For CR9114, we obtained the somatic heavy chain nucleotide sequence from Dreyfus et al., 2012 (GenBank JX213639.1) and reconstructed the germline nucleotide sequence using IMGT (Giudicelli et al., 2006) and IgBLAST (Ye et al., 2013). Both methods assigned the same V-gene and J-gene alleles (IGHV1-69*06 and IGHJ6*02), but there is ambiguity in the D-gene assignment and at the V-D junction, particularly at site 109. The preferred IMGT junction alignment assigns a mutation here, S109N, while a different junction alignment from IgBLAST does not. Because of the inherent difficulty of reconstructing mutations in the junction region, especially in antibodies with a short D region, we chose the alignment without the mutation at site 109. Our reconstructed germline nucleotide sequence is available in Supplementary file 12. We then took the resulting germline and somatic amino acid sequences, as shown in Figure 1A, and constructed new nucleotide sequences codon-optimized for yeast.

For CR6261, the somatic and reconstructed germline heavy chain amino acid sequences were published in Lingwood et al., 2012. We used these sequences, similarly constructing codon-optimized nucleotide sequences for expression in yeast. The original somatic nucleotide sequence is also available (GenBank HI919029.1).

We note that all antibody libraries and clonal strains were constructed using somatic forms of the light chain, as these antibodies were isolated by combinatorial phage display (Throsby et al., 2008; Dreyfus et al., 2012), and so it is not possible to infer the naturally paired germline light chain sequence. Additionally, the CR9114 and CR6261 light chains were previously determined not to impact binding (Lingwood et al., 2012; Dreyfus et al., 2012; Ekiert et al., 2009). The somatic light chain sequence for CR9114 was obtained from Dreyfus et al., 2012 (GenBank JX213640.1), and that for CR6261 was obtained from Throsby et al., 2008 (GenBank HI919031.1).

CR9114 contains a total of 18 amino acid substitutions between the somatic variant and the reconstructed germline sequence. However, a library of 218=262,144 variants would be costly and time-consuming to produce and assay via our methods. We therefore identified two mutations that were distant from antigen contacts in the crystal structure: A25S and E51D (Dreyfus et al., 2012). We measured binding affinities for somatic sequences with and without these two mutations, and found that these variants had comparable affinities for both H1 and H3 (Figure 1—figure supplement 7). Although these mutations may have some small impact on binding, especially in combination with others, excluding them allowed for a simpler cloning strategy and a more manageable library size.

Similar to the CR9114 library design, we reduced the number of mutations present in the CR6261 library by excluding three mutations that were distant from antigen contacts in the crystal structure: 6QE, L50P, and V101M (Ekiert et al., 2009). We validated the marginal contribution of these mutations to binding by measuring the binding affinities for the somatic sequence with and without these mutations reverted to the respective germline residue (Figure 1—figure supplement 7).

To generate clonal yeast display strains and libraries for CR9114, we cloned scFv constructs (V_L -Ser(Gly₄Ser)₅-V_H-Myc) into the pCT302 plasmid (Midelfort et al., 2004) (kind gift from Dane Wittrup; Addgene, Watertown, MA, #41845). For the clonal CR9114 somatic and germline strains, gene blocks corresponding to the somatic or inferred germline sequences were cloned into pCT302 by Gibson Assembly (Gibson et al., 2009) (plasmid maps in Supplementary files 4–5). For producing the plasmid backbone required for Golden Gate library generation (described below), we removed an existing Bsa-I site from the pCT302 plasmid by site-directed mutagenesis (Agilent, Santa Clara, CA, #200521) and replaced the V_H domain with the ccdB gene. To generate clonal yeast strains, Gibson Assembly products were transformed into electrocompetent DH10B E. coli cells, and the resulting plasmids were mini-prepped and Sanger sequenced. Following sequence confirmation, plasmids were transformed into EBY100 yeast cells (ATCC #MYA-4941) as described in the high-efficiency yeast transformation protocol (Gietz and Schiestl, 2007). Transformants were plated on SDCAA-agar (1.71 g/L YNB without amino acids and ammonium sulfate [Sigma-Aldrich #Y1251], 5 g/L ammonium sulfate [Sigma-Aldrich #A4418], 2% dextrose [VWR #90000–904], 5 g/L Bacto casamino acids [VWR #223050], 100 g/L ampicillin [VWR #V0339], 2% Difco Noble Agar [VWR #90000–774]) and incubated at 30°C for 48 hr, single colonies were restruck on SDCAA-agar and again incubated at 30°C for 48 hr, and the resulting clonal yeast strains were verified to have the construct of interest by colony PCR. Construction of the yeast libraries is described below. All yeast strains were grown to saturation in SDCAA at 30°C, supplemented with 5% glycerol, and stored at −80°C.

CR6261 clonal yeast display strains and libraries were generated in an identical manner to that of CR9114, except where noted below (see Supplementary files 6–7 for plasmid maps corresponding to the germline and somatic sequences).

For CR9114, due to the number of mutations required and their positions along the heavy chain coding sequence, we designed a library cloning strategy using Golden Gate combinatorial assembly (Engler et al., 2008). We divided the heavy chain coding region into five roughly equal fragments, ranging from 79 to 85 bp and each containing between 1 and 5 mutations. We added BsaI sites and additional overhangs to both ends of each fragment sequence, with cut sites carefully chosen so that the five fragments will assemble uniquely in their proper order within the plasmid backbone. For each fragment with n mutations, we then ordered 2n individual DNA duplexes with each possible combination of mutations (ranging from 2 to 32 versions for each fragment, a total of 66 fragments) from IDT (Coralville, IA) (see Supplementary file 2). By pooling the versions of each fragment in equal volumes, then pooling the five fragment pools in equimolar ratios, we obtained a randomized fragment mix containing all 2¹⁶ sequences present at approximately equal frequencies.

