Thermodynamic analysis of the GASright transmembrane motif supports energetic model of dimerization

The GASright motif, best known as the fold of the glycophorin A transmembrane dimer, is one of the most common dimerization motifs in membrane proteins, characterized by its hallmark GxxxG-like sequence motifs (GxxxG, AxxxG, GxxxS, and similar). Structurally, GASright displays a right-handed crossing angle and short inter-helical distance. Contact between the helical backbones favors the formation of networks of weak hydrogen bonds between Cα–H carbon donors and carbonyl acceptors on opposing helices (Cα–H∙∙∙O=C). To understand the factors that modulate the stability of GASright, we previously presented a computational and experimental structure-based analysis of 26 predicted dimers. We found that the contributions of van der Waals packing and Cα–H hydrogen bonding to stability, as inferred from the structural models, correlated well with relative dimerization propensities estimated experimentally with the in vivo assay TOXCAT. Here we test this model with a quantitative thermodynamic analysis. We used FRET to determine the free energy of dimerization of a representative subset of 7 of the 26 original TOXCAT dimers using FRET. To overcome the technical issue arising from limited sampling of the dimerization isotherm, we introduced a globally fitting strategy across a set of constructs comprising a wide range of stabilities. This strategy yielded precise thermodynamic data that show strikingly good agreement between the original propensities and ΔG° of association in detergent, suggesting that TOXCAT is a thermodynamically driven process. From the correlation between TOXCAT and thermodynamic stability, the predicted free energy for all the 26 GASright dimers was calculated. These energies correlate with the in silico ΔE scores of dimerization that were computed on basis of their predicted structure. These findings corroborate our original model with quantitative thermodynamic evidence, strengthening the hypothesis that van der Waals and Cα–H hydrogen bond interactions are the key modulators of GASright stability. Secondary Abstract We present a thermodynamic analysis of the dimerization of the GASright motif, a common dimerization motif in membrane proteins. Previously, we found that the stability of GASright is modulated by van der Waals packing and weak hydrogen bonds between Cα–H carbon donors and carbonyl acceptors on opposing helices. The experimental dimerization propensities were obtained with an in vivo assay. Here we assess this model quantitatively by measuring the free energy of dimerization of a subset of the original dimers. The thermodynamic data show strikingly good agreement between the original propensities and their ΔG° of association, confirming the model and strengthening the hypothesis that van der Waals and Cα–H hydrogen bond interactions are the key modulators of GASright stability.

understand the factors that modulate the stability of GASright, we previously presented a computational and experimental structure-based analysis of 26 predicted dimers. We found that the contributions of van der Waals packing and Cα-H hydrogen bonding to stability, as inferred from the structural models, correlated well with relative dimerization propensities estimated experimentally with the in vivo assay TOXCAT. Here we test this model with a quantitative thermodynamic analysis. We used FRET to determine the free energy of dimerization of a representative subset of 7 of the 26 original TOXCAT dimers using FRET.
To overcome the technical issue arising from limited sampling of the dimerization isotherm, we introduced a globally fitting strategy across a set of constructs comprising a wide range of stabilities. This strategy yielded precise thermodynamic data that show strikingly good agreement between the original propensities and ΔG° of association in detergent, suggesting that TOXCAT is a thermodynamically driven process. From the correlation between TOXCAT and thermodynamic stability, the predicted free energy for all the 26 GASright dimers was calculated. These energies correlate with the in silico ΔE scores of dimerization that were computed on basis of their predicted structure. These findings corroborate our original model with quantitative thermodynamic evidence, strengthening the hypothesis that van der Waals and Cα-H hydrogen bond interactions are the key modulators of GASright stability.

Introduction
Membrane protein oligomerization is a fundamental process in the life of a cell.
Oligomerization is especially important for bitopic proteins, i.e. the membrane proteins that contain a single transmembrane (TM) helix. The association of these TM helices can be optimized for stability in constitutive dimers, such as the case of glycophorin A (GpA) 1 . In other instances, stability is tuned appropriately to support dynamic association, which can be critical for regulating signal transduction or activation in important biological systems, such as integrins 2,3 and receptor tyrosine kinases [4][5][6] , to name a few. Understanding the interplay between the forces involved in TM helix oligomerization could support the prediction of structure and stability, the identification of potential conformational changes, and the interpretation of the effect of mutations in these system, providing insight into biological function and regulation.
