We first focused on analyzing how mutations and combinations of mutations lead to complete escape (defined as K_D,app above our limit of detection) for specific sets of antibodies. To do so, we analyze the enrichment of specific mutations among non-binders (Figure 2A). We find a largely orthogonal set of one or two mutations are enriched among variants that do not bind each antibody: almost all variants that do not bind LY-CoV016 contain K417N, almost all variants that do not bind REGN10987 contain G446S, and many also contain N440K, and E484A and Q493R are highly enriched among variants that do not bind LY-CoV555. These escape mutations were already identified on the Wuhan background (Tada et al., 2022; Starr et al., 2021c; Zhou et al., 2022). The fact that different sites are involved for each antibody suggests that the RBD can evolve to independently escape antibodies with each distinct epitope, and mutations can to some extent act independently on binding to each antibody.

To analyze this further, we calculate the percentage of genetic backgrounds on which each mutation leads to complete escape from a specific antibody (i.e. that mutation converts a variant with measurable K_D,app to a K_D,app above our limit of detection). We see that for each antibody, one or two mutations abrogate binding. These sets of mutations are largely orthogonal among antibodies (Figure 2B), consistent with the enrichment analysis (Figure 2A). Specifically, K417N always abrogates binding to LY-CoV016, G446S and N440K often abrogate binding to REGN10987, and E484A and Q493R often abrogate binding to LY-CoV555.

However, we note that this orthogonality is not complete. For example, G446S is slightly enriched among LY-CoV555 binders, while mutation E484A is slightly depleted among variants that do not bind REGN10987 (Figure 2A). Consistent with this, G446S sometimes abrogates binding to LY-CoV555 (Figure 2B). In addition, some apparently smaller-effect mutations can be involved in abolishing binding to multiple antibodies. For example, S375F is weakly enriched among variants that do not bind REGN10987 and LY-CoV555 and often abrogates binding to these two antibodies, with G496S, Q498R, and N501Y also playing a role.

To summarize how these different mutations can act individually or in combination to lead to antibody escape, we inferred a decision tree to classify variants as binders or non-binders. To do so, for each antibody, we calculate the mutation that maximally partitions the variants into binders or non-binders. If this partitioning is not perfect, we then calculate the second mutation that maximally partitions the variants conditional on each possible state of the first site. We then proceed to further partition variants based on additional mutations in the same way until the variants are perfectly partitioned or no further mutations can significantly improve the partitioning (see Methods). We show the corresponding decision trees for LY-CoV016, REGN10987, and LY-CoV555 in Figure 2C, D and E, respectively. As expected, the tree associated with LY-CoV016 is very simple: the mutation K417N perfectly partitions the variants into binders and non-binders. In contrast, the trees for REGN10987 and LY-CoV555 have more complex structures, reflecting the fact that it is possible to abrogate affinity to these antibodies via multiple distinct combinations of mutations. For example, variants can escape REGN10987 by acquiring G446S and N440K (100%), or alternatively, with S375F and G446S (89%, as additional mutations may also be required). For LY-CoV555, different sets of mutations can lead to escape (e.g. Q493R and G446S or E484A and S375F). Some of these mutations partially overlap with those for REGN10987 (i.e. they are not fully orthogonal), suggesting that selection pressure from one antibody could promote subsequent escape of another.

In addition to large-effect mutations which lead to complete escape of specific antibodies, a variety of other sites contribute to more subtle but potentially important changes in binding affinities. To analyze these subtle effects as well as the large-effect mutations leading to escape, we defined a linear model for −log(K_{D, app}) as the sum of single (additive) mutational effects plus interaction terms up to a specified order (note that because −log[K_{D, app}] is proportional to the free energy of binding, we expect it to behave additively in the absence of epistatic interactions). Because non-binding variants have −log(K_{D, app}) beyond our limit of detection, we fit a Tobit model (a class of regression model capable of handling truncated measurements; see Methods for details) using maximum likelihood with an L2-norm Lasso regularization. Specifically, we partition our data into training (90%) and test (10%) sets and use the training dataset to fit epistatic coefficients to a linear model truncated at each order (e.g. truncating to first-order yields additive mutational effects, second-order includes both additive effects and pairwise terms, etc.). We then evaluate performance (as the coefficient of variation) of each model on the held-out test dataset and compare the model performance using −log(K_{D, app}) for each of the antibodies and ACE2 (Figure 3A).

