Contribution of k_on and k_off to the binding free energy.

Our novel approach of using the ΔΔGs of single-point alanine mutations to characterize off-rates is based on two generalizations. The first one being that, the change in binding free energy is mostly reflected through a change in the off-rate rather than the on-rate. If this is so, any prediction algorithm designed for ΔΔGs, may to some extent also be used for the prediction of Δk_off and vice-versa. For the 713 off-rate mutations used in this work, the corresponding experimental values for ΔΔG and Δlog₁₀(k_on) are also extracted (see Dataset S1), and the PCCs between them are shown in Table 3A. The correlations are calculated for single-point alanine mutations, single-point non-alanine, multi-point, and on all mutations. Namely, ΔΔG, shows a correlation of R = 0.83 with Δlog₁₀(k_off) (Scatter Plot in Figure 2A) and R = −0.6 with Δlog₁₀(k_on). More notably is that the ΔΔG of multi-point and non-alanine mutations is strongly reflected through a change in Δlog₁₀(k_off) (R = 0.96, R = 0.92 respectively). Other lines of evidence also show that the change in binding free energy is largely explained through a change in off-rate; Namely, mutagenesis studies in [44], [45] show that the increases in dissociation rate constants were the dominant cause for a decrease in binding affinity. Work on the related phenomenon of protein-DNA binding shows that 78% of the variance of log₂(k_off) is explained by the variance of information of the binding site sequence as opposed to 49% of the variance of log₂(k_on) [46]. In a somewhat similar line of reasoning, work on the enhancement of the protein-protein association rate shows that mutations that affect binding free energy, as a result of affecting the on-rate with no change in the off-rate, are found at surface-exposed sites and located at the vicinity of, but outside, the binding site - as those within the binding site are generally off-rate modulating [47]. With this in mind, for the 713 off-rate mutation dataset, only 25% of the mutants are located at the edges (Rim) or outside the binding site (Surface), hence we also expect that the larger portion of mutants in our data, to predominantly affect the off-rate, as is also confirmed by the correlations in Table 3A.

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Figure 2. Relationship of off-rate changes upon mutation with change in binding free energy and change in interface hotspot energy.

(A) The relationship between experimental values for Δlog₁₀(k_off) and ΔΔG for all the 713 mutations in the SKEMPI off-rate dataset. (B) The relationship between changes in interface hotspot energies, as predicted by RFSpot_KFC2 hotspot predictor, and change in Δlog₁₀(k_off) for all the 713 mutations in the SKEMPI off-rate dataset. Note that 50% of off-rate mutants in this dataset involve mutations to non-alanine residues and include multi-point mutants. In turn Int_HS_Energy characterizes these changes with the use of single-point alanine ΔΔGs as highlighted in Figure 1.

https://doi.org/10.1371/journal.pcbi.1003216.g002

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Table 3. Relationship between experimental ΔΔG, Δlog₁₀(k_off), Δlog₁₀(k_on) and change in interface hotspot energy (Int_HS_Energy) for 713 mutations in SKEMPI.

https://doi.org/10.1371/journal.pcbi.1003216.t003

Explaining off-rate changes using ΔΔG energies from single-point alanine mutations.

Given this link between ΔΔG and Δlog₁₀(k_off) we can use ΔΔGs as a starting point for the prediction of off-rates. The second generalization being, that we can do so using only hotspot energies from hotspot predictors. Hotspot predictors, although being ΔΔG predictors, are limited to characterizing ΔΔGs of single-point mutations to alanine, as the definition of a hotspot requires. Therefore, if one were to use the ΔΔGs of a hotspot predictor directly to estimate Δk_off, it is not able to predict the effects of multi-point mutations and non-alanine mutations (which form 49% of the 713 off-rate mutations in our dataset). The main motivation behind the hotspot descriptors designed in this work is therefore to be able to map the effects of multi-point mutations and non-alanine mutations into energies involving only single-point alanine mutations, where the latter can be predicted by off-the-shelf hotspot predictors. To assess this proposition, we make use of a representative hotspot descriptor Int_HS_Energy,(1)where the effect of a mutation on the off-rate is calculated as the change in the sum of the hotspot energies across the interface pre- and post-mutation (See Figure 1). The PCC of Int_HS_Energy with Δlog₁₀(k_off), Δlog₁₀(k_on) and ΔΔG are shown in Table 3B. Given that Int_HS_Energy is generated separately using 6 hotspot prediction algorithms, all correlations presented are the average over its 6 instances. Int_HS_Energy provides a reasonable starting-point estimate of changes in both ΔΔG and Δlog₁₀(k_off), where the PCC of Int_HS_Energy with Δlog₁₀(k_off) (See Figure 2B) and ΔΔG are R = −0.51 and R = −0.53 respectively. The anti-correlation confirms that an increase in wild-type to mutant interface hotspot energies results in a lower off-rate and hence more stable complex.

The strength of correlation at R = −0.51, and those achieved by other hotspot descriptors which also use single-point alanine ΔΔGs to describe off-rate changes, is better understood in the context of other descriptors for off-rate estimation. The absolute PCCs with Δlog₁₀(k_off) for both the molecular (110 in total, including physics-based energy terms and knowledge-based potentials) and hotspot descriptors (including 6 sets of 16 hotspot descriptors as generated using the 6 hotspot predictors) are calculated (See Text S5, Table S1). At n = 713, all absolute correlations of |R|>0.1 are highly significant with p<0.001. A ranked list for both sets is superimposed and shows that hotspot descriptors can explain changes in Δlog₁₀(k_off) with markedly higher correlations than a diverse set of molecular descriptors (see Figure 3A). One should also note that, this is only an assessment of the raw predictive power of the descriptors. Once such a hotspot descriptors are fed into machine learning models and trained on off-rate data, their predictive power can be combined synergistically with that of others to achieve correlations as high as R = 0.79 with Δlog₁₀(k_off), as is shown in subsequent sections.

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Figure 3. Hotspot and molecular descriptors for estimating change in off-rate.

The hotspot descriptors designed in this work are benchmarked against a set of 110 molecular descriptors; both in their ability to estimate Δlog₁₀(k_off) and in their ability to detect stabilizing mutations of Δlog₁₀(k_off) <−1. The performance measures shown here enable us to assess the raw predictive power of the descriptors independent of any learning models. Green and black bars highlight descriptors from the hotspot and molecular descriptor sets respectively. (A) Comparison of the distribution of the absolute PCC values for the hotspot descriptors designed in this work against that for the molecular descriptors. The related list of descriptor names and their respective PCCs is found in Text S5. (B) Top 10 hotspot descriptors and top 10 molecular descriptor according to absolute PCC with experimental Δlog₁₀(k_off). (C) Mann Whitney U-Test rankings for all descriptors where values are ranked according to −log₁₀(pval) and represent the discrimination ability of the descriptors for the detection of stabilizing mutants (Δlog₁₀(k_off) <−1) from neutral to destabilizing mutants (Δlog₁₀(k_off) >0) (Referred to as CDS1). This dataset contains 31 stabilizing mutants and 503 neutral to destabilizing mutants. (D) Matthew's Correlation Coefficient (MCC) rankings for all descriptors on same dataset. (E) and (F) are identical to (C) and (D) except that results are for off-rates that satisfy |Δlog₁₀(k_off)| >1. This dataset contains 31 stabilizing mutants and 213 destabilizing mutants (referred to as CDS2).

https://doi.org/10.1371/journal.pcbi.1003216.g003

Accounting for experimental conditions.