In the present study, we focus on the energetics of association of an important structural dimerization motif known as GASright 7 , which is best known as the fold of the prototypical GpA dimer 1 . The GASright motif is one of the structural motifs most commonly observed in dimeric transmembrane complexes 8,9 . It is characterized by the presence of the small amino acids, Glycine, Alanine, and Serine (GAS) at the dimer interface, and a right-handed crossing angle of approximately −40° between the two helices. The small residues are separated by three amino acids and arranged on the same face of the helix to form GxxxG-like sequence motifs (GxxxG, GxxxA, SxxxG, etc.) [10][11][12] (Fig. 1). They form a flat face that allows the helical backbones to come in close contact, promoting tight packing. The contact between the backbones at the specific geometry of GASright promotes the formation of networks of weak hydrogen bonds between Cα-H carbon donors and carbonyl acceptors on opposite helices (Cα-H•••O=C) 7 .
C-H groups are typically weak hydrogen bond donors unless they are activated.
In the case of the Cα-H group, the flanking amide groups act as electronwithdrawing substitutes with respect to the Cα carbon, which, in turn, results in significant polarization of the Cα-H bond. Indeed, quantum mechanics calculations have estimated the energy of Cα-H hydrogen bonds in proteins to be approximately one-half of that of N-H donors in vacuum 13,14 . For this reason, they are likely to contribute significantly to the free energy of dimerization of GASright dimers, considering that multiple instances of these hydrogen bonds occur at the same interface in this motif 7,15 .
To test the hypothesis that Cα-H hydrogen bonds, along with van der Waals (VDW) packing, are major drivers of stability in GASright dimers, we previously used a combined computational experimental approach 16 . Specifically, we predicted the structure of a series of 26 GASright dimers using the program CATM 7 and compared the energy score that was calculated with the experimental dimerization propensities obtained with TOXCAT 17 , a genetic assay that measures oligomerization in the Escherichia coli membrane (Fig. 2a). To reduce some of the variability that is typical of a biological assay, we redesigned the constructs by "stitching" the 8 positions predicted by CATM to be at the dimer interface into a standardized 6 Fig. 2. Construct selection. a) In our previous study 16 , we used TOXCAT to study the dimerization of a set of predicted GASright dimers. TOXCAT is an in vivo reporter assay that provides propensities of dimerization relative to a known standard. In the assay, the TM domain under investigation is fused to the ToxR transcriptional activator. TM helix association results in the expression of a reporter gene in E. coli cells (CAT), which can be quantified. b) To reduce variability in TOXCAT, the eight amino acids that formed the interface as predicted by CATM were "stitched" into a standardized poly-Leu sequence. c) In our previous study, we found a correlation between the energy score predicted with the program CATM and the dimerization propensity measured with TOXCAT assay for a series of 26 standardized GASright dimers. Plot reproduced here from Anderson et al. 16 . The data suggest that two of the primary interactions that contribute to modulating the dimerization stability of these constructs are a combination of VDW interactions and weak Cα-H hydrogen bonding. The 8 constructs selected for the present study are highlighted in color. d) List of the subset of eight GASright dimers analyzed in vitro in this study to measure their ΔG° of dimerization. The subset covers a range of TOXCAT homodimerization propensities and CATM energy scores. 21 amino acids poly-Leu backbone (LLLxxLLxxLLxxLLxxLILI, where the x represents the variable interfacial amino acids) 16 . Such standardization was important to reduce the differences in protein expression level. It also allowed us to focus on the forces that play a role at the interaction interface, isolating them from other variables that could contribute to dimerization stability, such as the length of the hydrophobic region of the TM helices and the position of the crossing point in the dimer, which were features shared by all constructs (Fig.   2b). The predicted structures were partially validated by mutating the critical interfacial glycine at C1 (the position at the interface that is in the closest contact with the opposing backbone) to a large isoleucine, a mutation that is expected to completely abolish dimerization by creating steric clashes 16 .