In addition to the fragment mix, we prepared the plasmid backbone for the Golden Gate reaction. We created a version of the yeast display plasmid with the counter-selection marker ccdB in place of the heavy chain sequence, with flanking BsaI sites (see above). We performed Golden Gate cloning using BsaI-HFv2 (NEB, Ipswich, MA, #R3733) following the manufacturer recommended protocol, with a 5:1 molar ratio of the fragment insert pool to plasmid backbone.

We transformed the assembly mix into electrocompetent E. coli (DH10B) via electroporation in 10 x 50 μL cell aliquots. We recovered each transformation in 5 mL SOC (2% tryptone, 0.5% yeast extract, 10 mM NaCl, 2.5 mM KCl, 10 mM MgCl₂, 10 mM MgSO₄, 20 mM glucose) at 37°C for 1 hr, and then transferred each to 100 mL of molten LB (1% tryptone, 0.5% yeast extract, 1% NaCl) containing 0.3% SeaPrep agarose (VWR, Radnor, PA #12001–922) spread into a thin layer in a 1L baffled flask (about 1 cm deep). The mixture was allowed to set on ice for an hour, after which it was kept for 18 hr at 37C to allow for dispersed growth of colonies in 3D. We observed ∼3×105 colonies per aliquot, for a total of ∼3 million transformants. After mixing the flasks by shaking for 1 hr, we pelleted the cells and prepared plasmid by standard midiprep (Zymo Research, Irvine, CA, D4201), from which we obtained >120 μg of purified plasmid.

For CR6261, we designed a library cloning strategy also using Golden Gate combinatorial assembly, but with fragments created by PCR instead of purchased. We divided the heavy chain coding region into three roughly equal fragments, each containing between 2 and 5 mutations. We designed these fragments such that the mutations they contain are close to the 3’ or 5’ ends and can thus be easily incorporated by PCR. PCR primers included mutations, BsaI sites, and unique overhangs chosen so that the three fragments would assemble uniquely in their proper order within the plasmid backbone. For each version of the three fragments, we generated dsDNA by PCR (52 PCR reactions in total; see Supplementary file 3 for primer sequences). By pooling all versions of each fragment in equal volumes, then pooling the three fragment pools in equimolar ratios, we obtain a randomized fragment mix that, when ligated in the Golden Gate reaction, produces all of the 2¹¹ sequences present at approximately equal frequencies.

In addition to the fragment mix, we prepared the plasmid backbone for the Golden Gate reaction. We created a version of the yeast display plasmid with the counter-selection marker ccdB in place of the three-fragment sequence, with flanking BsaI sites. We performed Golden Gate cloning using BsaI-HFv2 (NEB #R3733) following the manufacturer recommended protocol, with a 7:1 molar ratio of fragment inserts to plasmid backbone.

The transformation of the CR6261 library into E. coli was conducted in a similar fashion to that of CR9114, except that 8x50 μL cell aliquots were transformed, and 600,000 colonies were pooled for plasmid midiprep.

We then transformed the CR9114 plasmid library into EBY100 cells by standard high-efficiency protocols (Gietz and Schiestl, 2007). We recovered transformants in molten SDCAA (1.71 g/L YNB without amino acids and ammonium sulfate (Sigma-Aldrich #Y1251), 5 g/L ammonium sulfate (Sigma-Aldrich, St. Louis, MO, #A4418), 2% dextrose (VWR #90000–904), 5 g/L Bacto casamino acids (VWR #223050), 100 g/L ampicillin (VWR # V0339)) containing 0.35% SeaPrep agarose (VWR #12001–922) spread into a thin layer (about 1 cm deep). The mixture was allowed to set on ice for an hour, after which it was kept for 48 hr at 30°C to allow for dispersed growth of colonies in 3D. From five such flasks, we obtained ∼700,000 colonies (>10 times the library diversity). After mixing the flasks thoroughly by shaking for 1 hr, we grew cells in 5 mL tubes of liquid SDCAA for five generations and froze the saturated culture in 1 mL aliquots with 5% glycerol.

The CR6261 yeast library was generated in a manner identical to that of CR9114, except that ∼60,000 colonies were pooled due to the smaller library size.

In addition to the full library, for both CR9114 and CR6261 we assayed a small number of variants by low-throughput flow cytometry for Tite-Seq validation. Any individual variant in the library can be produced in the same manner as described above: we simply selected the DNA duplex fragments corresponding to each desired variant and set up an individual Golden Gate reaction. The resulting assembled plasmid was transformed into E. coli, mini-prepped, and transformed into EBY100 in the same manner as described above. We verified the sequence identity of each variant by Sanger sequencing the entire scFv sequence.

We also constructed isogenic strains for validation experiments with genotypes that are not present in the full library. For CR9114, to test the impact of excluding mutations A24S and E46D, we constructed a strain containing the remaining 16 somatic mutations by cloning a gene block of the corresponding V_H sequence into the germline CR9114 pCT302 plasmid via Gibson Assembly (Figure 1—figure supplement 7). For CR6261, we similarly constructed a strain with the Q6E, L50P, and V101M mutations reverted.