It must be noted that, as with the ΔΔG values used to parameterize hotspot prediction algorithms, the k_off values in the SKEMPI data set are taken from different sets of experiments and were thus measured in a range of experimental conditions. Therefore, we performed an assessment of how severely variations in experimental temperature, ionic strength and pH can introduce noise into log₁₀(k_off) and Δlog₁₀(k_off). Firstly, 635 of the 713 values come from experiments reported to be performed in the 295–298K range, and 72 values either did not have their temperature reported, or were reported as ‘room temperature’ or ‘standard conditions’, corresponding to the 293–298K range [41]. The remaining six experiments were performed at 323K. Thus, only 0.8% of the data lies outside of a 5K temperature range. Although not reported in the SKEMPI database, most of the rate constants were determined using surface plasmon resonance or stopped-flow fluorescence in a relatively narrow range of standard buffer conditions. Further, ionic strength and pH predominantly affect the rate of association rather than the rate of dissociation; electrostatic shielding and changes in protonation state influence the long-range forces which drive protein association, rather than the short-range forces which keep the buried surfaces of the binding partners together. For instance, in the M3-XCL1 complex, in which ionic strengths in the 0.2 to 1.5 M NaCl range were investigated, the rate of association varied by over 70-fold, while the rate of dissociation varied by less than 3 fold (Figure 2C and Table III of [48]). Similarly, in a study of a VEGF-antibody interaction, varying pH in the 6.5–8.5 range resulted in around 30% variation in dissociation rate, while varying the ionic strength in the 10–1000 mM range produced a two-fold change in k_off [49]. Even assuming a large three-fold standard error in k_off, this would result in a standard error of 3/ln10≈1.3 in log k_off [49]. Lastly and most importantly, we make the assumption that though reference states may change across experimental methods and studies, within a given experiment the reference state is constant for the experimental determination of the wild-type and its mutants, which tend to be generated within the same experimental work. Given that we train on values for Δlog₁₀(k_off) = log₁₀(k_off)^Mut - log₁₀(k_off)^WT, any systematic variations associated with experimental conditions are eliminated, and thus we believe that this issue is less prominent for mutation prediction as it is for wild-type. Given the above assessment, we believe that the noise introduced by merging data from the different experiments that make up the SKEMPI data set is significantly less than the variation in Δlog₁₀(k_off) values which we are investigating and span a range of 15.3.

Two-step complex dissociation.

Complex association/dissociation of two proteins A and B can be described using a two-step reaction, where an encounter complex (AB*) is formed before the final complex (AB) [50], [51],(2)

Both stable and unstable encounter complexes have been suggested [50] and rate limiting steps may vary depending on the complex in question [52]. In turn, experimental characterization of these encounter states remains a major challenge [50]. In the two-step model,(3)k₋₁ represents the movement of proteins against an electrostatic field [51]. However, from experiments on a wide range of associating proteins, ionic concentration and hence long-range electrostatic forces have been shown to have a minimal effect on the off-rate [48], [49], [53], which suggests that in such cases, k₋₁ is not the rate-limiting step during complex dissociation. Here, k₋₁>>k₂ and hence k_off =k₋₂. Therefore, for such cases, the breakdown or stability of the final complex (AB) is the rate limiting step in complex dissociation [51]. Under this assumption, which is the one taken in this work, the encounter complex (AB*) need not be taken into consideration.

Hotspot and molecular descriptors and fine-grained detection of stabilizing mutations.

The top 10 ranking descriptors according to PCC with Δlog₁₀(k_off) from both the hotspot and molecular descriptor sets are superimposed and are presented in Figure 3B. The highest ranked descriptors all relate to energetic changes in hotspots suggesting that changes in hotspot counts is not sufficient to characterize changes in off-rate. The most prominent being; the change in hotspot energies at the core region (CoreHSEnergy) and the change in the total hotspot energies at the interface (Int_HS_Energy). Given that these descriptors show up when generated using different hotspot prediction methods, indicates that they are insensitive to prediction biases of the hotspot predictors generating them. Further analysis on the sensitivity of the hotspot descriptors generated by each hotspot predictor is presented in Text S2. Other descriptors which show high PCCs with Δlog₁₀(k_off), include the change in total hotspot energy at the interface on accounting for positive hotregion cooperativity (HSEner_PosCoop_RFSpot R = −0.57, see Figure 4C) and the change in hotspot energies in the support region (SuppHSEnergy_KFC2a R = −0.62, see Figure 4A). Apart from the DARS atomic potential (AP_MPS [54], see Figure 4B) designed for protein-protein docking with |R| = 0.46, the top 10 molecular descriptors (Figure 3B (black)) are dominated by coarse-grain statistical potentials.

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Figure 4. Hotspot and molecular descriptor scatter plots.

The relationship between experimental values for Δlog₁₀(k_off) and (A) hotspot descriptors showing highest correlation with Δlog₁₀(k_off) (SuppHSEnergy_KFC2a - changes in hotspot energies in the support region as predicted by KFC2a [30]), (B) molecular descriptor showing highest correlation with Δlog₁₀(k_off) (AP_MPS - the DARS atomic potential [54]), (C) top performing hotspot descriptor for the detection of stabilizing mutants (HSEner_PosCoop_RFSpot – changes in hotspot energies on accounting for positive cooperativity in hotregions) and (D) top performing molecular descriptor for the detection of stabilizing mutants (CP_TB – coarse grained protein-protein docking potential).

https://doi.org/10.1371/journal.pcbi.1003216.g004

Hotspot and molecular descriptors and coarse-grained detection of stabilizing mutations.

Understanding and predicting the consequences of mutations on protein-protein interactions is a precursor to two important endeavors in biology. At one end of the spectrum, destabilizing mutations at protein-protein interfaces are a hallmark of many cancers and other complex diseases [12]. At the other end, the identification of stabilizing mutations is central to computational protein design strategies [8]–[10], [40]. To assess the discriminatory power of the hotspot and molecular descriptors, the dataset is partitioned into (Δlog₁₀(k_off)<−1), representing the stabilizing portion of the dataset, and (Δlog₁₀(k_off)>0), representing the neutral to destabilizing portion of the dataset (referred to as CDS1 –Classification Dataset 1). Another dataset which removes the neutrals, as detailed in Material and Methods, is also used (referred to as CDS2). For an unbiased assessment of descriptor discrimination ability, a number of discrimination performance measures are calculated; the Mann Whitney U-Test (Figure 3C, E and Table S8 in Text S5), the Matthew's Correlation Coefficient (MCC) (Figure 3D, F), the Area Under the Curve (AUC) (Table S9 in Text S5) together with a number of classification performance measures (Table S4, Table S5) as described in Materials and Methods.