As shown in Fig. 2c, in our previous study we identified a statistically significant relationship between experimental stability estimated with TOXCAT and the computational energy scores of the constructs (data reproduced from Anderson et al. 16 ). However, neither CATM nor TOXCAT are rigorous methods for assessing the free energy of association of TM dimers. CATM is a program that can predict the structures of known GASright dimers with high accuracy, but its energy function is relatively simple, consisting of an unweighted sum of only three terms: VDW, hydrogen bonding, and an implicit solvation function. The ΔE score it produces is obtained by calculating the differential of these three terms between static monomeric and dimeric models. TOXCAT reports relative propensities of association relative to know standards (glycophorin A and its monomeric G83I variant), but it does it indirectly, through the stimulation of the expression of a reporter gene (Fig. 2a). It indicates semiquantitatively whether a dimer is strong, moderately stable or weak but it does not provide free energies of association. The assumption is that the signal is proportional to the amount of dimer present in the membrane, and thus the response is thermodynamically driven 18 , an assumption that was yet to be fully tested. Furthermore, the assay could be affected by the variability of a living biological system and in particular by differential expression levels of the constructs. Despite these limitations, by sampling a large set of validated GASright constructs of different stabilities with these two methods, in our previous study we were able to identify features that correlated statistically with the apparent stability of the dimers 16 . The data indicated that packing and Cα-H hydrogen bonds were major factors that modulated the stability of these constructs and suggested that stability can be governed by inter-helical geometry, which in turn is enabled by the sequence 16 .
In this study, we take an important step forward towards formulating a quantitative model of the energetics of GASright, with a quantitative thermodynamic analysis of dimerization in vitro. We selected a representative subset of the original 26 GASright dimers 16 and used F rster Resonance Energy Transfer (FRET) to measure their free energy of association in ӧ detergent. We found a strong correlation exists between the free energy of association in detergent and the dimerization propensity obtained previously with TOXCAT. The data also suggest that there is a striking correspondence between the ΔΔG° of association measured in vitro and the same quantities in the membrane of E. coli. The study provides a thermodynamic foundation for the model that identifies a combination of VDW packing and weak hydrogen bonding as the primary drivers and modulators of stability of GASright dimers.  Table S1) when reacted with either Cy5-maleimide (acceptor), or Cy3-maleimide (donor) and was thus selected for the study (Fig. 3a).

Results and Discussion
From the 26 constructs from our previous TOXCAT study 16 we excluded five that contained a cysteine in their sequence to avoid possible issues with non-specific labeling.
The constructs that did not clone efficiently or expressed poorly were also eliminated. This yielded a final set of eight constructs that cover a wide range of homodimerization propensities in TOXCAT and energy scores predicted by CATM (Fig. 2d). These constructs were expressed, purified in n-decyl-β-D-maltopyranoside (DM) detergent, and labeled. assumption that the equilibrium of the dimerization reaction is governed not by the total volume but by the "hydrophobic volume", given that the TM helices are constrained to reside in the interior of a detergent micelle 33 . Protein concentration is thus expressed as protein:detergent mole fraction, which was explored over a range of approximately 1:10 -5 to 1:10 -3 to drive the reaction from monomer to dimer states. The mole fraction was corrected by subtracting the critical micellar concentration from the detergent concentration, to account for the presence of monomeric (non-micellar) detergent.

Theoretical framework of the FRET experiments and global fitting analysis
An example series of donor-acceptor fluorescence spectra at increasing protein mole fraction is illustrated in Fig. 3b. The spectra were first corrected by subtracting the contribution to emission of acceptor-only samples with the same concentration and then normalized using the isosbestic point (640 nm for our system), to correct for changes in signal that are not related to the transfer of energy between the fluorophores, such as small errors in the estimate of concentration as well as differences in labeling efficiencies 34 .
Original and normalized spectra of all constructs are shown in supplementary Fig. S1.
The were also included as controls. The gray lines correspond to the global fit of the FRET data, which yielded the dissociation constants (KD). The construct OPA1 (h) was excluded from the global fitting because it showed a larger emission range and its stoichiometric analysis indicated it forms higher-order oligomers (supplementary Fig. S3).