CR9114 was isolated from pooled PBMC from three donors who had received the trivalent 2006 influenza vaccine (Throsby et al., 2008; Dreyfus et al., 2012), which contained A/New Caledonia/20/1999 (H1N1), A/Wisconsin/67/2005 (H3N2), and B/Malaysia/2506/2004 (Victoria lineage) (Ekiert et al., 2011). CR6261 was isolated from pooled PBMC from the same three donors, plus an additional seven donors who did not receive the vaccine (Throsby et al., 2008). Because PBMC were isolated only 7 days after vaccination, although it is possible that CR6261 and CR9114 matured in response to these specific antigens, it is more likely that the vaccine elicited memory recall of these antibodies (Victora and Nussenzweig, 2012). Here, we chose to measure binding affinities to diverse antigens spanning the range of breadth for both CR9114 and CR6261. CR9114 neutralizes strains across influenza A (groups 1 and 2) and influenza B, so we measured affinities to one strain from each of these groups, and selected vaccine-like strains: A/New Caledonia/20/1999 (H1N1), A/Wisconsin/67/2005 (H3N2), and B/Ohio/1/2005 (Victoria lineage). CR6261 neutralizes strains across influenza A group 1, thus we measured affinities to two strains from distinct subtypes within group 1: A/New Caledonia/20/1999 (H1N1) and A/Hong Kong/1073/1999 (H9N2). We note that CR9114 indeed binds A/Hong Kong/1073/1999 (H9N2) (Dreyfus et al., 2012), but CR9114 variant affinities for this strain were not measured here, as we prioritized measurements to antigens that span the breadth of each antibody.

Trimeric hemagglutinin (HA) antigen was produced as previously described (Ekiert et al., 2011; Dreyfus et al., 2012; Margine et al., 2013). Briefly, the HA ectodomain (Influenza A: residues 11–329 of HA1 and 1–176 of HA2 (H3 numbering); Influenza B: residues 1–523) of Influenza A/New Caledonia/1999 H1, Influenza A/Hong Kong/1999 H9, Influenza A/Wisconsin/2005 H3, and Influenza B/Ohio/2005, with N-terminal gp67 signal peptide and C-terminal biotinylation site (GGGLNDIFEAQKIEWHE), thrombin cleavage site, trimerization domain and His₆ tag, were cloned into pFastbac (plasmid maps in Supplementary files 8–11). Recombinant bacmid was generated using the ThermoFisher Bac-to-Bac kit (ThermoFisher, Waltham, MA, #10359016). Sf9 cells (ThermoFisher #B82501, not authenticated but verified to be mycoplasma-negative) were then transfected (ThermoFisher #A38915, not authenticated but verified to be mycoplasma-negative) with the resulting bacmids, and P0 HA-baculovirus was harvested 7 days post-transfection by clarifying viral supernatant at 1000 x g for 10 min. HA-baculovirus was then amplified twice by successively infecting 187 million Sf9 cells with 100 μL of viral supernatant and incubating in a humidified incubator at 28°C for 12 days. To induce HA expression, 105 million High-Five cells (ThermoFisher #B85502) were resuspended with 15 mL P2 HA-baculovirus, incubated for 20 min at room temperature, and then transferred to a 1 L non-baffled flask with 200 mL Corning Express-Five media (ThermoFisher #10486025) supplemented with 18 mM L-glutamine (VWR #45000–676). Expression cultures were incubated in a shaking incubator at 28C and 110 rpm for 48 hr, after which HA-containing media was clarified by spinning first at 1000 x g for 5 min at 4°C, and then by spinning the resulting supernatant again at 4000 x g for 20 min at 4C. The clarified media was then dialyzed into PBS (VWR #45000–448) by performing 4 x 2 hr 10-fold buffer exchanges to remove metal chelators from culture media. Dialyzed media was then combined with 10 mL equilibrated NiNTA resin (ThermoFisher #R90101), gently shaken for 3 hr at 4°C, and loaded onto a column. The resin was washed first with 15 column volumes Wash Buffer 1 (50 mM Tris pH 8 at 4C, 300 mM KCl, 10 mM imidazole) and subsequently with 15 column volumes Wash Buffer 2 (50 mM Tris pH 8 at 4C, 300 mM KCl, 20 mM imidazole). HA was eluted from the resin after 10 min incubation with Elution Buffer (50 mM Tris pH 8 at 4°C, 300 mM KCl, 250 mM imidazole). HA was then buffer exchanged into PBS using 10 KDa Amicon Ultra Centrifugal Filters (Millipore Sigma, Burlington, MA #UFC901008) and concentrated to at least 1 mg/mL for downstream biotinylation.

BirA was expressed and purified as previously described (Ekiert et al., 2011). Briefly, pET21a-BirA expression plasmid (Howarth et al., 2005) (kind gift from Alice Ting; Addgene #20857) was transformed into BL21 (DE3). Transformed BL21 cells were grown in 4 L baffled flasks with 1 L low-salt LB medium (5 g/L NaCl, 5 g/L yeast extract (VWR #90000–722), 10 g/L tryptone (VWR #90000–286)) at 37°C to an OD (600 nm) of ∼0.8. The culture was then moved into cold water to bring it to 23°C, IPTG was added to a final concentration of 1 mM, and the culture was incubated at 23C for ∼16 hr. The culture was then harvested by centrifugation (3000 x g, 10 min), resuspended in 30 mL lysis buffer (50 mM Tris pH 8 at 4C, 300 mM KCl, 10 mM imidazole, EDTA-free protease inhibitor cocktail tablet (Millipore Sigma #4693159001)), lysed by sonication (Branson Sonifier 450), and shaken at 4°C for 30 min. Lysate was clarified by spinning at 25,000 x g for 1 hr, and then the supernatant was incubated with 5 mL NiNTA resin at 4°C for 3 hr with gentle shaking. The resin was pelleted by spinning at 500 x g for 5 min and washed twice by gentle shaking with 35 mL lysis buffer at 4°C for 30 min. Protein was eluted with 20 mL Elution Buffer (50 mM Tris pH 8 at 4°C, 300 mM KCl, 250 mM imidazole), buffer exchanged into Storage Buffer (50 mM Tris pH 7.5 at 4°C, 200 mM KCl, 5% glycerol) using 10 KDa Amicon Ultra Centrifugal Filters (Millipore Sigma #UFC901008), flash frozen in liquid nitrogen, and stored in single-use aliquots at −80°C.