We find that adding epistatic interactions improves the predictive power of the model for all four antibodies as well as for binding to ACE2, though the optimal order varies (Figure 3A). This indicates that epistasis does play a significant role in all cases (up to second order for REGN10987, to third order for LY-CoV555, and to fourth or higher order for LY-CoV016, S309, and ACE2). The additive, pairwise, and higher-order coefficients resulting from these models are summarized in Figure 3—figure supplement 1. In general, we find many strong interactions across several positions in each antibody, involving both the sites that strongly determine escape variants for that antibody (e.g. between N440K and G446S for REGN10987) as well as others.

Notably, we find that the higher-order epistasis plays a much stronger role in determining affinity for ACE2 than for the three antibodies fully escaped by BA.1 (Figure 3B). This reflects the impact of a few strong-effect mutations in determining affinity for LY-CoV555, LY-CoV016, and REGN10987 and the role of compensatory epistasis in determining ACE2 affinity. In other words, while epistasis is relevant for all measured phenotypes, antibody escape is more simply determined by the additive effects of individual mutations, while maintaining ACE2 affinity involves more complex epistatic interactions.

High-order epistatic interactions are also important in determining affinity to S309. In Figure 3C, we highlight four neighboring mutations (G339D, S371L, S373P, and S375F) which interact non-additively to produce the reduction in affinity observed in BA.1 relative to Wuhan Hu-1. Each of these mutations weakly reduces affinity on their own, and specific combinations of these mutations can reduce affinity by up to two orders of magnitude, but the reduction in affinity resulting from all four mutations is less than some sets of three mutations. These patterns emerge from a complex set of high-order epistatic interactions among the mutations. For example, S371L reduces affinity on the Wuhan Hu-1 background but increases affinity on the background containing G339D, S373P, and S375F (and without G339D, S371L increases affinity in the presence of S373P if S375F is absent but not if it is present). Thus, some variants lacking S371L evade S309 more effectively than BA.1, and interestingly we note that this mutation is absent in BA.2 and BA.3 and replaced instead by S371F.

In previous work, we found that antibody escape mutations (as defined in earlier studies) typically reduce ACE2 affinity, suggesting that viral evolution is constrained by a tradeoff between immune evasion and the ability to enter host cells. Consistent with this, we find here that variants that escape one or more antibodies (as defined by the data reported in this work) but have few additional mutations have reduced ACE2 affinity relative to Wuhan Hu-1. However, as additional BA.1 mutations are accumulated, the ACE2 binding affinity tends to increase until it exceeds the Wuhan Hu-1 value even in the presence of multiple antibody escape mutations (Figure 4A). This suggests that the evolution of the BA.1 variant is driven both by immune escape and the need for compensatory mutations that mitigate the negative effects of the escape mutations on ACE2 binding.

The strength of this tradeoff and the potential importance of compensatory evolution are distinct between the different antibodies (Figure 4B and C). For example, escape from LY-CoV016 or LY-CoV555 reduces ACE2 binding affinity in the absence of compensatory mutations (Q498R and N501Y) but not in their presence (Figure 4B). In contrast, REGN10987 escape does not strongly reduce affinity to ACE2, whether or not Q498R and N501Y are present. However, this tradeoff is likely relevant overall, as escaping all three antibodies substantially reduces ACE2 affinity in the absence of Q498R and N501Y, while the reduction in ACE2 affinity is minimal in their presence. Consistent with this general picture, the frequency of most escape mutations (G446S, E484A, and Q493R) is higher across the SARS-CoV-2 phylogeny in the presence of compensatory mutations (Figure 4D), though we note that this is not universally true (e.g. the frequency of N440K is only slightly higher in the presence of compensatory mutations, and the frequency of K417N is lower with the compensatory mutations).

