a matrix of the same size as Nχ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaaqaaiabd6eaobaacqaHhpWydaahaaWcbeqaaiabikdaYaaaaaa@2FDD@. We used the difference between Nχ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaaqaaiabd6eaobaacqaHhpWydaahaaWcbeqaaiabikdaYaaaaaa@2FDD@ and the background matrix as a final measure of dependence (or statistical coupling), which can be interpreted as the extra Nχ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaaqaaiabd6eaobaacqaHhpWydaahaaWcbeqaaiabikdaYaaaaaa@2FDD@ due to the actual arrangement of amino acids in the MSA columns. Note that χ² in this procedure was not used in its usual statistical sense, but simply as a measure of normalized "distance" between two joint distributions. Obviously, physicochemical properties of amino acids were not considered in this analysis. A comparable strategy has been undertaken by Kass and Horovitz 26. Finally, possible contributions of evolutionary noise to the covariance information 27 were disregarded in the present analysis, since the results of COREX analysis (as explained below) were used as a source of independent information to judge the relevence of covariance data. The entire procedure is available upon request as a Matlab script.

For the present analysis, we used a non-redundant multiple sequence alignment consisting of 233 samples of α-subunits of heterotrimeric G protein sequences, obtained from the pfam database (please see Availability & requirements section below), from which redundancies were eliminated manually (the entire data set is available upon request). We excluded those sites for which the available number of amino acid samples was less than 90% of the size of the MSA. Highly conserved sites were also disregarded in the analysis since the covariance with or between such sites is statistically ambiguous as they possess little or no variance.

In order to identify the network of alike sites with respect to their coupling strength and pattern, we did a two-way hierarchical cluster analysis with the corrected Nχ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaaqaaiabd6eaobaacqaHhpWydaahaaWcbeqaaiabikdaYaaaaaa@2FDD@ matrix after eliminating the sites that did not fulfill the sampling criterion mentioned above. We used complete linkage function and city-block distance metrics (sum of absolute differences) as the simplest and most natural choice 28. We tested the efficiency of the entire procedure by using simulated multiple sequence alignments, in which different types of couplings with different strengths were imposed. Simulated patterns were perfectly recovered by the above procedure, whereas the original measure of Lockless and Ranganathan failed [see Additional file 1].

In thermodynamic equilibrium, the folded state of a protein comprises an ensemble of energetically close conformational states 29. Each state in this ensemble emerges with a certain probability depending on its free energy. Equivalently, this ensemble can be viewed as the "state repertoire" of a single molecule that is continuously explored by the molecule with different mean dwell-times, the latter being eventually related to the state probabilities when the molecule is observed in the time scale of thermodynamic averaging. In any case, the native state of a protein for a given set of conditions can be specified by a probability distribution defined over this conformational space 30. It can be theoretically shown that modification of this probability distribution by different perturbations made at distant sites on a protein results in energetic coupling of the perturbations when they are made simultaneously (e.g. coupling between two ligand binding processes at two distant binding sites) 3132, which constitutes the microscopic basis of two equivalent linkage theories that have been formulated differently by Wyman 33 and Weber 34. Thus, allosteric linkage between distant sites on a protein molecule can be analyzed by using such probability distributions 1735, provided that the equilibrium probabilities associated with specific conformers in the native ensemble are available.

In a series of studies, Hilser, Freire and their co-workers have developed an effective strategy to model the native ensemble of folded states of proteins 30363738. The procedure, known as the COREX algorithm, can be summarized as follows. The native ensemble of the protein is modeled as a collection of partially folded states of the protein: Each residue is considered as either folded (native-like as in the high-resolution structure) or unfolded (devoid of structure), and folding blocks of consecutive (usually 6 to 12) residues, in which residues are collectively folded or unfolded, are defined. The native ensemble is obtained by combinatorial folding and unfolding of these blocks. Each microstate in the ensemble thus generated is then assigned a Gibbs free energy by means of an empirically parameterized energy function 38. This function, which is based on solvent accessible surface area and conformational entropy, has been previously calibrated and validated for globular non-membrane proteins 37. Free energy distribution over the states (ΔG_i) gives the probability distribution of interest:

K i = e − Δ G i / R T Q = ∑ i = 1 n K i p i = K i Q . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabdUealnaaBaaaleaacqWGPbqAaeqaaOGaeyypa0Jaemyzau2aaWbaaSqabeaacqGHsislcqqHuoarcqWGhbWrdaWgaaadbaGaemyAaKgabeaaliabc+caViabdkfasjabdsfaubaaaOqaauaabeqabiaaaeaacqWGrbqucqGH9aqpdaaeWbqaaiabdUealnaaBaaaleaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHris5aaGcbaGaemiCaa3aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpjuaGdaWcaaqaaiabdUealnaaBaaabaGaemyAaKgabeaaaeaacqWGrbquaaGccqGGUaGlaaaaaaaa@4EDE@

K_i is the Boltzman weight of state i, Q is the partition function of the ensemble, and p_i is the probability of state i. The summation runs over the entire ensemble.