Purified hemagglutinin was biotinylated as previously described (Fairhead and Howarth, 2015; Ekiert et al., 2011). Briefly, 100 μL HA (> 1 mg/mL) was incubated with 0.5 μL 1 M MgCl₂, 2 μL 100 mM ATP, 0.5 μL 50 mM biotin, and 2.5 μL BirA (10 mg/mL). This was mixed by gentle pipetting and incubated at 30C with gentle rocking. After 1 hr incubation, equivalent amounts of ATP, BirA, and biotin were added to the reaction, which was incubated for an additional hour at 30°C. Following the 2 hr incubation, the 100 μL reaction was exchanged thrice into 15 mL PBS using a 50 KDa MWCO buffer exchange column (Millipore Sigma #UFC905008). The degree of biotinylation was then assessed by a streptavidin gel-shift assay, as previously described (Fairhead and Howarth, 2015). Briefly, 10-fold molar excess streptavidin (Millipore Sigma #189730) was added to 4 μg biotinylated HA and incubated at room temperature for 5 min prior to running on SDS-PAGE. Gels were transferred to nitrocellulose membranes and probed with mouse anti-His monoclonal antibodies (ThermoFisher #R930-25) and Goat-anti-mouse secondary antibodies (LiCor, Lincoln, NE, Cat#925–32210). HA was verified to be > 80% biotinylated by densitometry.

On day 1, yeast scFv libraries, as well as germline and somatic clonal strains, were thawed by inoculating 5 mL SDCAA (1.71 g/L YNB without amino acids and ammonium sulfate (Sigma-Aldrich #Y1251), 5 g/L ammonium sulfate (Sigma-Aldrich #A4418), 2% dextrose (VWR #90000–904), 5 g/L Bacto casamino acids (VWR #223050), 100 g/L ampicillin (VWR # V0339)) with 150 μL glycerol stock (saturated culture with 5% glycerol) and rotated at 30°C for 20 hr. On day 2, yeast cultures were back-diluted to OD600 = 0.2 in 5 mL SDCAA and rotated at 30°C for approximately 4 hr, or until reaching log phase (OD600 = 0.4–0.8). 1.5 mL log-phase cells were then pelleted, resuspended in 4 mL SGDCAA (1.71 g/L YNB without amino acids and ammonium sulfate (Sigma-Aldrich #Y1251), 5 g/L ammonium sulfate (Sigma-Aldrich #A4418), 0.2% dextrose (VWR #90000–904), 1.8% galactose (Sigma-Aldrich #G0625), 5 g/L Bacto casamino acids (VWR #223050), 100 g/L ampicillin (VWR #V0339)), and rotated at room temperature for 20–22 hr.

On day 3, 20–22 hr post-induction, yeast cultures were pelleted, washed twice with 0.1% PBSA (VWR #45001–130; GoldBio, St. Louis, MO, #A-420–50), and resuspended to an OD600 of 1. A total of 700 μL of OD1 yeast cells were labeled with biotinylated HA at each of eleven antigen concentrations (half-log increments spanning 1 pM – 100 nM for H1 and H9, and 10 pM – 1 μM for H3 and influenza B, as well as no HA), with volumes adjusted such that the number of antigen molecules was in 10-fold excess of antibody molecules (assuming 50,000 scFv/cell). Yeast-HA mixtures were rocked at 4°C for 24 hr.

On day 4, yeast-HA complexes were pelleted by spinning at 3000 x g for 10 min at 4°C, washed twice with 5% PBSA + 2 mM EDTA, and simultaneously labeled with Streptavidin-RPE (1:100, Thermo Fisher #S866) and anti-cMyc-FITC (1:50, Miltenyi Biotec, Somerville, MA, #130-116-485) at 4°C for 45 min. Following secondary labeling, yeast were washed twice with 5% PBSA + 2 mM EDTA, and left on ice in the dark until sorting.

Yeast were sorted on a BD FACS Aria Illu, equipped with 405 nm, 440 nm, 488 nm, 561 nm, and 635 nm lasers, and an 85 micron fixed nozzle. Prior to sorting, single-color controls were used to compensate for the minimal FITC overlap with PE. Single cells were gated by FSC vs SSC, and then this population was sorted either by expression (FITC) or by expression and binding (PE). For all sorts, at least ten-fold excess of the library diversity was sorted (∼1.6 million cells for CR9114; ∼500,000 cells for CR6261). For the expression sorts, singlets were sorted into eight equivalent FITC log-spaced gates. For the binding sorts, FITC-positive cells were sorted into 4 PE bins (the PE-negative population comprised bin 1, and the PE-positive population was split into three equivalent log-spaced bins 2–4; see Figure 1—figure supplement 6). Polypropylene collection tubes were coated and filled with 1 mL YPD supplemented with 1% BSA and placed on ice until recovery. Sorted cells were pelleted by spinning at 3000 x g for 10 min, and supernatant was removed by pipette to avoid disturbing the pellets. Pellets were then resuspended in 4 mL SDCAA, a small amount was plated on SDCAA-agar to quantify recovery efficiency, and cultures were rocked at 30°C until reaching late-log phase (OD600 = 0.6–1.2).