Similar to the correlations with Δlog₁₀(k_off) (Figure 3 A, B), the changes in hotspot descriptors show better discrimination abilities than changes in molecular descriptors (Figure 3C–F). This confirms that stabilizing mutations of multi-point and non-alanine nature, may be also be detected using simply the energies of single-point alanine mutations. Therefore, the more destabilizing are single-point mutations to alanine on the mutated interface, compared to the wild-type interface, the more stable is the interaction as a result of the mutation in question. Scatter plots of representative hotspot and molecular descriptors (HSEnerPosCoop_RFspot and CP_TB [55], see Figure 4C, D) which do well on both CDS1 and CDS2 highlight a tendency to underestimate stabilizing mutations. For both CDS1 and CDS2, the positive cooperativity descriptors HSEner_PosCoop/HS_PosCoop dominate the ranked lists (Figure 3C–F) and RimHSEnergy/RimHS for CDS1 (Figure 3D). For example RimHSEnergy_KFC2a achieves a TPR/FPR/MCC of (0.52/0.09/0.51) on CDS1 where neutrals are present. In turn, HSEner_PosCoop_RFSpot achieves a TPR/FPR/MCC of 0.58/0.05/0.62 for the detection of stabilizing mutants on CDS2. Given that HSEner_PosCoop_RFSpot supersedes Int_HS_Energy (additivity within hotregions assumption) and HSEner_NegCoop (negative cooperativity within hotregions assumption) suggests that applying the general assumption of positive cooperativity between hotspots within a hotregion, and accounting for it, provides higher detection rates of stabilizing mutations (i.e. Δlog₁₀(k_off)<−1). It should be noted however, that out of the three hotspot predictors which generate the most discriminatory hotspot descriptors (i.e. RFSpot, RFSpot_KFC2 and KFC2a), the positive cooperativity descriptors which show high discrimination abilities, are limited to those generated by RFSpot and RFSpot_KFC2. The relationship of Δlog₁₀(k_off) and cooperative effects within hotregions is addressed more specifically in the subsequent sections (see Effects of hotregion size, count and cooperativity on the off-rate).

Prediction patterns and data region analysis.

To gain a better understanding of the stronger and weaker regions of correlation in the off-rate dataset, and how dependent this correlation is on the off-rate predictor in question, the predictions of each off-rate predictor are also assessed at different regions of the dataset referred to as data regions (see Materials and Methods) and presented in Figure 5B. All off-rate predictors obtain good correlation on core mutations, less so for support region mutations, and the weakest correlations are found on rim region mutations. The addition of molecular descriptors to the models, as presented in the lower half of the matrix, increases the accuracy of the predictors both at the core and support regions, though rim regions are still inadequately characterized. The hotspot descriptor predictors are better at capturing effects of mutants on Small-Interface-Area (SIA) than Large-Interface-Area (LIA) complexes. This discrepancy is alleviated with the addition of molecular descriptors to the models, where mutations in LIA complexes are characterized with similar correlations to SIA complexes. Single-point mutations to alanine are generally better characterized than single-point mutations to non-alanine, and the addition of molecular descriptors to the model again reduces this discrepancy. RFHotpoint2_Off-Rate, KFC2a_Off-Rate and KFC2b_Off-Rate models show weak correlation for mutations to polar or charged residues, and better able to characterize mutations to hydrophobic residues. This discrepancy is not however observed in RFSpot_Off-Rate and RFSpot_KFC2_Off-Rate models. The low correlations for mutations to polar or charged residues for Hotpoint2_Off-Rate, KFC2a_Off-Rate and KFC2b_Off-Rate are alleviated once the molecular descriptors are added to the models; and the highest correlations on these residue types are achieved by RFSpot+Mol_Off-Rate and RFSpot_KFC2+Mol_Off-Rate. In line with the PCCs shown in Table 3B, multi-point mutations are notably better characterized than single-point mutations, where in the former; correlations as high as R = 0.9 with Δlog₁₀(k_off) are achieved with certain models. This suggests that the subtleties of single-point mutations are harder to characterize than the collective effort of multi-point mutations. Note that, though theoretically, multi-point mutations have the potential to cause off-rate changes of larger magnitudes, this is not so in the present dataset, where the mean and standard deviation of|Δlog₁₀(k_off)| for multi-point mutations is 0.96 and 1.4 compared to 1.17 and 1.48 for SP mutations. Therefore, we cannot conclude that the reason for better prediction of multi-point mutations is related to being able to predict extreme changes in Δlog₁₀(k_off) better than subtle changes in Δlog₁₀(k_off). Results for more stringent forms of cross-validation and model predictions on data regions, which collect mutations on related complexes and interfaces together, are also presented in Text S3. Here it is observed that mutations on unseen complexes are markedly harder to predict, though on controlling for conformational changes, this difficulty is alleviated. Using specialized feature-selection models trained only on mutations from related complexes, and analyzing their descriptors shows that these are highly specific to certain classes of complexes. Therefore, such descriptors cannot generalize to unseen and unrelated complexes.

Prediction of stabilizing mutations.

Similar to the regression RF models, several RF classification models are also built for the detection of stabilizing (i.e. Δlog₁₀(k_off)<−1) mutants and results are presented for both Classifier Dataset 1 (CDS1) and Classifier Dataset 2 (CDS2). The MCC for the 20-Fold CV test predictions are presented in Figure 5C (Blue: CDS1, Red: CDS2) and related classifier performance measures are presented in Tables 4 A–B (See Table S2 and Table S3 for list of predictions for each model). As expected our ability to detect stabilizing mutants is diminished when neutral mutations are present. The highest MCC obtained for CDS1 is achieved by RFSpot_KFC2_Off-RateC (MCC = 0.60, TPR = 0.45, FPR = 0.01) and RFSpot+Mol_Off-RateC for CDS2 (MCC = 0.82, TPR = 0.84, FPR = 0.02). Figure 7A shows the list of 31 stabilizing mutants (Δlog₁₀(k_off)<−1) sorted according to the number of classifiers which detect the given mutation as stabilizing. Of particular interest are those stabilizing mutations which go undetected and therefore only data from CDS2 is used, as all mutations undetected in CDS2 were also undetected in CDS1, though not the contrary. Two stabilizing mutants go undetected by all the predictors, namely the double alanine mutant VA216A-YB50A on 1JTG and the 4-point mutant CB161L-CB299F-KB287C-KB294C on 1MQ8. Stabilizing mutations which are the harder to predict, as shown by the inability of a number of different off-rate classifiers to detect them (Figure 7B), generally involve a mutation to an alanine residue. Complex stabilizing alanine mutations have been previously reported [56], [57], and the likely interpretation is that several side-chains may sometimes hinder binding. For example, several alanine-shaving experiments have show an increase in binding affinity between protein binding partners, as found for an octa-alanine mutant of the hGH receptor which binds its hGHbp ligand 50-fold times tighter than the wild-type [56], [57].

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Figure 7. Detection of rare complex stabilizing mutations using off-rate classification models.

(A) Ranked list of 31 stabilizing mutations (Δlog₁₀(k_off) <−1) in SKEMPI off-rate dataset. The list is ranked according to the number of off-rate prediction classification models that detect the mutation in question as stabilizing. Detections per model (B) are highlighted in white, and non-detections highlighted in black. The lower portion of (A) is dominated by single-point mutations to alanine residues, which suggests that the stabilizing effects of these mutations, as opposed to their more common neutralizing/destabilizing effects, are much harder to characterize.

https://doi.org/10.1371/journal.pcbi.1003216.g007

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Table 4. Performance of off-rate classification models for the detection of stabilizing mutations.

https://doi.org/10.1371/journal.pcbi.1003216.t004

Off-rate prediction using molecular descriptors.