Three parameters were derived from the fitting. Two of them were global parameters, NFD and NFM, i.e. the normalized fluorescence intensities at the limits in which the samples are fully dimeric (at infinite concentration) or monomeric (at infinite dilution), respectively. The third parameter was the desired dissociation equilibrium constant KD, which was individually fitted to the data of each construct.
A major challenge presented by this type of FRET analysis is obtaining enough coverage of the association range (from mainly monomeric to mainly dimeric) so that all three parameters can be calculated with sufficient confidence. As illustrated in Fig. 4, the weakest dimers (MCL1, 1A32-2, and 1A31) cover primarily the monomeric region, whereas the strongest dimers (CP8B1 and the control GpA) cover most of the transition and approach a fully dimeric state but lack the monomeric baseline. Fitting the NFD and NFM parameters globally solves this problem since these baselines are expected to be similar across all constructs.
It should be noted that the treatment requires the labeling efficiencies of the donor to be consistent across samples, as well as those of the acceptor (however, the labeling efficiencies of the donor and acceptor do not need to be identical). The normalization by the isosbestic point takes care of differences if labeling differences occurs in a limited range, particularly of the labeling efficiency of the donor and the establishment of the monomeric baseline NFM. In theory, a second rescaling term should be introduced to correct for the relative variation in acceptor efficiency relative to the donor (the Cy5% / Cy3% ratio), which affect more the dimeric baseline NFD. However, we found that this second factor did not improve our global fitting results, and it thus was not possible to determine whether it is beneficial when these ratios are sufficiently close.
In practice, inconsistent labeling can often be mitigated simply by dilution. Highly labeled samples can be diluted with unlabeled protein, bringing the set of samples closer to a common value. It is also possible to combine donor and acceptor samples of the same construct in different ratios than 50:50, to increase the labeling of one species and decrease the other. For example, if the donor's efficiency is 90% and the acceptor's efficiency is 60%, mixing them in a 4:6 ratio would bring them both at a 72% labeling level. It should also be kept in mind that for good quality FRET data, ideally, all samples should be preferably >70% labeled for both donor and acceptor. Correspondingly, the two constructs that approach a fully associated state at their highest concentrations (GpA and CP8B1) are consistent with the calculated dimeric baseline (NFD = 2.35 ± 0.03). Finally, the data points of the construct that span a significant portion of the monomer/dimer transition (GpA, CP8B1, ITA2, K132L, and 1A32) fit the curves well. Only the construct OPA1 displayed a larger emission range than the others, suggesting it may form higher-order oligomeric states. This was confirmed by stoichiometry analysis by varying donor-acceptor ratio 35,36 (supplementary Fig. S3). For this reason, OPA1 was not included in the global fitting and was no longer considered in the analysis.

The selected GASright dimers cover a wide range of thermodynamic stabilities in vitro
The standard association free energy that we obtained for GpA was -6.07 ± 0.07 kcal/mol.
This value is comparable with previous thermodynamic studies of GpA in other detergents by analytical ultracentrifugation, which reported free energies of -5.7 ± 0.3 and -9.0 ± 0.1 kcal/mol in C14 betaine 27 and C8E5 26 , respectively. As expected, the GpA G83I monomeric variant is much destabilized (-2.51 ± 0.12 kcal/mol). These results demonstrate that the global fit strategy across a set of constructs of varied stability enables the determination of the dissociation constant for a series of samples that individually do not cover a sufficient range from dissociated to associated states.