A total of 1.5 mL of late-log yeast cultures were pelleted and scFv plasmid was extracted using Zymo Yeast Plasmid Miniprep II (Zymo Research # D2004), per the manufacturer’s instructions, and eluted in 10 μL elution buffer. Heavy-chain amplicon sequencing libraries were prepared by a two-step PCR as previously described (Nguyen Ba et al., 2019). In the first PCR, unique molecular identifiers (UMI), inline indices, and partial Illumina adapters were appended to the heavy chain through 3–5 amplification cycles to minimize PCR amplification bias. In the second PCR, the remainder of the Illumina adapter and sample-specific Illumina i5 and i7 indices were appended through 35 amplification cycles (see Supplementary file 1 for primer sequences). The first PCR used 5 μL plasmid DNA as template in a 25 μL reaction volume, with Q5 polymerase according to the manufactuer’s instructions (NEB # M0491L), and was incubated in a thermocycler with the following program: 1. 60 s at 98°C, 2. 10 s at 98°C, 3. 30 s at 66°C, 4. 30 s at 72°C, 5. GOTO 2, 2-4x, 6. 60 s at 72°C. PCR products were then combined with carrier RNA and purified by 1.1X Aline beads (Aline Biosciences #C-1003–5), and eluted in 35 μL elution buffer. 33 μL of the elution was used as input for the second PCR, in a total volume of 50 μL using Kapa polymerase (Kapa Biosystems #KK2502) according to the manufacturer’s instructions, and incubated in a thermocycler with the following program: 1. 30 s at 98°C, 2. 20 s at 98°C, 3. 30 s at 62°C, 4. 30 s at 72°C, 5. GOTO 2, 34x, 6. 300 s at 72°C. The resulting sequencing libraries were purified by 0.85X Aline beads, amplicon size was verified to be ∼500 bp by running on a 1% agarose gel, and amplicon concentration was quantified by a fluorescent DNA-binding dye (Biotium, Fremont, CA, #31068, per manufacturer’s instructions). Amplicons were then pooled for each gate according to the number of sorted cells to ensure even sequencing coverage. The pool was further size-selected by a two-sided Aline bead cleanup (0.55–0.85X), and the final pool size was verified by Tapestation 5000 HS and 1000 HS. Final sequencing library concentration was determined by Qubit fluorometer and sequenced on an Illumina NovaSeq S2 or Miseq v3 (2x150) with 5% PhiX.

We first processed our raw sequencing reads to identify and extract the indexes and mutational sites, discarding priming regions and the constant regions between mutations. To do so, we developed custom Python scripts using the approximate regular expression library regex (Barnett, 2013), which allowed us to handle complications in sequence parsing that arise from the irregular lengths of the indices and from sequencing errors. We accept sequences that match the entire read (with no restrictions on bases at mutational sites) within the following mismatch tolerances: two mismatches in the multiplexing index, two mismatches in the priming site, and 15 substitution mismatches within the 170 bases of constant antibody sequence.

We then examine the mutational sites to call germline or somatic alleles, producing binary genotypes (‘0’ for germline or ‘1’ for somatic at each position). We require the exact germline or somatic sequence at every site: if there are any substitution errors in any of the mutation sites, the entire read is rejected. While it is possible to perform error correction based on Hamming distance to rescue reads with a few substitution errors, we find that on average only <8% of reads per sample contain any errors, and so we adopt the conservative approach of requiring perfect matching.

We next discarded sequencing reads with any mismatched indices (four total indices from the two PCR reactions), as well as reads with duplicate UMI sequences. Counts for each genotype were then tabulated, producing the final counts used for binding affinity inference (see below). On average, across all antigens and replicates, we obtain a mean coverage of ∼350 for CR9114 and ∼950 for CR6261, and a median coverage of ∼250 for CR9114 and ∼900 for CR6261.

Induction of scFv surface display, primary labeling, and secondary labeling of isogenic strains were performed identically to the Tite-Seq assay, except yeast cell and antigen volumes were scaled down by a factor of 10. Yeast cell FITC (scFv expression) and R-PE (HA binding) fluorescence intensity was assayed on a BD LSR Fortessa equipped with four lasers (440, 488, 561, and 633 nm). The equilibrium binding affinities (KD) for each variant are inferred by fitting the log of a Hill function to the mean log R-PE fluorescence of scFv-expressing (FITC+) singlet yeast cells:(1)meanlogfluorescence=log10⁡(Ascc+KD,s+Bs),where c is the antigen concentration in molar units, As is the increase in fluorescence due to saturation with antigen, Bs is the background fluorescence, and KD,s is the equilibrium binding affinity. All isogenic measurements were performed in two to three biological replicates; see Figure 1—source data 3 for isogenic -log10⁡KD.

To infer binding affinities using a simple mean-bin approach (Peterman and Levine, 2016), we incorporate sequencing data (the unique read counts of each genotype sequence s in bin b at concentration c, Rb,s,c) with flow cytometry data (the mean and standard deviation of log₁₀-fluorescence of sorted cells in each bin b at concentration c, Fb,c and σFb,c respectively, and cell counts for each bin b at each concentration c, Cb,c).

The mean log-fluorescence of each genotype sequence at each of the 12 antigen concentrations is calculated as:(2)F¯s,c=∑bFb,c⁢pb,s|c,where pb,s|c is the probability a cell with sequence s would be sorted into bin b at concentration c is estimated from the sequencing read counts as:(3)pb,s|c=Rb,s,c∑s′Rb,s′,c⋅Cb,c∑b′(Rb′,s,c∑s′Rb′,s′,c⋅Cb′,c),in other words, the fraction of total reads in the bin corresponding to sequence s, scaled by the number of sorted cells in that bin, normalized over the four bins for each concentration.

The uncertainty in the mean bin inference was propagated as:(4)δ⁢F¯s,c=∑b(δ⁢Fb,c2⁢pb,s|c2+Fb,c2⁢δ⁢pb,s|c2).

Here, δ⁢Fb,c represents the spread in log-fluorescence values of cells sorted into the same bin b. While we could estimate this value using the bin width, in practice we find that the distribution of cell log-fluorescence values in a bin is far from uniform across the bin width. The distribution is often not normal either, but we find that approximating δ⁢Fb,c≈σFb,c, or the standard deviation in log₁₀-fluorescence of cells sorted into bin b at concentration c, adequately captures the typical variation. The error in pb,s|c arises largely from the sampling process of sequencing, which can be approximated as a Poisson process when read counts are relatively high. This gives(5)δpb,s|c=pb,s|cRb,s,c.

Thus, δ⁢F¯s,c can be written as(6)δ⁢F¯s,c=∑b(σFb,c2⁢pb,s|c2+Fb,c2⁢pb,s|c2Rb,s,c).