The off-rate prediction models investigated so far concentrated on the use different off-rate prediction models which use hotspot descriptors generated from different hotspot predictor algorithms. Here the performance of models created from different categories of molecular descriptors is shown. These include Atomic Potentials (AP), Coarse-grain Potentials (CP) and Physics-Based energy terms (PB) (See Text S1). The same protocol as in the previous sections is followed, where regression models are trained on the 713 mutant dataset and 20-Fold CV results are analyzed as a whole and also separately on different regions of the data set. Classification models are also built on CDS1 and CDS2. As a benchmark comparison, the results of the best performing off-rate predictors built on hotspot descriptors (RFSpot_KFC2_Off-Rate) and the best off-rate predictor built on both hotspot and molecular descriptors (RFSpot_KFC2+Mol_Off-Rate) are also presented with those of the molecular descriptor models (See Figure 5D–F). The physics-based descriptors' model (PB_Off-Rate, R = 0.72) which includes CHARMM [58], FoldX [43] and PyRosetta [59] energy terms performs better than the models with coarse-grain (CP_Off-Rate, R = 0.68) and atomic (AP_Off-Rate, R = 0.61) statistical potentials alone or combined (CP_AP_Off-Rate, R = 0.69). RFSpot_KFC2_Off-Rate (R = 0.76) built on hotspot descriptors only, achieves higher PCC than a model with all molecular descriptors combined (CP_AP_PB_Off-Rate, R = 0.72), whereas the highest correlation is achieved when combining both molecular and hotspot descriptors (RFSpot_KFC2+Mol_Off-Rate, R = 0.79) as already highlighted in the previous section. On analysis of the various regions of the off-rate dataset (Figure 5E) we observe that on all data regions, either the hotspot descriptor model or the molecular and hotspot descriptor models combined always perform better than the molecular descriptor models. This is most notable for SIA complexes and charged residues. Again mutations at the rim regions are the least accurately predicted, and multi-point mutants are better characterized than singe-point mutants. This is consistent with what is observed in Figure 5B for the hotspot descriptor models. The highest discriminatory power for the detection of stabilizing mutants, for both CDS1 and CDS2, is achieved by the off-rate models which make use of hotspot descriptors (Figure 5F).

Broadly predictive and highly specific descriptors for off-rate data regions.

To uncover links between descriptors and certain classes of mutations, the descriptors important for the prediction of mutations within each data region are also analyzed. For each region, the descriptors which are part of the final model in at least half of the total number of runs are singled out for analysis and presented in heat maps which indicate their importance to the given data region (Figure 8A: GS-FS (LR) and Figure 8B GS-FS (SVM)). On the y-axis, the singled out descriptors are listed and categorized according to descriptor type (CP, AP, PB, and Hotspot Descriptors from top to bottom), and each data region shown on the x-axis. Globally, it is observed that whereas for LR models, top features are distributed throughout the four main feature categories, for the non-linear SVM models, 61% of the features are hotspot descriptors, suggesting that non-linear relationships between hotspot descriptors can be better exploited for the predictions of off-rates. To visualize the interconnections between descriptors and data regions, descriptor-data region networks are generated for both the LR (Figure 8C) and SVM (Figure 8D) GA-FS runs. An edge between a descriptor and a data region is shown if the given descriptor is part of the final GA-FS model in at least 50% of the GA-FS runs for the given data region (with increasing edge weight for >50%). For the LR model, two statistical potentials (AP_T1 [55] , CP_MJ2 [60]) are highly specific to rim region mutations, whereas others such as HS_PosCoop, as highlighted by its high degree, are broader in their predictive value and can explain off-rate changes in a number of data regions collectively. Interestingly, for the support region, MaxClusterSize is invoked which suggests that larger hotregions in the support regions may be important for complex stabilization. Whereas certain descriptor-data region relationships hold for both LR and SVM models, such as electrostatic contribution (CHARMM_elec [58]) for mutations on complexes of LIA, the ability to model non-linearities between features, invokes some different descriptors. Most notably, a key observation specific to the SVM descriptor-data region network, is a central cluster of highly interconnected hotspot descriptors and data regions, which involve HS_PosCoop, HSEner_PosCoop, Int_HS_Energy and RimHSEnergy.

Having looked at both the use of specialized models (GS-FS SVM/LR) for different types of complexes, mutations, and regions on the interface, and the use of a global one-fits-all models (such as RFSpot_KFC2+Mol_Off-Rate) for off-rate prediction, it is important to highlight whether there is any advantage in having such specialized models for the prediction of off-rate mutations which fall under a given data region. In Figure 8E, the correlations for the various data regions are compared. For most of the regions, having one-fits-all model suffices, however, for mutations in the rim region, and mutations to charged or polar residues, having a specialized model markedly increases our ability to characterize off-rates in these data regions.

Stability regions in SIA and LIA complexes.

To investigate whether the relative importance of these three regions of stability change when considering complexes of different interface areas, the dataset is divided into Small-Interface-Area (SIA) complexes (<1600 Ų buried surface area) and Large-Interface-Area (LIA) complexes (>1600 Ų buried surface area). The threshold of 1600 Ų is such that both subsets are of similar number of examples. The mean PCC for the CoreHSEnergy, SuppHSEnergy and RimHSEnergy for LIA and SIA complexes is calculated and shown in Figures 9A and 9B respectively. For the LIA complexes, a dominant contribution from the changes in core hotspot energies (CoreHSEnergy |R| = 0.48) and minimal contribution from SuppHSEnergy (|R| = 0.37) and RimHSEnergy (|R| = 0.20) is observed. Therefore, even though a given set of mutations might be affecting support or rim regions, it is the changes in hotspot energies at the core region which show up as the dominant changes in the off-rate (|R| = 0.48). For SIA complexes, changes in hotspot energies in both the in the rim regions show a highly significant 2-fold increase in correlation (p<<0.01). This renders all three regions with similar contributions to complex stability (CoreHSEnergy |R| = 0.56, SuppHSEnergy |R| = 0.46, RimHSEnergy |R| = 0.40). For LIA complexes, the ratio of mutations applied in positions that affect the core to those that affect the rim is 2∶1. On considering SIA complexes this ratio increases to 3∶1. Therefore, we negate the possibility that the increased presence of the rim hotspot energies from LIA to SIA is due to an increase in the number of mutations affecting these regions. Rather, we see an increase in correlation of RimHSEnergy in spite of a reduction in mutations affecting these regions. As an additional test which accounts for biases in the number of examples affecting each region, the correlations are calculated for only the mutations which make changes in the respective region, again taking an average over all 6 hotspot predictors' descriptors. Here no significant changes in correlation are observed in LIA and SIA complexes for the core and support region. For LIA complexes, changes in rim hotspot energies have minimal effect on the off-rate with |R| = 0.29, whereas for SIA complexes, a 1.75-fold increase (p<0.01) in correlation is observed (|R| = 0.51).

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Figure 9. Stability regions, interface-area and complex-size.