Strong correspondence of relative stabilities in detergent and the E. coli membrane
After obtaining the thermodynamic stability of the constructs in detergent in vitro, we compared it with their dimerization propensities in biological membranes obtained previously with TOXCAT 16 . As shown in Fig. 5a 16 ), therefore this discrepancy may be possibly due to  S4). The coefficient is maximized at a 1.8×10 -4 mole fraction. The region in which R 2 ≥ 0.95 is shaded, corresponding to concentrations between 2×10 -5 and 6×10 -4 mole fractions. The outlier CP8B1 was not included in this linear fit. c) Linear fit of TOXCAT vs fraction of dimer at 1.8 × 10 -4 mole. The outlier CP8B1 was not included in this linear fit. d) The predicted fraction of dimer in the biological membrane of E. coli, based on the linear relationship observed between TOXCAT and the fraction of dimer, under the assumption that CAT expression is directly proportional to fraction dimer. The data suggest the constructs are mostly monomeric in TOXCAT, ranging from 2% for the weakest dimer (MCL1) to 35% for the strongest (K132L) from the conditions of maximum correspondence with the free energy data. The region that covers the predicted fraction dimer for all concentrations that produced an R 2 ≥ 0.95 is also indicated (shaded). significant differences in protein concentration of CP8B1 in the E. coli membrane compared to the other constructs.
The overall good correlation with the FRET data suggests that the response of TOXCAT is indeed governed by the thermodynamic stability of the dimers and that the expression of the CAT reporter gene is directly proportional to the amount of dimer that occurs in the membrane 16 . It also suggests that there is a close relationship between the stability of the constructs in detergent and the membrane. This relationship cannot be resolved directly from the TOXCAT data because of two unknown quantities. First, the free energies of dimerization of the constructs in the two environments are likely different 37 . In addition, the concentration of constructs in the E. coli membrane is unknown. However, if we make the assumption that the expression level in TOXCAT is relatively similar across this set of standardized poly-Leu constructs (as assessed by Western blotting 16 ), we can ask whether the relative stability (ΔΔG°) of the series of constructs is conserved between the two environments.
We asked this question by checking if there is a correlation between the amount of dimer in detergent and their TOXCAT signal, which we assume is a proxy for the amount of dimer in We found that TOXCAT and dimer fraction correlate extremely well at certain concentration regimes. The regression coefficient plotted against protein:detergent mole fraction reaches a maximum of R 2 = 0.98 at a mole fraction of 1.8×10 -4 (Fig. 5b,c). At this mole fraction, the amount of dimer ranges from approximately 2% for the weakest dimer to 35% for the strongest (Fig. 5d). It should be noted that the correlation remains very strong for a range of concentrations. As a reference, the range in which R 2 ≥ 0.95 is between 2×10 -5 and 6×10 -4 mole fraction (shaded region in Fig. 5b). We conclude that the monomer/dimer equilibria of the six constructs occur in the membrane at levels of dimerization that are close to the regime observed at 1.8×10 -4 in detergent (dashed line in Fig. 5d) and most likely falling on the left half of the binding curve (the shaded area in Fig. 5d). In this region, most constructs go from all nearly monomeric to reaching halfway in the saturation curve for the most stable. These data, therefore, are in reasonable agreement with a previous suggestion that TOXCAT constructs exist in mostly monomeric state in the membrane 18 .
These results indicate that there is a striking correspondence between relative stability (ΔΔG°) in detergent and the E. coli membrane for this set of constructs. They confirm for the hypothesis that the TOXCAT process is governed by thermodynamic stability 18 . Finally, the findings provide validation for the experimental data that was used to base the model that a combination of VDW packing and weak hydrogen bonding act as the primary drivers and modulators of stability of GASright dimers 16 .

The CATM energy score captures predicted stabilities in detergent
After examining the relationship with TOXCAT, we proceeded to compare the free energy of association in vitro with the energy score we previously calculated with the structural prediction program CATM 16 . We first compared the experimental ΔG° of association in detergent of the subset of seven GASright constructs analyzed here with their CATM energy score (Fig. 6a). The linear regression did not produce a significant correlation (R 2 =0.26 p-value=0.2446), indicating that CATM does not capture well the energetics of association of this particular subset of constructs. We then took advantage of the derived relationship between TOXCAT signal and the ΔG° of association to back-calculate predicted ΔG° values in detergent for all 26 original constructs (supplementary Table S2) and assess the correlation with their CATM energy score. In this case, we found a reasonable correlation (R 2 =0.46) with highly significant p-value = 0.0001354 (Fig. 6b). The majority of the points form a clear trend dispersed around the line, with only two data points performing as clear outliers (TNR12, ROMO1). This discrepancy between Fig. 6a and b is not unreasonable: although a clear correlation can be identified with a larger set of 26 constructs, the same correlation may not become apparent with a smaller number of data points due to the noisy relationship and the loss of statistical power.