The equilibrium binding affinities (KD) for each variant are inferred by fitting the logarithm of a Hill function to the resulting mean log₁₀-fluorescence across the twelve antigen concentrations:(7)F¯s,c=log10⁡(As⁢cc+KD,s+Bs),where c is the antigen concentration in molar units, As is the increase in fluorescence due to saturation with antigen, Bs is the background fluorescence, and KD,s is the binding affinity. Fitting was performed with the curve_fit function of the Python package scipy.optimize. Reasonable bounds on the values of A(10³-10⁵), B(10⁰-10³) and K_D(10^-14-10^-5) were imposed. Sequences leading to a failed optimization were deemed ‘non-binding’.

Inferred KD outside of the titration boundaries were then pinned to the boundaries (10-12 and 10-7 for H1 and H9; 10-11 and 10-6 for H3 and FluB). Inferred KD with high error (standard deviation of log10⁡KD>1.0) or resulting from a poor fit (r2<0.8) were removed from the data set prior to averaging -log10⁡KD values across biological replicates.

We also explored an alternative maximum-likelihood framework for inferring binding affinities (see Appendix 1), but found it to be less accurate than the mean-bin approach when compared to isogenic flow cytometry measurements. Thus, we restricted our analysis to the simpler and more robust mean-bin inference presented here.

To represent the high-dimensional binding affinity landscape in two dimensions, we use a force-directed graph layout approach. Each sequence in the antibody library is a node, connected by edges to its single-mutation neighbors (sequences that can be reached by one additional somatic mutation). An edge between two sequences s and t is given the weight(8)ws,t=10.01+|log10⁡(KD,sag)−log10⁡(KD,tag)|,where KDag represent binding affinities to a particular antigen, ag. In the layouts shown in the main text, we use binding affinities to H1 for both CR6261 and CR9114. In force-directed layouts, edge weights correspond to the effective spring constant that tends to pull nodes closer together. Thus, a mutation from sequence s to t that has little impact on binding will cause that edge weight to be large, and the nodes will be pulled strongly together. A mutation from sequence s to t that causes a large difference in binding affinity (positive or negative) to the antigen will reduce the edge weight, moving those nodes further apart. 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.

As noted above, antibody libraries were sorted into eight bins along the FITC-A fluorescence axis (where FITC-A fluorescence is proportional to expression), each comprising 12.5% of the total singlet population (Figure 1—figure supplement 6). The mean expression log-fluorescence was computed for each variant using the corresponding variant counts and fluorescence data, as described above for the mean-bin KD inference. These expression values were then averaged across all biological replicates for each antibody (nine replicates for CR9114, six replicates for CR6261), and correlation between biological replicates, as well as with -log10⁡KD values, are illustrated in Figure 1—figure supplement 5. For the isogenic flow cytometry measurements, variant expression was computed as the mean log FITC-A fluorescence.

To infer specific mutational effects, we begin with simple linear models where the effects of mutations (and mutation combinations) add to produce phenotypes. Our log-transformed phenotypes for each variant s, ys=-log10⁡(KD,s), are proportional to free-energy changes, and thus a natural null expectation is that they combine additively (Wells, 1990; Olson et al., 2014) (although we also consider nonadditive epistatic interactions between individual loci here, and analyze the effects of an overall nonlinear transformation of this data in Appendix 2). Our additive-only model is(9)ys=β0+∑i=1Lβi⁢xi,s+ε,

 where L is the number of mutations for a given antibody, β0 is an intercept term, βi is the effect of the mutation at site i, xi,s is the genotype of variant s at site i, and ε represents independently and identically distributed errors. Our general linear interaction models are(10)ys=β0+∑iβi⁢xi,s+∑i<jLβi⁢j⁢xi,s⁢xj,s+∑i<j<kLβi⁢j⁢k⁢xi,s⁢xj,s⁢xk,s+…+εwhere βi⁢j represent second-order interaction coefficients between distinct sites i and j, βi⁢j⁢k represent third-order interaction coefficients, and so on up to the desired maximum order of interaction.

There are multiple alternative coding systems for the binary genotypes xi,s that affect the values of inferred effects β as well as their interpretation. Two common choices are (1) xi,s∈{0,1}, often called ‘biochemical’ or ‘local’ epistasis, and (2) xi,s∈{-1,1}, often called ‘statistical’ or ‘ensemble’ epistasis (Poelwijk et al., 2016). These frameworks are equivalent and related by a simple linear transformation, but the values of the coefficients vary between frameworks and have different interpretations. For ease of interpretation, in the Main Text and Figures we always show results obtained from inference in the biochemical epistasis framework. In Appendix 2, we discuss the differences between these two frameworks, and present results from inference in the statistical epistasis framework.

For an antibody with L mutations, there are L possible orders of interactions, with a total of 2L epistatic coefficients β. From a measurement of y for all 2L possible sequences, there is a simple linear transformation to calculate the resulting 2Lβ parameters (Poelwijk et al., 2016). This is a simple and fast approach to the calculation of epistasis that is widely used (Sailer and Harms, 2017b; Poelwijk et al., 2019), and we explore this approach in Appendix 2. However, we may instead wish to restrict our model to a lower order and examine whether it can explain the data with far fewer than 2L parameters, as a conservative approach to detecting high-order epistasis.

Specifically, we truncate the model above at a maximum order n and then fit and evaluate the resulting model. We begin with n=1 and continue to increase n until the optimal model has been identified. There are multiple strategies for selecting between models with different numbers of parameters, such as AIC and BIC; here we take a cross-validation approach. For each fold, we hold out 10% of the dataset, train models at each maximum order on the remaining 90%, and evaluate the prediction performance (R2) of the model on the held-out test set. After averaging the performances across all 10 folds for each truncated model, we choose the order that maximizes the test set performance as the optimal maximal order of interaction. We then re-train the model truncated at this order on the full dataset to obtain the final coefficients. We find that the optimal model identified by cross-validation for each antibody-antigen pair satisfies p<N by ∼1 order of magnitude, where p is the total number of model coefficients and N the number of data points with measurable binding affinity. This gives confidence that our parameter estimates are well constrained by the data, even in the absence of other regularization (such as Lasso or Ridge regularization approaches).