The probability distribution p can be used to estimate a descriptor of the residue-specific equilibrium; the residue-specific stability constant (k_f). This quantity is the ratio of total probability of the states where residue j is in folded conformation to that of the states where residue j is in unfolded conformation:

k f , j = ∑ i = index of states where j is folded p i ∑ i = index of states where j is unfolded p i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4AaS2aaSbaaSqaaiabdAgaMjabcYcaSiabdQgaQbqabaGccqGH9aqpjuaGdaWcaaqaamaaqafabaGaemiCaa3aaSbaaeaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabbMgaPjabb6gaUjabbsgaKjabbwgaLjabbIha4jabbccaGiabb+gaVjabbAgaMjabbccaGiabbohaZjabbsha0jabbggaHjabbsha0jabbwgaLjabbohaZjabbccaGiabbEha3jabbIgaOjabbwgaLjabbkhaYjabbwgaLjabbccaGiabbQgaQjabbccaGiabbMgaPjabbohaZjabbccaGiabbAgaMjabb+gaVjabbYgaSjabbsgaKjabbwgaLjabbsgaKbqabiabggHiLdaabaWaaabuaeaacqWGWbaCdaWgaaqaaiabdMgaPbqabaaabaGaemyAaKMaeyypa0JaeeyAaKMaeeOBa4MaeeizaqMaeeyzauMaeeiEaGNaeeiiaaIaee4Ba8MaeeOzayMaeeiiaaIaee4CamNaeeiDaqNaeeyyaeMaeeiDaqNaeeyzauMaee4CamNaeeiiaaIaee4DaCNaeeiAaGMaeeyzauMaeeOCaiNaeeyzauMaeeiiaaIaeeOAaOMaeeiiaaIaeeyAaKMaee4CamNaeeiiaaIaeeyDauNaeeOBa4MaeeOzayMaee4Ba8MaeeiBaWMaeeizaqMaeeyzauMaeeizaqgabeGaeyyeIuoaaaaaaa@9772@

It follows that the local folding free energy of residue j is -RTln(k_f,j). Although this quantity is designated as "local", it actually depends on all the other residues through the probability distribution, which is determined by the properties of all microstates in the ensemble to which every part of the protein contributes. Hence, in the context of a given ensemble, k_f is local in the sense that it is a true physical descriptor of the equilibrium state of a single residue. However, this does not mean that it is independent of the state of other residues in the protein. It has been shown by extensive analysis of various proteins that these residue-specific constants reproduce a number of experimental observables, suggesting that the actual ensemble is well-represented by the COREX ensemble 38.

Another interesting piece of information that can be drawn from the COREX analysis is the degree of dependence of local folding free energies among different sites of the protein, which can be interpreted as an additional sign of allosteric linkage within the molecule 17. This information can be obtained by evaluating folding correlations between different sites in a subset of the ensemble which contains the most probable states (e.g. with 0.99 probability):

r i j = cov ⁡ ( F i , F j ) var ⁡ ( F i ) var ⁡ ( F j ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOCai3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpjuaGdaWcaaqaaiGbcogaJjabc+gaVjabcAha2naabmaabaGaemOray0aaSbaaeaacqWGPbqAaeqaaiabcYcaSiabdAeagnaaBaaabaGaemOAaOgabeaaaiaawIcacaGLPaaaaeaadaGcaaqaaiGbcAha2jabcggaHjabckhaYnaabmaabaGaemOray0aaSbaaeaacqWGPbqAaeqaaaGaayjkaiaawMcaaiGbcAha2jabcggaHjabckhaYnaabmaabaGaemOray0aaSbaaeaacqWGQbGAaeqaaaGaayjkaiaawMcaaaqabaaaaOGaeiOla4caaa@4F65@