Although antibody escape mutations do tend to reduce ACE2 affinity, antibody binding affinity (but not complete escape) is not strongly correlated with ACE2 affinity (Figure 4C). The details of this relationship vary by antibody. For LY-CoV016 and LY–CoV555, there is a weak overall positive correlation (i.e. lower antibody affinity also tends to correspond to reduced ACE2 affinity). However, this correlation is dominated by the variants that lack the compensatory mutations at sites 498 and 501; in the presence of Q498R and N501Y, the correlation is reduced, especially for LY-CoV016. For REGN10987 and S309, there is a similar weak overall positive correlation, which is not dependent on Q498R and N501Y. We also note that while compensatory mutations Q498R and N501Y largely drive the variance in ACE2 affinity, they minimally impact antibody binding affinities (Figure 4D).

We used the same library and strains as produced in Rotem et al., 2018. In brief, to generate clonal yeast strains for the Wuhan Hu-1 and Omicron BA.1 variants, we cloned the corresponding RBD gblock into a pETcon vector via Gibson Assembly. We then extracted and transformed Sanger-verified plasmids into the AWY101 yeast strain (kind gift from Dr. Eric Shusta; Wentz and Shusta, 2007) as described in Gietz and Schiestl, 2007. To produce the RBD variant library, we employed a Golden Gate combinatorial assembly strategy. We constructed full RBD sequences from five sets of dsDNA fragments of roughly equal size. Each set contains versions of the fragments that differ by the mutations included. Following bacterial transformation of this Golden Gate assembly product, we extracted and transformed the library into AWY101 yeast strain, from which we inoculated and froze a library containing obtained ∼1.2 million colonies.

We performed Tite-seq assay as previously described (Starr et al., 2020; Moulana et al., 2022; Adams et al., 2016; Phillips et al., 2021), with two replicates for each antibody (LY-CoV016, LY-CoV555, REGN10987, and S309 [Genscript, Gene-to-Antibody service]) assay on different days, for a total of eight assays.

Briefly, we thawed yeast RBD library and the Wuhan Hu-1 and Omicron BA.1 strains by inoculating the corresponding glycerol stocks in SDCAA (6.7 g/L YNB without amino acid [VWR #90004–150], 5 g/L ammonium sulfate [Sigma-Aldrich #A4418], 2% dextrose [VWR #90000–904], 5 g/L Bacto casamino acids [VWR #223050], 1.065 g/L MES buffer [Cayman Chemical, Ann Arbor, MI, #70310], and 100 g/L ampicillin [VWR # V0339]) at 30°C for 20 hr. The cultures were then induced in SGDCAA (6.7 g/L YNB without amino acid [VWR #90004–150], 5 g/L ammonium sulfate [Sigma-Aldrich #A4418], 2% galactose [Sigma-Aldrich #G0625], 0.1% dextrose [VWR #90000–904], 5 g/L Bacto casamino acids [VWR #223050], 1.065 g/L MES buffer [Cayman Chemical, Ann Arbor, MI, #70310], and 100 g/L ampicillin [VWR # V0339]) and rotated at room temperature for 16–20 hr.

Following overnight induction, we pelleted, washed (with PBS + 0.01% BSA [VWR #45001-130; GoldBio, St. Louis, MO #A-420-50]), and incubated the cultures with mAb at a range of concentrations (10^–6 to 10^–12 with 0.75-log increments for CoV555, 10^–7 to 10^–12 with 0.5-log increments for S309, 10^–6 to 10^–12.7 with 0.75-log increments for REGN10987, 10^–6 to 10^–12 with 0.75-log increments for SB6). The yeast-antibody mixtures were incubated at room temperature for 20 hr. The cultures were then pelleted washed twice with PBSA and subsequently labeled with PE-conjugated goat anti-human IgG (1:100, Jackson ImmunoResearch Labs #109-115-098) and FITC-conjugated chicken anti-cMmyc (1:100, Immunology Consultants Laboratory Inc, Portland, OR, #CMYC-45F). The mixtures were rotated at 4°C for 45 min and then washed twice in 0.01% PBSA.