To train a model of given order on a set of sequences, we use ordinary least squares (OLS) regression with the Python package statsmodels. From this, we obtain the coefficient values β with their standard errors and p-values. To define significance of coefficients, we use a p-value cutoff of 0.05 with Bonferroni correction by the total number of model parameters. Coefficients, standard errors, p-values, and Bonferroni-corrected 95% confidence intervals are reported in Figure 1—source data 1 and 2. We also predict phenotypes y^ for each sequence from the coefficients and use these values in Figure 5A,B.

For CR9114 binding to influenza B, the number of sequences used for inference is far fewer than other antibody-antigen pairs (N=256), due to the large number of required mutations. We therefore use a fivefold rather than 10-fold split to reduce the test set noise. Nevertheless, the cross-validation procedure identifies a first-order (additive) model as optimal, due to the smaller sample size.

To examine the structural context of linear and pairwise coefficients, we performed three simple analyses. (1) First, we used ChimeraX (Pettersen et al., 2021) to calculate the buried surface area between HA and each mutated residue in CR9114 and CR6261, using the measure buriedarea function and the default probeRadius of 1.4 angstroms to approximate a water molecule. We plot this ‘contact surface area’ vs the linear effect of the corresponding mutation on HA binding (Figure 2C; Figure 2—figure supplement 1A). (2) We used PyMol (Schrodinger, LLC, 2015) to count the number of HA residues within six angstroms of each antibody mutation site. Six angstroms was chosen as an upper limit to capture potential antibody-antigen interactions (Bondi, 1964; Baker and Hubbard, 1984; Israelachvili and Pashley, 1982; Ekiert et al., 2009; Dreyfus et al., 2012), although we note that this analysis is robust to other distance thresholds. (3) We also used PyMol to measure the distances between α-carbons for all mutation pairs, and plotted these distances against the corresponding pairwise epistatic terms (Figure 2F; Figure 2—figure supplement 1B). We note that each of these analyses were performed with co-crystal structures of the somatic antibodies with HA (PDB ID: 4FQI (CR9114 –H5; CR9114 –H1 crystal structure not available) Dreyfus et al., 2012; 4FQY (CR9114 –H3) Dreyfus et al., 2012; 3GBN (CR6261 –H1) Ekiert et al., 2009).

To study the likelihood of various mutational pathways leading from the germline to the somatic sequence, we must assume a selection model. Selection in germinal centers is considerably more complex than in classical population genetics models, involving spatial structure, changing population sizes, and T-cell-mediated selection, among other factors (Mesin et al., 2016). Capturing these aspects in quantitative models is an active field of research (Amitai et al., 2017). However, here we wish to adopt an extremely simple model of selection as a first step in understanding the impacts of the binding affinity landscape on antibody selection, with the goal of understanding the implications of the expectation that mutational steps become more probable as their effect on binding affinity becomes more positive. Combining the more realistic models of immune selection with our detailed characterization of mutational effects on antigen binding affinity remains an interesting avenue for future work.

Here, we restrict to the weak-mutation regime where mutation fixation events occur independently of one another. Selection proceeds as a Markov process, where the population is characterized by a single sequence that acquires a single mutation at each discrete step (McCandlish, 2011). We choose a simple form for the fixation probability of a mutation from sequence s to sequence t, as discussed below. This then determines the transition probability for the population to move from s to t. We assume that sequences cannot back-mutate (i.e. a residue changing from the somatic allele to the germline allele), and do not acquire multiple mutations in the same step. The absence of back-mutation is justified by the relatively large number of possible mutation sites compared to the total number of mutation events.

We define the transition probability of a single mutational step from the classical fixation probability for a mutation with selection coefficient σ in a population of size N(Kimura, 1962):(11)pstep(σ,N)=1−e−σ1−e−Nσ.

Here, we define the selection coefficient σ to be proportional to the difference in log binding affinities to a particular antigen between the two sequences s and t:(12)σ=γ⁢Δs,tag=γ⁢(-log10⁡KD,tag-(-log10⁡KD,sag)).

This model has two tunable parameters: N represents the effective population size and γ represents how strongly differences in binding affinity impact fitness. We chose three parameter values to span a range of selection strengths (see Figure 5—figure supplement 1): moderate, with N=1000 and γ=1; weak, with N=20 and γ=0.5; and strong, with N→∞ and γ→∞ such that pstep reduces to a step function (one if Δ>0 and 0 otherwise). These three models all show similar results, with differences between selection scenarios becoming more exaggerated with stronger selection and less exaggerated with weaker selection, as expected (see Figure 5—figure supplement 1).

From the fixation probabilities for a given parameter regime, we have the transition probability (up to a constant factor) for all sequences s,t over all antigens ag,(13)Ps,tag={pstep⁢(Δs,tag,γ,N)if t has one more somatic mutation than s0otherwise,which we use for all the calculations described below for results presented in Figure 5 and supplements.

It is particularly useful to store the probabilities Ps,tag as (sparse) transition matrices Pag of dimension 2N×2N for each antigen, where entries are nonzero only where sequence t has one more somatic mutation than s.

First, we wish to obtain a measure of total probability for a particular antigen scenario, as shown in Figure 5E,F. We calculate this by computing the matrix product over all mutational steps i for a particular sequence of antigen contexts {ag1,…,agL}:(14)𝒫t⁢o⁢t=∑paths(∏stepsPstep)=[∏i=1LPagi]sg,ss,where [⋅]s,s′ corresponds to taking the matrix element in the row corresponding to variant s and column corresponding to variant s′. In the right-most term, the products are matrix operations and sg, ss are respectively the indices of the germline and somatic variants.