The changes in hotspot energies upon mutation are assessed at three interface regions, which enable us to explore changes in the distribution of stability for complexes of different size and interface-area. CORE, RIM and SUPP represent the PCCs of CoreHSEnergy/RimHSEnergy/SuppHSEnergy averaged for the 6 hotspot prediction algorithms with Δlog₁₀(k_off).(A) PCCs for mutants on Complexes with interface-area >1600 Ų (LIA). (B) PCCs for mutants on complexes with interface-area <1600 Ų (SIA). (C) PCCs for mutants on complexes with size <500 residues (SCS). (D) PCCs for mutants on complexes with size >500 residues (LCS). (E) LIA-SCS, (F) LIA-LCS, (G) SIA-SCS, (H) SIA-LCS. (I) Scatter plot of complex size vs. interface area for all complexes in off-rate mutant dataset. Here it is observed that complex stability is distributed across all three regions for small-size complexes (C, E and G), whereas the core becomes a localized region of stability for large-complex sizes (D, F, H). On analysis of the interface-area vs. complex-size subsets (E–H), the distribution of stability regions is affected primarily through complex-size irrespective of interface-area.

https://doi.org/10.1371/journal.pcbi.1003216.g009

Complex size, interface area and stability regions.

To probe further the difference in the distribution of stability regions in SIA and LIA complexes, the size of the complexes is also taken into consideration. The dataset is divided into the mutations which are found on Large-Complex-Size (LCS) (with 231 mutations) and Small-Complex-Size (SCS) complexes (with 482 mutations). The PCC for CoreHSEnergy, RimHSEnergy, SuppHSEnergy averaged over the descriptors from all hotspot predictors is calculated for both LCS and SCS (Figure 9C, D). Core hotspots are critical to the stability of LCS complexes whereas for SCS complexes, all three regions are important. This effect is synonymous with what is observed in LIA and SIA complexes, though the increase in correlation for RimHSEnergy (R = 0.07 to R = −0.36 p<<0.001) is more pronounced for complex size. Noting that fewer mutations are on LCS complexes, the percentage of mutants affecting each region in LCS, compared to that for SCS, is similar across the three regions (61%, 52% and 46% for core, rim and support regions respectively) and therefore shows no relationship to the changes seen in the PCC of the three regions from LCS to SCS. On the 50 complexes considered in the 713 off-rate mutant dataset, complex size and interface size show a correlation of R = 0.55 (Scatter plot in Figure 9I). The correlation is higher (R = 0.74) for complexes sizes of less than 500 residues, and becomes insignificant (R = 0.18) beyond complex sizes of 500 residues. The dataset is therefore further divided into four regions (Scatter Plot I), which include: SIA-SCS (191 mutations) , SIA-LCS (67 mutations), LIA-SCS (191 mutations), LIA-LCS (164 mutations) and again the PCC for CoreHSEnergy, RimHSEnergy, SuppHSEnergy averaged over the descriptors of all hotspot predicators is calculated and shown in Figure 9G: SIA-SCS, Figure 9H: SIA-LCS, Figure 9E: LIA-SCS and Figure 9F: LIA-LCS. Here it is observed that given a fixed complex size (SCS or LCS), moving from small interface areas to larger interface areas, the landscape for the contributions of the core, rim and support regions is unchanging. Therefore, independent of the interface area size, for low complex sizes the off-rate has the propensity to be affected equally from all regions of the interface, whereas for high-complex sizes, stability is primarily emanating from core hotspots. Further analysis of SCS and LCS complexes shows a greater sensitivity in off-rate changes upon mutations for SCS complexes; the mean |Δlog₁₀(k_off)| is 1.4 and 0.69 for SCS and LCS complexes respectively. Though the latter result is intuitive, in that changes on large complexes are less likely to have effects as significant as those on small complexes, the key finding here is that on dissection of the three interface regions, the reduction in the ability to make significant changes in LCS is not equally shared on the three regions. Rather, mutations at the core can still have notable effects on the stability of large complexes as in the case of smaller complexes.

From our findings we confirm that the higher sensitivity of SCS complexes to mutations manifests as an increase in the role of the rim regions and also possibly the support regions. Support regions represent residues which are generally buried both in the unbound and bound structures; therefore, hotspots in this region are primarily responsible for monomer stability as their disruption is likely to affect intra-protein contacts. Given the correlations observed between protein-size and protein stability [61], it is likely that the possible increased role of support regions in SCS complexes, though not as evident here as is for rim regions, is related to monomer size. Rim region residues on the other hand are generally exposed both in the unbound and bound states, but form inter-protein contacts in the complex state. Of particular interest is the observation that rim hotspots are unimportant for the stability of large complexes, even for small-interface-areas. However, given the small number of mutations affecting these complexes (67 mutations), to substantiate this observation, further experimental data may be required. Also for additional validation, analysis of the flexibility of rim regions and the contribution such flexibility is likely to make to the dissociation process, using for example MD simulations on large and small complexes with small-interface-areas, may give further insights.

Hotregion size, count and complex stability.

Analysis of the mean PCCs for No_Clusters (the change in the number of hotregions upon mutation, R = −0.15) and MaxClusterSize (the change in size of the largest hotregion R = −0.09), show no notable contribution to changes in the off-rate (Table 2). Both the change in interface hotspot energy, and change in the number of hotspots show higher correlations (R = −0.51 and R = −0.44 respectively). For RFSpotKFC2, both No_Clusters and MaxClusterSize show higher correlations than the average (R = −0.29, for both), and the combination of the two descriptors into one using multiplication increases the PCC with log₁₀Δ(k_off) to R = −0.48. Nevertheless, its correlation of R = 0.6 with the change in hotspot energies (Int_HS_Energy), suggests that the underlying mechanism might still be the change in hotspot energies, irrespective of hotregion size and count. Note also that this does not imply that larger hotregions do not provide added stability to the complex, but rather their disruption is not critical to complex stability. Understanding if there is any advantage, when attempting to increase complex stability, in having larger hotregions, or more hotregions, would ultimately require analysis which controls for the number of hotspots, varies the number of hotregions or their size and assesses changes in the off-rate. However, current experimental data is limited in size and diversity for this to be performed comprehensively.

Hotregion cooperativity and complex stability.

Probing the importance of the tendency for hotspots to cluster into hotregions, and for that matter, the importance of both size and number of hotregions for complex stability, has also to be done in the light of hotspot cooperativity. Cooperativity within hotregions has been suggested to be a natural consequence of the tight packing ratios found for hotspot residues in hotregions [37]. This adds another layer of complexity in validating the role of hotregions, as under cooperativity, larger hotregions do not necessarily contribute more to complex stability. In turn, this knowledge is critical in order not to overestimate or underestimate the contribution of hotspot energies within hotregions. There are two caveats to this, firstly we need to address the question of what type of cooperativity exists within the hotregions and complexes in the dataset, and secondly we need to have a function which can model or in this case account for it. To our knowledge, this is the first work to include energetic descriptors which account for potential cooperative effects in an empirical scoring function.

Diversity of cooperative effects.