F_i and F_j indicate the folding state (i.e. F = 0 or 1) of residue i and j, respectively, across the most-probable states. An alternative strategy to reveal such linkages is to calculate the propagation of thermodynamic mutations made at a specific residue to all other residues 36. This is achieved by artificially increasing (~1 kcal/mol) the free energy of all states in which the residue of interest is folded in the representative ensemble, and then recalculating residue-specific stability constants. Such a virtual perturbation results in re-sorting of the probabilities associated with the states of the ensemble, and thus changes the value of residue-specific stability constants. Changes in the stability constants indicate how the perturbation made in a single residue propagates throughout the molecule. In principle, this information is similar to what the simple correlation analysis provides. Nevertheless, we performed both analyses, but provided only the results of the former as they indeed gave similar results.

In the present analysis we used high resolution structures of G_αi1 and G_αt, which have been determined in their GDP- or GTPγS-bound forms (PDB ID: 1BOF, 1GIA for G_αi1 and 1TAG(A and B) for G_αt). We used the web server at UTMB (please see Availability & requirements section below) for Monte-Carlo sampling of COREX states (with their calculated free energies) of G_αt. We performed high state density (i.e., exhaustive enumeration) COREX calculations only for G_αi1. In both cases a window size of 8 residues per folding blocks was used. In high-resolution correlation calculations we evaluated the top 5000 states of G_αi1.

In order to test whether an overlap between the two subsets of amino acid sites selected from the entire sequence was a result of pure coincidence, we used the following statistics (for the context of the problem, see results). The statistics is based on the probability distribution defined over the number of elements x in the intersection of two subsets selected randomly from a parent set. The null hypothesis is that the two subsets (whose sizes are given) are selected randomly. Given the size N of the parent set and the sizes of selected subsets K and M, the size of intersection x uniquely defines the partition as N = n+k+m+x, where k = K-x, m = M-x, n = N-(k+m+x). The number of possible such partitions for a particular x is q_x = N!/(n!k!m!x!), and for all allowed values of x, from 0 to min(K,M), provided that K+M ≤ N, the total number of possible partitions is Ω = Σq_x, {x = 0...min(K,M)}. Given the null hypothesis, all events in Ω are equally likely. Hence, the probability of selecting K and M randomly with a particular number of overlaps x is p_x = q_x/Ω. The statistics for the actually observed number of overlaps x₀ then reads as p{x ≥ x₀} = Σp_x, {x = x₀...min(K,M)}.

We used p{x ≥ x₀} < 0.05 as a one-tail rejection criterion for the null hypothesis that the observed overlap is coincidental (or equivalently, the observed overlap is not significantly different from the expected overlap under the null hypothesis). When the alternative hypothesis is the hypothesis of interest, the left-hand tail of the same distribution can obviously be used to test whether the observed lack of overlap is coincidental or not.

Statistical coupling analysis in MSA provides a symmetrical matrix (361 × 361) of coupling values for all possible pairs of individual sites in G protein as described in the method section (actual number of couplings is 361 × 360/2 due to the symmetry of the matrix). However, in order to acquire a general picture of statistical coupling, we first evaluated the couplings between secondary structure blocks by averaging the coupling of individual sites in a given secondary structure element. Overall, 35 such elements as helices, sheets, loops, linkers etc. are identifiable in the molecule (13 α-helices, 6 β-sheets and 16 connecting loops and linkers). Hence, this procedure yielded a 35 × 35 symmetric matrix, entries of which represent average pairwise coupling between the secondary structure blocks (net number of couplings is 35 × 34/2). A hierarchical cluster analysis of the coupling matrix identified four clusters of elements with similar patterns and extents of couplings. The coupling matrix, rearranged according to these clusters, is shown in Figure 2A. Clusters #2 and #3 in Figure 2A consist of mainly uncoupled elements, although a mutually coupled spot is apparent in cluster #2. This spot, which is relatively isolated from the rest of the molecule in terms of statistical coupling, mainly consists of very conserved sites, whose functional importance is relatively well known (i.e. α2-helix, β4-sheet, and β4-α3 linker sequence, which correspond to switches II and III, respectively). Therefore, we did not further discuss these regions in the context of the present analysis. On the other hand, clusters #1 and #4 are formed by relatively less conserved sites. The elements in #4 are strongly, and those in #1 are moderately intra-coupled. The two clusters are also coupled to each other (see Figure 2A). Secondary structure elements that form these two clusters are shown on the structure of Gαi in Figure 2A. They cover mostly those domains that are known or predicted to be involved in the interaction of Gα with its partners, namely Gβγ, nucleotide, receptor, effector, RGS and AGS proteins (switches II and III are not present in this picture for reasons explained in the Methods section). Note that the sites in cluster 1 and 4 are located in the opposite faces of the molecule, including a part of the nucleotide binding cleft (β6-α5, β5-αG). Therefore, the observation that the elements of cluster 1 are mutually coupled to those of cluster 4 implies a statistical coupling between the two faces of the molecule. Incidentally, a mechanical coupling between the two faces of the protein has also been revealed by molecular dynamics simulations 39. These results imply and confirm that, on average, all the functional domains of Gα are statistically (or allosterically) intercoupled. Although this may seem to be trivial at first sight, a global intercoupling of these domains implies a more complex and flexible picture of allosteric regulation than a sequential interaction scheme would suggest. For example, statistical coupling between receptor, effector and Gβγ binding sites raises the possibility of direct allosteric interactions between these sites which may by-pass the nucleotide exchange.