We note that the transition probabilities Pagi are not normalized at each step. In practice, this means that mutations are optional: many outcomes will not reach the somatic sequence and the likelihood encodes the probability of reaching the somatic state. This makes it possible to compare different scenarios, as some scenarios are more likely than others to reach the somatic state. However, because these values do not represent true probabilities — the units are arbitrary — they cannot be compared between antibodies or between selection models. The exception is for the strong scenario, where the total probability for each path is one if all steps are uphill (Δs,tag>0) and 0 otherwise. Thus, here 𝒫t⁢o⁢t has a natural interpretation as the total number of uphill paths. When we present results from the strong model (Figure 5C,D, Figure 5—figure supplement 1, and numbers of uphill paths for H1-only scenarios as discussed in the text), we represent uphill path numbers on a linear scale without log-transforming.

Although there are many possible antigen exposure scenarios, we restrict our analysis to several classes. First, in single-antigen scenarios, all steps i use the same antigen. Second, for sequential scenarios, antigen exposures must occur in non-repeating segments (for example, H1 - H3 - H1 is not allowed), although we consider all possible lengths and orders of segments.

Mixed scenarios are more complicated, as we do not fully understand the nature of B cell interactions with multiple antigens in the same germinal center (Wang et al., 2015; Wang, 2017; Kuraoka et al., 2016). One option is to assume that the B cell engages the antigen for which it has the highest affinity and define Δ by the maximum binding affinity across all possible antigens at each step, but this definition would trivially imply that the mixed scenario has the highest probability. Instead, we choose two alternatives: first, ‘average’ mixed, where we assume the B cell engages all antigens and use the average binding affinity change over all three (for CR9114) or two (for CR6261) antigens, Δmixed=1Nag⁢∑agΔag; and second, ‘random’ mixed, where we assume the B cell randomly engages a single antigen and hence the antigen at each mutational step is chosen randomly. For the latter definition, we calculate 𝒫t⁢o⁢t as described above for 1000 randomly drawn scenarios and average the resulting log probability. When we illustrate mutational paths and mutation orders, we choose a representative scenario (with close to median probability) from the 1000 random draws.

We estimate the error of these probabilities by bootstrapping. Specifically, for 10 bootstrap iterations, we resample each binding affinity -log10⁡KD,sag from a normal distribution according to its value and standard deviation. We then recalculate the total probability 𝒫t⁢o⁢t, average over the 10 values to obtain mean and s.e.m. values, and transform by the natural log for plotting, as shown in Figure 5 and Figure 5—figure supplement 1. We note that for the strong selection scenario (where probabilities represent total numbers of uphill paths), values are not log-transformed, and many scenarios have total path numbers of exactly zero. We refrain from studying the ‘average’ mixed scenario for strong selection because it is essentially equivalent to choosing the antigen with maximum improvement: the quantitative effect of averaging is undone when the transition probability is binarized. For CR6261, all mutations at the first mutational step are neutral (with the exception of one mutation that improves affinity for H1 only), and so we allow all mutations with equal probability for the first step in the strong selection model.

Next, to identify the most likely paths under a given exposure scenario, we reframe this Markov process as a directed weighted graph. Each sequence s is a node, and a directed edge exists toward all sequences t that can be reached by one additional somatic mutation. The edge weight is calculated from the transition probability, ws→t=-log⁡(Ps,tag+ϵ), where ϵ is an extremely small value to ensure weights are finite. In this graph framework, we can use fast algorithms to obtain the ‘shortest’ paths from the germline to the somatic node (those for which the sum of weights is lowest, that is the total probability is highest). Specifically, we use the shortest_simple_paths function from the Python package networkx (Hagberg et al., 2008) to compute the k shortest paths, as shown in Figure 5G,H. This method is exact and uses the algorithm described in Yen, 1971.

Next, we wish to obtain the probability that a mutation at site m happened at a specific step j (Figure 5I,J). As we are focusing on one antigen context, we can normalize the transition matrices and define:(15)P~s,tag=Ps,tag×(∑tPs,tag)-1,if Ps,tag≠0 and 0 otherwise. We can further restrict the transition matrix at step j, P~agj, to have nonzero the mutation that occurs is at a particular residue α, P~αagj. The total relative probability for that site at that mutational step under an antigen exposure scenario is then(16)℘j,α=[(∏i=1j−1P~agi)⋅P~αagj⋅(∏i=j+1LP~agi)]sg,ss,where, again, products are matrix operations. Because a sequence of L steps starting from the germline can only lead to the somatic state, P~ verifies [∏i=1LP~agi]sg,ss=1. With the relation ∑αP~αagj=P~agj this implies that these probabilities are already normalized: ∑α𝒫j,α=1.

Finally, we wish to determine the total probability of each variant (Figure 5—figure supplement 2), that is the sum of probabilities of all paths passing through that variant, for a given selection scenario. For a variant s that contains j somatic mutations, we calculate(17)𝒫s=([∏i=1jP~agi]sg,s)⋅([∏i=j+1LP~agi]s,ss),where the first term is the probability of reaching sequence s at mutational step j, and the second term is the probability of reaching the somatic sequence after passing through sequence s. When representing this number we add an additional normalisation factor, 𝒫s′=𝒫s×nj, where nj=(Lj) is the number of sequences with j mutations, so that variants with different numbers of mutations have comparable values. 𝒫s′ thus represents the ratio of the probability in a selective model to the probability in a neutral model (which is 1/nj). Thus, sequences with log10⁡(𝒫s′)>0 are favored by the given selection scenario, and those with log10⁡(𝒫s′)<0 are disfavored, as shown in Figure 5—figure supplement 2 for moderate selection under the optimal sequential scenario.