The approach taken here is that no assumption is made before hand for any type of cooperativity prevalent in the complexes and hotregions within our dataset. Rather the two hypotheses of positive cooperativity (HSEner_PosCoop) and negative cooperativity (HSEner_NegCoop) are investigated and compared to the baseline hypothesis of additive hotspot energies (Int_HS_Energy) – (We refer to these three descriptors as the cooperativity descriptors). The motivations and design of these descriptors are detailed in Materials and Methods, but effectively, the higher the PCC of these descriptors with the off-rate, the more likely it is that hotregions on the complexes of the 713 off-rate mutant dataset, show the given type of cooperative/additive effect. In Figure S2, the PCCs of cooperativity descriptors with Δlog₁₀(k_off) are highlighted for every hotspot predictor investigated. From these results, we find no evidence for a prevalent form of cooperativity in hotregions, as the additivity assumption works generally better than positive or negative cooperativity assumption. Several alanine scanning experiments on protein-protein interactions indicate that mutations are, to a large extent, naturally additive [62]–[64]. Eleven residues in the helix-turn-helix motif of the N-terminal domain of Gamma repressor, found in a region important for DNA binding, were substituted to alanine using binomial mutagenesis [64]. The additivity of two mutations was tested by comparing the observed and expected frequencies of the pairwise substitutions. Most of the pairwise substitutions occurred among the active sequences at frequencies close to the product of their single frequencies, thus confirming their additive nature. More so, a model assuming additive interactions was able to predict the activity class of mutants with 90% accuracy [64]. In similar fashion, nineteen residues within the hGH site 1 for binding to the hGHR were randomized using a combinatorial, shotgun alanine-scanning library [62]. On comparison of the counts of double alanine-mutations in hGH site 1 variants selected for binding to the hGHR , from the 144 pairwise combinations, only 15 pairs (10%) behave in a cooperative manner. Still, the analysis of such experiments is not limited to only hotspot residues, and therefore cannot be generalized to those of hotregions; for example combinatorial mutant analysis of the TEM1-BLIP complex which is performed on residues in tight packed modules, and hence more akin to hotregions, shows that residues within a cluster tend to show strong positive cooperativity [65]. With this in mind, our results in Figure S2 are dependent on both the definition of a contact and that of a hotregion. There is no rule of thumb on how to define a contact or hotregion; in one example, the distance between radii balls, with origins set on each C-α atom of the residues in question, is used to define a contact between two hotspot residues [37]. A hotspot residue is added to a hotregion cluster if it is in contact with at least two existing hotspot residues. Our definition uses a more fine-grain approach as a contact between two hotspot residues is created if any of their atoms are at a distance less than their van der Waals radii +0.5 Å. Though for hotspot residues to be added in an existing hotregion, it only needs to be in contact with any other of the hotspot residues, and therefore might be a more lenient way of adding hotspot residues to a hotregion cluster, which in turn may render less packed hotregions. Other contact methods also include weighted contacts according to whether side-chain or backbone atoms are in contact [65]. Most importantly, these different definitions generate different clusters of different packing ratios depending on their leniency and stringency, and therefore may affect the levels of cooperativity observed.

Another factor which may account for the inconclusiveness regarding the more prevalent form of cooperativity, is the modeling of cooperativity functions itself. Finding the right weights to apply to hotregions to account for cooperativity is not trivial as experimental data (such as [65]) is not common enough to be able to learn cooperativity functions from experimental data. Last but not least, the diversity of interactions within the dataset may be better characterized with different cooperativity functions. Interestingly, this diversity of cooperative effects is also observed in the GA-FS runs performed on subsets of related complexes (see Figure S4 in Text S3). Namely we observe that HSEner_PosCoop, HSEner_NegCoop and Int_HS_Energy tend to be important for different sets of related complexes in a mutually exclusive manner. This re-stresses the importance of detecting when a given type of cooperativity is present as much as it is important to model or account for it accurately.

Effects of cooperativity on effective energetic contribution of hotregions.

In order to understand better the effects of our cooperativity descriptors on the distribution of average hotspot energy, and the average hotregion energy for different hotregions sizes, we compare the average hotspot and hotregion energies of different hotregions sizes with no cooperative weightings (i.e. additive hotspot energy assumption), to those after positive cooperativity and negative cooperativity weightings are applied (Figure 10A–B). Analysis of the mean hotspot energies predicted by each hotspot predictor (first row in Figure 10A) shows a constant mean energy profile of hotspot energies within different hotregions. For the additive energy assumption (first row in Figure 10B) and the negative cooperativity assumption (second row in Figure 10B), a linear and exponential-like increase of energetic contribution from larger hotregions is shown respectively. For the positive cooperativity assumption, application of a linearly decreasing function on increasing hotregion sizes which have constant hotspot energies to start off with, results in a bell-shape contribution from hotregions. This suggests that maximum stability is provided by HR sizes of around 5; therefore, a saturation of HR contribution is achieved, beyond which larger hotregions do not necessarily increase complex stability.

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Figure 10. Effects of cooperativity on effective energetic contribution of hotregions.

The summation of single-point alanine ΔΔGs of a hotregion may underestimate/overestimate its contribution if negative/positive cooperative effects are at play respectively. In this work, in order to account for potential cooperative effects, hotspot descriptors HSEner_PosCoop, HSEner_NegCoop apply linearly decreasing and increasing weights respectively to single-point alanine ΔΔGs within a hotregion. In turn Int_HS_Energy, based on the assumption the hotspot residues within the hotregion can be assumed to be additive, does not apply any weights. Here, the effects of accounting for cooperative/additive effects on the predicted hotspot and hotregions energies on all mutated complexes used in this work, is shown. (A) The mean hotspot energies for hotregion sizes of 1 to 8 hotspot residues. Each column shows the predictions of different hotspot predictors. (A) First row (blue), shows the raw mean hotspot energies, which essentially assumes all hotspots are additive within a hotregion. (A) Second row (red), assumes negative cooperativity within hotregions. To account for negative cooperativity, a linearly increasing weight is applied to the hotspot energies according to the size of the hotregion they are in (see Materials and Methods). (A) Third row (green), assumes positive cooperativity within hotregions and a linearly decreasing weight is applied to the hotspot energies according to the size of hotregion. (B) is similar to (A) but values are now the mean of the total hotregion energy of the given size. Effectively, the additive hotspot energy assumption results in hotregions contributing in a linearly increasing manner according to their size, the negative cooperativity assumption results in hotregions contributing in an increasing exponential-like manner as the hotregions increase in size, and the positive cooperativity assumption results in hotregions reaching a maximum contribution at around a hotregion size of 5, with their contribution decreasing beyond.

https://doi.org/10.1371/journal.pcbi.1003216.g010

RFSpot and RFSpot_KFC2.