It is clear that secondary structure elements do not necessarily coincide with energetic or functional units in proteins 40. Therefore, we further refined the picture given in Figure 2A by applying a cluster analysis to the coupling matrix of individual sites. The analysis identified 12 clusters of alike coupling patterns. Figure 2B shows the rearrangement of the original matrix according to these clusters. Out of twelve, two clusters designated as c1 and c2 in Figure 2C are of particular interest. These two clusters together consist of 50 moderately or poorly conserved sites (average conservation in the MSA is 40%, ranging from 19% to 86%). Moreover, these sites are slightly, albeit significantly, coupled to a considerable portion of the molecule. The sites that form cluster c1 and c2 are shown on the structure of Gαi in Figure 2B. The pattern of coupling between secondary structural elements shown in Figure 2A is well represented by the distribution of the sites that belong to c1 and c2 (compare Figure 2A and Figure 2C). Two other highly coupled clusters in the coupling matrix are marked with yellow arrows in Figure 2C. Sites in these clusters mainly correspond to the highly inter-coupled spot in cluster #2 shown in Figure 2A (i.e. mainly those sites that form switches II and III).

In summary, cluster analysis of the coupling matrix revealed a set of individual sites that were all strongly coupled to each other and moderately coupled to the rest of the molecule in the statistical sense. The overall coupling between the two domains shown in Figure 2A is well represented by these sites. Therefore, we considered these sites as the core of allosteric interactions in Gα in the statistical sense. This core, which constitutes only about 15% of the entire molecule, is designated for brevity as the "allosteric core cluster".

We evaluated equilibrium ensembles of Gαi and Gαt in their GDP- or GTPγS-bound forms that were modeled as partially folded states by COREX algorithm. Ensemble average of fractional folding of Gαi or Gαt showed that ~4% of the residues were unfolded in equilibrium. However, GDP-bound forms of both Gαi and Gαt were slightly less folded compared to their GTPγS-bound forms (average fractional unfolding was 5% vs. 3% in Gi, 4% vs. 3% in Gt for GDP- and GTPγS-bound forms, respectively). This was reflected by increased average conformational entropy and solvent accessible (polar and apolar) surface area in the GDP-bound form of the proteins, indicating that the GDP-bound form of Gα is slightly more "flexible" than the GTPγS-bound form in the thermodynamic sense.

Evaluation of folding states of individual residues in the ensemble (residue-specific stability constants) showed that the local folding free energy was not distributed uniformly over the residues of Gαi or Gαt (Figure 3). The pattern of local free energy distribution was roughly similar in Gαi and Gαt, as one might expect from the structural similarity of the two proteins. In Figure 3, it is worth noting that some of the low stability regions coincide with switches I, II and III (indicated in the picture), and with the αB-αC loop, which has also been designated as a switch region in Gαi (switch-IV) that assumed nucleotide-dependent conformations.