Sorting, recovery, and sequencing library preparation followed Moulana et al., 2022; Phillips et al., 2021; Gietz and Schiestl, 2007. In short, we sorted ~1.2 million yeast cells per concentration, gated by FSC vs. SSC (forward vs. side scatter) and then by expression (FITC) and/or binding fluorescence (PE) on a BD FACS Aria Illu. The machine was equipped with 405 nm, 440 nm, 488 nm, 561 nm, and 635 nm lasers and an 85 micron fixed nozzle. Sorted cells were then pelleted, resuspended in SDCAA, and rotated at 30°C until late-log phase (OD600=0.9–1.4). The cultures were then pelleted and stored at –20°C for at least 6 hr prior to extraction using Zymo Yeast Plasmid Miniprep II (Zymo Research # D2004), following the manufacturer’s protocol. The sequencing amplicon libraries were then prepared by a two-step PCR as previously described (Moulana et al., 2022; Phillips et al., 2021; Nguyen Ba et al., 2019). In brief, we added to the amplicon unique molecular identifies (UMIs), inline indices, and partial Illumina adapters through a seven-cycle PCR which amplifies the RBD sequence in the plasmid. We then used the cleaned product from the first PCR in the second PCR to append Illumina i5 and i7 indices accordingly (see https://github.com/desai-lab/compensatory_epistasis_omicron/tree/main/Supplementary_Files for primer sequences). The products were then cleaned using 0.85× Aline beads, verified using 1% agarose gel, quantified on Spectramax i3, pooled, and verified on Tapestation 5000HS and 1000HS. Final library was quantitated by Qubit fluorometer and sequenced on Illumina Novaseq SP, supplemented with 10% PhiX.

Following Moulana et al., 2022, we processed raw demultiplexed sequencing reads to identify and extract the indexes and mutational sites. Briefly, for each antibody, we utilized a snakemake pipeline (https://github.com/desai-lab/omicron_ab_landscape; Moulana, 2022) to parse through all fastq files and group the reads according to inline indices, UMIs, and sequence reads. We accepted sequences based on criteria previously determined (10% bp mismatches) and converted accepted sequences into binary genotypes (‘0’ for Wuhan Hu-1 allele or ‘1’ for Omicron BA.1 allele at each mutation position). Reads containing errors at mutation sites were removed. Finally, the pipeline collated genotype counts based on distinct UMIs from all samples into a single table.

We fit the binding dissociation constants K_D,app for each genotype as previously described (Nguyen Ba et al., 2019). Briefly, using sequencing and flow cytometry data, we calculated the mean log-fluorescence of each genotype s at each concentration c, as follows:F̄s,c = ∑bFb,c pb,s|c

where, Fb,c is the mean log-fluorescence of bin b at concentration c, and pb,s|c is the inferred proportion of cells from genotype s that are sorted into bin b at concentration c, which is estimated from the read counts as:pb,s|c=Rb,s,c∑sRb,s,cCb,c∑b(Rb,s,c∑sRb,s,cCb,c)

Here, Rb,s,c represents the number of reads from genotype s that are found in bin b at concentration c, and Cb,c refers to the number of cells sorted into bin b at concentration c.

We then computed the uncertainty for the mean log-fluorescence:δF̄s,c =∑b( δFb,c2 pb,s|c2+ Fb,c2δpb,s|c2)

where, δFb,c is the spread of the log fluorescence of cells sorted into bin b at concentration c. The error in pb,s|c emerges from the sampling error, which can be approximated as a Poisson process, such that:δpb,s|c=pb,s|cRb,s,c

Finally, we inferred the binding dissociation constant (K_D,s) for each variant by fitting the logarithm of Hill function to the mean log-fluorescence F̄s,c , as a function of concentrations c:F̄s,c=log10(cc+KD,sAs+Bs)

where, As is the increase in fluorescence at antibody saturation, and Bs is the background fluorescence level. The fit was performed using the curve_fit function in the Python package scipy.optimize. Across all genotypes, we imposed bounds on the values of As to be 10²–10⁶, Bs to be 1–10⁵, and K_D,s to be 10⁻¹⁴–10⁻⁵. We then averaged the inferred K_D,s values across the two replicates for each antibody after removing values with poor fit (r2<0.8 or SE>1). Variants were defined as non-binders if the difference between the maximum and the minimum of their estimated log-fluorescence over all concentrations was lower than 1 (in log-fluorescence units). This value was set by measuring the distribution for known non-binders (see Figure 1—figure supplement 1).