It should be noted that in Fig. 6b there is a large difference in the scale of the two axes, with approximately 17 kcal/mol of CATM energy corresponding to just 2 kcal/mol of predicted ΔG° in DM. This difference is not straightforward to interpret physically. It appears that CATM somehow over-estimates the interactions, highlighting a need for recalibrating its energy function, which is not unexpected. The CATM function is simple, as an unweighted sum of only three terms, VDW interactions (CHARMM 22 38 ), hydrogen bonding (SCRWL 4 39 ), and implicit solvation (IMM1 40 ). The score ignores other potentially important terms, such as, for example, electrostatics or entropic contributions. With the availability of a sufficiently large data set of validated GASright TOXCAT constructs, it would be possible to derive (and rigorously test) an effective energy function to predict the relative stability of these dimers in detergent and potentially in the biological membrane. Nevertheless, it is notable that the simple energy function of CATM in its current form, appears to be a reasonable predictor of the relative stability of a large set of GASright dimers, confirming the model.

Conclusions
Understanding the physical basis of folding and association in membrane proteins remains challenging because of the technical difficulties posed by these systems, together with the complexity of a process occurring in an anisotropic and highly heterogeneous milieu 41 . Many factors have been proposed to play a role in membrane protein folding and association. Among them are lipid-specific effects (such as solvophobic exclusion, specific lipid binding, lateral pressure, and hydrophobic matching [42][43][44][45][46][47][48][49] ) and a variety of physical interactions (for example, aromatic and cation-π interactions [50][51][52]  Therefore, measuring their contribution of Cα−H hydrogen bonds association has been exceedingly difficult 65,66 . In this study, we advance these debates as they apply to the GASright motif, with a thermodynamic analysis of association of eight representative candidates from the original pool of 26 constructs we previously studied with TOXCAT 16 . Of these constructs, one formed higher oligomers whereas a second stood out as a clear outlier in the apparent relationship between ΔG° of association and TOXCAT signal. For the remaining six constructs, we found that there is a striking correspondence between their relative stability (ΔΔG°) in detergent and their predicted stability in the biological membrane of E. coli calculated from TOXCAT ( Fig. 5c). Additionally, we found that when the free energy of association of the entire original pool of 26 constructs is back-calculated from TOXCAT using this apparent relationship, a significant correlation is observed with the ΔE calculated with CATM based on their predicted structural models (Fig. 6b). Therefore, the present data provide quantitative thermodynamic validation that is consistent with our previous findings 16 and supports them.
The GASright motif is a versatile dimerization motif found both in structural proteins that require maximal stability, as well as in dynamic proteins, such as receptors, in which stability needs to be finely modulated to support their function. Combined, the present and previous studies provide insight into how sequence, structure, and energetic factors modulate this stability. The analysis of the predicted structural models identified significant trends that suggested that geometry can affect stability 16 . Specifically, shorter interhelical distances and crossing angles near −40° tend to produce the most stable dimers. In turn, these ideal geometries are favored by the presence of GxxxG motifs (as opposed to other GxxxG-like motifs, such as AxxxG or GxxxA). In particular, dimerization appears to become most effective when a Gly residue is present at the interfacial position designated as N1, forming a GxxxG with the Gly at C1 (Fig. 2b). None of these rules are stringent; it is possible for dimers containing GxxxG motifs to assume near-ideal geometry and, yet, their stability could be detuned if the shape of the residues at the dimer interface is not conducive to good packing 28 .
However, the analysis suggests that a geometric "sweet spot" exists, where weak hydrogen bonding and packing can be optimized, and which can be exploited by nature when maximal stability is required for function.  Samples were loaded on a 10DG desalting gravity column to remove the imidazole. The desalted fractions containing protein were then ultracentrifuged at 100,000 ×g for 30 min at 4°C

Experimental Procedures
to remove possible aggregates. The final protein concentration was determined via UV-Vis spectroscopy using calculated extinction coefficients (supplementary Table S3).