It is evident from Figure 3 that the local stability of many residues changes upon "nucleotide exchange", both in Gαi and Gαt. The patterns of these nucleotide-dependent changes (ΔΔG) in Gαi and Gαt are comparable. The present analysis can predict nucleotide-dependent changes in the local stability of the regions that have already been identified as switch regions in crystallographic studies (see Figure 3 for Gαt). Additional nucleotide-dependent changes that were not apparent in the crystal structure were also provided by the present analysis. The distribution of these nucleotide-dependent changes over the entire protein can be interpreted as the allosteric propagation of a free energy perturbation induced by nucleotide exchange at the guanine nucleotide binding site of Gα throughout the protein. Figure 4 shows the map of this propagation on the structure of Gαi: Perturbation upon nucleotide exchange propagates nonuniformly to distant sites in the protein, and it covers a considerable part of the molecule (especially the GTPase domain). A qualitatively similar pattern of propagation was also observable in Gαt (not shown). In addition to the switch regions, the following regions are also affected by nucleotide exchange (Figure 4): N-terminal of αG, αG-α4 loop, almost the entire sequence of α4-β6-α5, β2-β3 hairpin, and obviously a large part of the nucleotide binding site. Among these regions, αG, αG-α4 loop, α4, α4-β6 loop and α5 have all been suggested to be involved in the formation of the binding interface between Gα and receptor 14041.

We evaluated the relationship between the residues designated by the MSA analysis as "allosteric core cluster" and the residues identified by the Corex analysis to be involved in the propagation of nucleotide-induced thermodynamic effect. We used an arbitrary cut-off of 1.5 kcal/mol (comparable with an H-bond) for the local folding free energy difference (ΔΔG) between GDP- and GTPγS-bound Gαi to construct the set of residues that were affected by nucleotide exchange. The overlap between the two sets was remarkable (ca. 65% of the allosteric core intersected with the selected set of residues for which ΔΔG>1.5 kcal/mol). However, the set of residues that changed local stability (above the cut-off) upon nucleotide exchange was quite large (about 1/3 of the entire molecule), which raised the possibility of a coincidental overlap between the two sets. We tested this possibility by using the test statistics described in the method section, which yielded a low probability of coincidence (p = 0.0086 for N = 322, K = 144, M = 45 and x = 28). Therefore, we interpret the observed overlap as significant, and suggest that the residues in the "allosteric core cluster" are very likely to be mediating allosteric propagation of energetic perturbations in Gα.

Additionally, we searched for folding correlations as another potential indicator of allosteric coupling in the high resolution statistical ensemble generated by the COREX algorithm for Gαi. For this analysis, 5000 partially folded states of GDP-Gαi or GTPγS-Gαi that possessed the highest probability of occurrence in the ensemble, and which accounted for a cumulative probability of greater than 99% were used. In this analysis nucleotide-dependent couplings in the molecule were disregarded, as the propagation of nucleotide-induced perturbation was already considered in the above analysis. In order to exclude the effect of nucleotide, only the correlations that are common to GDP- or GTPγS-bound forms of Gαi were selected. Also excluded were high correlations between those residues that have a high level of stability in the representative ensemble. These correlations are statistically ambiguous for reasons similar to the correlations of highly conserved (invariant) residues in the statistical coupling analysis of MSA (as mentioned above).

In order to filter out the correlations that were not common in GDP-Gαi and GTPγS-Gαi, we constructed a correlation matrix whose entries were obtained through element-by-element multiplication of the two correlation matrices calculated for GDP-Gαi or GTPγS-Gαi (Figure 5). In this correlation matrix, an unambiguous correlation group that corresponds to residues forming α3-β5 and α4-β6 loops was identifiable (indicated with arrows in Figure 5). The same two loops were also represented in the "allosteric core cluster" by 8 residues, and the test statistics yielded a p value equal to 0.0018 (for N = 322; total, K = 45; allosteric core, M = 19; correlation group, x = 8; overlap) indicating that the intersection between the correlation group and the "allosteric core cluster" could not be explained by pure coincidence. This again confirms the convergence of the two independent analyses, as far as the allosteric coupling is concerned. Nucleotide-independent coupling between α3-β5 and α4-β6 loops may indicate a nucleotide-independent allosteric interaction between receptor and effector on Gα, as the two loops have been implicated to be involved in effector 10 and receptor 14142 interactions respectively. Such a nucleotide-independent coupling between effector and receptor sites on Gα may explain the experimental observation that β2-adrenoceptors are able to activate adenylate cyclase through Gs independently from nucleotide-exchange 43. Using the statistical ensembles of GDP-Gαi and GTPγS-Gαi, we also performed a mutual perturbation response analysis based on thermodynamic "mutation" in the ensemble 37. This analysis also identified the same coupling group as the one we found in the correlation analysis above (data not shown).