For each training example in the SKEMPI alanine dataset, and hence wild-type complex PDB structure, a number of molecular descriptors, describing various aspects of the interaction, were calculated. These descriptors have already proven successful in our previous work related to the prediction of wild-type binding free energies [42] and wild-type kinetic rate constants [19]. A full list and explanation of the molecular descriptors can be found in the Text S1. After calculation of the molecular descriptors on the wild-type complex PDB structure, each respective structural mutation was made using FoldX [43] and the same set of molecular descriptors recalculated. Each descriptor, fed into the learning model, is determined as the difference between the mutant and wild-type descriptor value:(4)

As a learning algorithm the RF classifier model is employed [75], using 1000 trees and an mtry (i.e. number of random variables sampled as candidates for a split) of 15. The RF learner is well suited for high dimension datasets, such as the one described here with 110 features. Throughout the manuscript, we refer to this RF hotspot classifier algorithm as RFSpot. RFSpot_KFC2 is a similar classifier model to RFSpot with the difference that it adds to the 110 molecular features set, 13 features from the original KFC2a and KFC2b models. These include: res_hp, pos_per, delta_tot, core_rim, rot5, plast4, plast5, fp10 from KFC2a and res_size, ratio5, rot4, hp5, fp9 from KFC2b. Details on the calculation of each specific descriptors are described in [30], most notably they include features which position the mutation using solvent accessibility. This enables the model to exploit the fact that hotspots tend to occur in regions of low solvent accessibility [24] and is the key difference between RFSpot and RFSpot_KFC2.

RFHotpoint1 and RFHotpoint2, KFC2a, KFC2b.

Similar to RFSpot and RFSpot_KFC2, RFHotpoint1 and RFHotpoint2, are RF hotspot classifiers trained on the SKEMPI alanine dataset which use only features from the original Hotpoint server as features, these include: relativeComplexASA, relativeMonomerASA, pairPotential, complexASA as described in [28]. RFHotpoint2 differs from RFHotpoint1 in that for the former, the threshold is lowered to allow for more hotspot detections at the cost of a higher FPR. The reason behind developing the RFHotpoint models is due to the fact that the original Hotpoint server does not associate an energetic or confidence value to its hotspot prediction, hence hotspot descriptors which make use of hotspot energies cannot be calculated. RFHotpoint models therefore enable us to use Hotpoint features, trained on a larger dataset of SKEMPI instead of ASEdB (as in the original Hotpoint algorithm) and most importantly, associated confidence values to our hotspot predictions using the RF model. To validate RFHotpoint1 and RFHotpoint2 as a representative alternative to Hotpoint, predictions from Hotpoint server were generated for the SKEMPI alanine dataset. Any predictions for mutations in ASEdB were also removed since Hotpoint uses ASEdB as training data. The predictions are compared to the 20-fold test predictions of RFHotpoint1 and RFHotpoint2 for the same mutations and classification results are presented in Table S15 in Text S4. Both RFHotpoint1 and RFHotpoint2 achieve higher MCCs than Hotpoint. For KFC2a and KFC2b, no models needed to be re-trained again, as the original predictions from the KFC2 server have associated with them an energetic value which can be directly used for the calculation of the hotspot descriptors.

Hotspot energies.

In the RF classifier model used in RFSpot, RFSpot_KFC, RFHotpoint1 and RFHotpoint2, each tree in the 1000-tree forest makes is own class prediction (Hotspot/Non-Hotspot) of the mutation in question. The class which accumulates the majority of tree-votes is the predicted class, and the difference in the number of votes for the hotspot class relative to the non-hotspot class (Votes_Hotspot- Votes_Non-Hotspot) indicates the model's confidence in the predicted class. In this work, these confidence values are used as an estimation of hotspot ΔΔGs. The rational is that the higher the confidence value, the more trees have predicted this to be a hotspot, implying that larger numbers of different feature subsets consider this to be a hotspot also. Given that several different aspects of the protein interaction have vouched for the example to be a hotspot, we expect the hotspot ΔΔG to be larger. To confirm this, RF regression models are trained on the same training data as RFSpot, RFSpot_KFC2, RFHotpoint1 and RFHotpoint2 RF classifiers, in order to generate true ΔΔG predictions and compared to the confidence values generated by each of them. Note that RFHotpoint1 and RFHotpoint2 use the same confidence values and only differ by their threshold on those confidence values; hence one correlation for the confidence values is presented for both. The confidence values of the classifier models, show correlations of R = 0.88, R = 0.86, R = 0.86 with the regression models' ΔΔG predictions for RFSpot, RFSpot_KFC2 and RFHotpoint1&2 respectively. Therefore, apart from the differences in their absolute values, the confidence values do provide relative values which have a direct linear relationship to ΔΔG. On assessment of the MCCs of the regression RF models at a threshold of > = 2 kcal/mol, the regression models achieve lower MCCs to that of the classifier models, all of which are the result of higher FPR. Given that 75% of the ΔΔG data is of the negative non-hotspot class, minimal increases in the FPR add a significant number of false-positives which would subdue the gain of additional hotspots correctly detected. Therefore, the use of a classifier in RFSpot, RFSpot_KFC2, RFHotpoint1 and RFHotpoint2, enables us to achieve a lower false positive rate, to that of a regression model, but still be able to have confidence values which relate directly to ΔΔG. For the sake of simplicity, we refer to the ΔΔG confidence values extracted by the method described here as ΔΔGs throughout the manuscript.

Performance of hotspot predictors on SKEMPI alanine dataset.

The predictive accuracy of the hotspot predictors from which the hotspot descriptors are generated from (i.e RFSpot, RFSpot_KFC2, RFHotpoint1, RFHotpoint2, KFC2a and KFC2b), is assessed on the SKEMPI alanine dataset using a number of classification performance measures. For RFSpot, RFSpot_KF2, RFHotspoint1 & RFHotpoint2, the predictions results from a 20–Fold CV are used, whereas for KFC2a and KFC2b the predictions from KFC2 [30] server are used. Note that for KFC2a and KFC2b, the predictions for the data which is in SKEMPI and not in ASEdB is presented, as KFC2 server algorithm uses ASEdB mutations for model design and training. The predictions are compared to a number of hotspot prediction algorithms (KFC2 [30] HotPoint [28], Robetta [35], RFMirror [32], and TSVM [31]). Details on each hotspot predictors and the sources of their predictions are presented in Table S10 in Text S4. The performance of each hotspot predictor is shown in Table S11 in Text S4 and a list of their predictions in Table S16. Note however that Table S11 in Text S4, though ranked according to MCC, shows their performance on different mutations, and therefore cannot be relatively compared. A relative comparison between two predictors can only be performed on the intersections of mutations for which both algorithms provide unbiased predictions, which is beyond the scope of this work. However, this comparison is made for the two hotspot predictors developed in this work namely RFSpot and RFSpot_KFC2. Table S13 in Text S4 shows the performance of RFSpot against all other hotspot predictors on their data intersection, similarly for RFSpot_KFC2 in Table S14 in Text S4. Though RFSpot excels at having a low false-positive-rate (FPR), its true-positive-rate (TPR) is compromised. We note that some hotspot prediction algorithms introduce features, which are based on statistical tendencies of hotspots [30], [32]. For example, the inclusion of W, Y, R as features, since all three show a predisposition to be hotspot residues [36], or the inclusion of the accessible surface area of the residue in question, where it is known that hotspots have a predisposition to occur at the core of the interface [24]. To maintain an unbiased prediction scheme, based purely on molecular and physical descriptors, we intentionally avoid the inclusion of such descriptors in RFSpot, and this is the probable reason for its low TPR. It is understood that the addition of descriptors which relate to solvent accessibility may increase the TPR of RFSpot as this would enable the RF learner to distinguish between mutations performed at the core as opposed to those at the rim, where less hotspots occur [24]. Indeed, it has been shown that a predictor with just 3 solvent accessibility features can result in a sensitivity of 0.87 (i.e. TP/(TP+FN)) for the BID test set [30]. This is also confirmed using RFSpot_KFC2 which introduces features that relate to solvent accessibility and upon setting the threshold to achieve the same FPR to that of RFSpot, the TPR is increased from 0.27 (in RFSpot) to 0.49 (in RFSpot_KFC2). With this in mind, low solvent accessibility is not a sufficient indicator of hotspots as most residues at the core are still non-hotspots [24], [30]. Such models are biased towards predicting hotspots at the core regions as a result of a statistical tendency, and may lay the risk of not being able to detect mutations in other regions as accurately. The risk is even higher if the training set is small and other regions outside the core are underrepresented. Though RFSpot_KFC2 uses such solvent accessibility descriptors, the model uses other molecular features and is trained on a more diverse alanine dataset of SKEMPI as opposed to the ASEdB. Therefore, though still present, the bias towards the prediction of hotspots towards the core is not as strong.