FRET experiments
Emission spectra measurements were collected in a Tecan infinite M1000 pro plate reader. Samples were excited at 500 nm and emission spectra were collected from 505 nm to 800 nm every 1 nm step. The acceptor-only sample emission spectra were subtracted from the FRET samples emission spectra. Then the data was normalized by dividing every data point by the isosbestic point (640 nm) (supplementary Fig. S1). Equilibrium constants of dissociation were obtained by fitting the normalized fluorescence intensity at 670 nm (NF670) as a function of the protein:detergent mole fraction (χT) using equation 1 (derived in supplementary material): Three parameters were derived from the fitting: NFD, the normalized fluorescence intensity at the limit in which the samples are fully dimeric (i.e., at infinite concentration), NFM, the normalized fluorescence baseline at the limit in which the samples are fully monomeric (at infinite dilution), and KD, the desired dissociation equilibrium constant. The NFM and NFD parameters were fit as global variables, whereas each KD, was fit individually to their respective construct's data.
The fraction of dimer ϕD as a function of protein:detergent mole fraction χT and dissociation constant KD was calculated with the following equation: The dissociation free energy was determined from KD using the following equation: where R is the gas constant, T is the temperature. Statistical analysis was performed using the R software 70 , with the aid of the tidyverse 71 and drc 72 packages.

Determination of the oligomeric state
Stoichiometry experiments were performed as previously described 35  Cy3-maleimide and unlabeled protein samples. All samples were prepared at a mole fraction of 4.8×10 -4 . The relative donor quenching Q was obtained using the following equation: where F is the experimental quenched fluorescence for a specific donor:acceptor ratio, and F0 is the experimental unquenched fluorescence for the same amount of donor. To estimate the oligomeric state, the relative donor quenching Q was plotted as a function of donor fraction PD, and fit to the following equation: to obtain the oligomeric state n. The parameter k is defined by the following equation: where   Table S2. The found relationship between TOXCAT and fraction of dimer was used to predict the standard free energy of the 26 constructs that we previously studied using TOXCAT and CATM (Predicted ΔG°). The relative free energy in the E. coli inner membrane was obtained with respect to poly-Leu GpA (ΔΔG°poly-Leu GpA).   Derivation of the equation used for global fitting of the fluorescence data to obtain the dissociation constants.
To derive the equation to fit the normalized fluorescence intensity at 670 nm (NF670) as a function of mole fraction to determine the equilibrium dissociation constant KD, we started from the following relationships: Where χM is the monomer mole fraction, χD is the dimer mole fraction, χT is the total protein mole fraction, FT is the total fluorescence, FD is the fluorescence intensity of a fully dimeric sample, FM is the fluorescence intensity of a fully monomeric sample and CP is the total protein concentration use in the sample.
Rearranging (1) for χD we get: (4) χ D = χ M 2 K D Using (4), we eliminate χD from (2): Rearranging (5) we get: Solving the quadratic equation (6) for χM we get: Rearranging (2) for χD we get: Using (8), we eliminate χD from (3): Rearranging (9) we get: Using (7), we eliminate χM from (10): At the isosbestic point (640 nm, in our case) the fluorescence of the donor and acceptor samples are equal by definition, therefore: and thus, at this wavelength, the quadratic term drops and the total fluorescence in (11) simplifies to: and substitute (15) and (16) in (14),  The expected inverse linear relationship between donor quenching and acceptor sensitization is observed. For all constructs that cover the transition from mostly monomeric to mostly oligomeric (GpA, ITA2, K132L, 1A32, CP8B1, and OPA1), consistent slopes and intercept values are observed. The slopes corresponds to the average decrease in the normalized fluorescence of the donor per unit increase in acceptor normalized fluorescence. The intercepts are the theoretical emissions for a completely monomeric (zero FRET) sample. OPA1 displayed a larger emission range when compared to the other constructs suggesting it may be forming higher order oligomers. The low R 2 for 1A32-2, and all the other weakly associating constructs (G83I, 1A31, MCL1) is expected. The data points for these nearly monomeric constructs are concentrated in a short range in the upper left corner, in the low-FRET region. As a result, the measurements do not contain sufficient spread for accurate linear regression.