We validated our high-throughput binding affinity method by measuring the binding affinities for the Wuhan Hu-1 and Omicron BA.1 RBD variants. For each isogenic titration curve, we followed the same labeling strategy as in Tite-seq, titrating each antibody at concentrations ranging from 10⁻¹²-10⁻⁷ M (with increments of 0.5 for the first replicate and 1 for the second one) for isogenic yeast strains that display only the sequence of interest. The mean log fluorescence was measured using a BD LSR Fortessa cell analyzer. We directly computed the mean and variances of these distributions for each concentration and used them to infer the value of K_D,app using the formula shown above.

To summarize mutations that drive the loss of binding (escape) for each antibody, we constructed a decision tree using package rpart in R (Therneau et al., 2013) with its default parameters. In brief, for each antibody (except for S309 where every sequence binds the antibody), we first categorized each genotype into a binary parameter with values ‘binding’ or ‘non-binding’. Then, the function rpart splits the tree based on any one of the 15 mutations by minimizing the Gini impurity for the binding parameter. The method continues to partition the tree if the cost complexity parameter of the split does not drop below 0.01. This parameter is the sum of all misclassifications (binding vs. non-binding) at every terminal node (analogous to residual sum of squares in regression), added by the product between the number of splits (analogous to degree of freedom) and a penalty term inferred through cross-validation performed by the rpart algorithm. The tree is then presented in Figure 2 using ‘ggparty’ package (Borkovec, 2019).

We used a linear model where the effects of combinations of mutations sum to the phenotype of a sequence. The logarithm of the binding affinity log⁡KD,app is proportional to change in free energy. Thus, without epistatic interactions, the effects of mutations are expected to combine additively (Wells, 1990; Olson et al., 2014). We describe here our analysis of epistatic effects that lead to departures from this additive expectation.

We could naively infer all 2¹⁵ epistatic coefficients (corresponding to each subset of mutations, including all possible orders of epistasis) since we have measured binding affinities for all possible combinations of the 15 RBD mutations. However, this approach is inherently unstable: such inference will tend to identify spurious and insignificant higher-order epistatic terms to compensate for measurement errors. To avoid this problem, we truncated our model at an optimal order. That is, we neglected all epistasis terms involving more than a certain number of mutations, as is common in other analyses of epistasis (Moulana et al., 2022; Phillips et al., 2021; Otwinowski et al., 2018). To determine which order is optimal, we used a 10-fold cross-validation strategy by training each model on 90% of the dataset and examining its performance on the remaining 10%, as shown in Figure 3A.

Some phenotypic variables log⁡KD,app are unavailable in our dataset due to the upper limit of the assay concentration: we are unable to precisely infer KD,app for the low-affinity (or non-binding) variants, particularly when the true −log⁡KD,app<6 (the highest concentration used). To address this issue, we augmented our linear model with a lower boundary, following a Tobit left-censored model (Tobin, 1958). In this model, the sampling probability of −log⁡KD,app<6 is modeled using a cumulative distribution which contributes to the maximum-likelihood. Thus, the full K-order model can be written as:-log⁡KD,s*= β0+∑i=1K∑c∈Ci βcxc,s+ϵs

where Ci contains all Li combinations of size i of the mutations and xc,s equal to 1 if the sequence s contains all the mutations in c and to 0 otherwise. Here, -log⁡KD,s=-log⁡KD,s* if −log⁡KD,s∗>6 and -log⁡KD,s=6 if -log⁡KD,s*≤6. Then, following the Tobit model approach, we compute the likelihood function to infer coefficient parameters β^MLE , given by:L(β,σ)= ∏j=1N1σφyj-(β0+∑i=1K∑c∈Ci βcxc,s)σI(yj)1-Φ(β0+∑i=1K∑c∈Ci βcxc,s)-6σ1-I(yj)

where, yj=-log⁡KD,app,j , and φ and Φ denote the standard normal cumulative distribution function and probability density function, respectively. Moreover, note that I(y)={0 if y≤61 otherwise. This optimization problem would include coefficients that are associated with the loss-of-binding phenotypes. Consequently, by the model, these coefficients do not have lower bounds, and the optimization would have resulted in deflated coefficients offset by inflated higher-order coefficients, or vice-versa. To resolve this problem, we add a lasso regularization term in the form of ϵ ∑|βc| to the likelihood, with ϵ=0.01. This term is small enough to reduce the magnitude of constrained coefficients but act as intended on the non-constrained ones. In particular, we do not use the correction to make any strong assumption on the sparsity of the epistatic coefficients but to reduce instability caused by the Tobit model inference. To maximize the log-likelihood function, which is a concave function, we used the optimize module in the scipy package, with the BFGS (Broyden–Fletcher–Goldfarb–Shanno) gradient-descent method.