Interface hotspot descriptors.

Int_Energy_1 is the difference in the sum of the single-point alanine ΔΔGs of all interface residues N, pre- and post-mutation.(6)Int_HS_Energy is the difference in the sum of the single-point alanine ΔΔGs of all hotspot residues N_HS, pre- and post-mutation.(7)No_HS is the change in number of hotspots predicted at the interface pre- and post-mutation. This can be considered to be a coarse-grained version of Int_HS_Energy.

Solvent accessible region hotspot descriptors.

To account for the different solvent accessible regions where hotspots may occur at the interface, the following hotspot ΔΔGs are summed separately for the core, rim and support regions and termed as CoreHSEnergy, RimHSEnergy and SuppHSEnergy respectively. Therefore, these hotspot descriptors are similar to Int_HS_Energy but limited to counting ΔΔG for hotspots that fall in the given region. In addition, CoreHS, RimHS and SuppHS desfcriptors, count the hotspot changes within each region. Again these can be considered as coarse-grained versions of their respective counterparts. The core, rim and support regions of the complex interface are defined according to [77]. Core residues are generally exposed in the unbound configuration but buried in the bound state. Rim regions are generally exposed in both the bound and unbound states whereas support residues are generally buried in both states. The thresholds chosen in defining these regions are such that each region has a similar number of residues [77].

Hotregion cooperativity descriptors.

The cooperativity of a pair of residues m₁ and m₂, can be calculated by comparing the gain of adding each residue separately from a neutral reference state of both wild-type residues mutated to alanine (ΔΔG_{A1,A2→A1,m2}+ΔΔG_{A1,A2→m1,A2}) to that of adding both residues concurrently, given the same reference state(ΔΔG_{A1,A2→m1,m2}) [78]. Namely, let A₁ and A₂ represent the alanine mutation of m₁ and m₂ respectively, then(8)

If ΔΔΔG is positive, this indicates positive cooperativity as the contribution of both residues together is more stabilizing than the sum of their parts. Conversely if the ΔΔΔG is negative, this indicates negative cooperativity, whereas if the ΔΔΔG is close to zero, then such pairs can be considered to be effectively independent of each other hence their contributions to be additive in relation to each other.

Expanding ΔΔG_{A1,A2→A1,m2} and ΔΔG_{A1,A2→m1,A2} we get(9)(10)

In this work, we only make single point-mutations during the alanine scan and calculate the energetics associated with such complex states as in eqn (10): ΔΔG_{m1,m2→A1,m2} and ΔΔG_{m1,m2→m1,A2}. The summation of these energies is then used as an estimate of the off-rate. If hotspots within a cluster are additive, then the summation of ΔΔG_{m1,m2→A1,m2}+ΔΔG_{m1,m2→m1,A2} would be a sufficient estimate of the cluster's contribution to the off-rate. However if m1 and m2 are positively cooperative, then their contribution towards the off-rate using the summation ΔΔG_{m1,m2→A1,m2}+ΔΔG_{m1,m2→m1,A2} would be an overestimate of the true contribution ΔΔG_{m1,m2→A1,A2}, hence the positive value for ΔΔΔG. Therefore in this case, to account for positive cooperativity we down-weight the summation of ΔΔG_{m1,m2→A1,m2}+ΔΔG_{m1,m2→m1,A2}. Conversely if m1 and m2 are negatively cooperative, then a positive weighting would be more suitable to account for the underestimation. Further, higher order cooperativity effects involving three or more residues are known [78] and it is likely that many binding modules exhibit such complexity, where it is not possible to decouple the contributions from each individual residues. However, if we assume that cooperativity effects are taking place, the weighting applied should also reflect the number of residues suspected to be cooperative. With this in mind, the cooperativity hotspot descriptors are designed as follows; given a set of predicted hotspots at the interface, each hotspot is categorized according to the hotregion cluster size it is found in. As Int_HS_Energy assumes hotspot contribution is additive, the sum of the hotspot energies is independent of the hotspot locations eqn (7). On the other hand, HSEner_PosCoop and HSEner_NegCoop are the sum of the hotspot energies downweighted/upweighted using simple linearly decreasing/increasing functions related to the size of the hotregion the given respective hotspot is in:(11)(12)where w_HR^Dec = (0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1) and w_HR^Inc = (1, 0.875, 0.75, 0.625, 0.5, 0.375, 0.25, 0.125) for hotspot n_HS in a hotregion of sizes (HR = 1, 2, 3, 4, 5, 6, 7, 8+) respectively. Though more complex non-linear weightings could be investigated, such as ones fitted to the off-rate data itself, this would require sacrificing parts of the data for fitting. With this in mind, all hotspot descriptors designed in this work were independent of any off-rate data. Coarse-grained versions HS_PosCoop and HS_NegCoop, which weight hotspot counts instead of energies, are also implemented in the model. One should note that since the energetic contribution of a hotregion taken as a whole is considered to be additive and independent of other hotregions [37], [65] we only aim to investigate and account for intra-hotregion cooperativity using these descriptors as opposed to inter-hotregion cooperativity.

Hotspot coverage related descriptors.

Other hotspot descriptors relate to the spread of hotspots across the interface. The intuition here is that a heterogeneous distribution of hotspots across the interface might be more beneficial to complex stability than if hotspots where concentrated onto a specific region of the interface only. AVG_HS_PathLength is the average path length between all possible pairs of hotspots at the interface, normalized to the average path length of all possible pairs of a random set of residues at the interface. The path length between two residues is calculated as the least number of contacting residues linking them together. Two residues are considered to be in contact if any of their atoms are at a distance smaller than the sum of their van der Waals radii +0.5 Angstroms. No_Clusters counts the number of unique hot regions, where it is likely that more hotregions may span the interface given that separate hotregions are not in contact. MaxClusterSize counts the change in the number of hotspots in the largest hotregion.

Definition of a hotregion.

Some of the hotspot descriptors use hotregion information within them (No_Clusters, MaxClusterSize, HSEner_PosCoop/HS_PosCoop and HSEner_NegCoop/HS_NegCoop). A hotregion is created whenever two or more hotspot residues are in contact. Two hotspot residues are considered to be in contact if any of their atoms are at a distance smaller than the sum of their van der Waals radii +0.5 Å. A hotspot residue is added to an existing hotregion, if any of its atoms makes contact with any of the hotspot residues already in the hotregion.