Although we have focused our analyses on idiosyncratic epistasis, we could also consider models of global epistasis. The significance of global epistasis in the context of protein binding is still disputed. However, global epistasis, as described by Sailer and Harms, 2017, might arise from changes in conformation or stability, leading the measured phenotypes to be a non-linear function of a simpler underlying linear phenotype. We have made use of this concepts and inferred global epistatic terms in previous other recent related work (Sailer and Harms, 2017). However, in this study, due to potential confounding effects arising from the non-binding variants (and corresponding lower boundary of detection), we chose not to perform such inference on this dataset. Moreover, we do not have any evidence that there are any changes in conformation or stability of the protein that could potentially lead to global epistatic effects in our data.

We used the reference structure of a 2.79 Å cryo-electron microscopy structure of Omicron BA.1 complexed with ACE2 (PDB ID: 7WPB). The contact surface area is determined by using ChimeraX (Pettersen et al., 2021) to measure the buried surface area between ACE2 and each mutated residue in the RBD (measure buriedarea function, default probe radius of 1.4 Å), whereas distance between α-carbons is measured using PyMol (Schrodinger LLC, 2015).

The high-dimensional binding affinity landscape can be projected in two dimensions with a force-directed graph layout approach (see https://desai-lab.github.io/wuhan_to_omicron/). Each node corresponds to each sequence in the library, connected by edges to a neighbor that differs in one single site. For each antibody, an edge between two sequences s and t is given the weight:ws,t=10.01+|log⁡KD,s-log⁡KD,t|

Additionally, we also constructed a different layout that includes affinities to all antibodies, where the weight between two sequences depends on the sum over the antibodies of the difference between their affinities:ws,t=10.01+∑a∈A|log⁡KD,a,s-log⁡KD,a,t|

where, A is the set of antibodies we used. In a force-directed representation, the edges pull together the nodes they are attached to proportional to the weight given to each edge. In our scenario, this means that nodes with a similar genotype (a few mutations apart) and a similar phenotype (binding affinity or total binding affinity) will be close to each other in two dimensions.

Importantly this is not a ‘landscape’ representation: the distance between two points is unrelated to how easy it is to reach one genotype from another in a particular selection model. Practically, after assigning all edge weights, we use the layout function layout_drl from the Python package iGraph, with default settings, to obtain the layout coordinates for each variant.

To analyze SARS-CoV-2 phylogeny, we used all complete RBD sequences from all SARS-CoV-2 genomes deposited in the Global Initiative on Sharing All Influenza Data (GISAID) repository (Khare et al., 2021; Elbe and Buckland-Merrett, 2017; Shu and McCauley, 2017) with the GISAID Audacity global phylogeny (EPI_SET ID: EPI_SET_20220615uq, available on GISAID up to June 15, 2022, and accessible at https://doi.org/10.55876/gis8.220615uq). We pruned the tree to remove all sequences with RBD not matching any of the possible intermediates between Wuhan Hu-1 and Omicron BA.1 and analyzed this tree using the python toolkit ete3 (Huerta-Cepas et al., 2016). We measured the frequency of each mutation by counting how many times it emerges in the tree, normalized by the total occurrences of other mutations. For frequency with Q498R and N501Y, we counted the occurrence of each mutation only on branches that already contains Q498R and N501Y and normalized similarly.

All data processing and statistical analyses were performed using R v4.1.0 (R Development Core Team, 2017) and python 3.10.0 (Van Rossum and Drake, 2009). All figures were generated using ggplot2 (Wickham, 2016) and matplotlib (Hunter, 2007).

Correspondence and requests for materials should be addressed to M.M.D. (mdesai@oeb.harvard.edu).

Raw sequencing reads have been deposited in the NCBI BioProject database under accession number PRJNA877045. All associated metadata are available at https://github.com/desai-lab/omicron_ab_landscape, (copy archived at swh:1:rev:9ab630decfa835b2551430ed693796ef366b1aff; Moulana, 2022).