We studied HIV-1 evolution accompanying the development of bnAbs in two donors, CH505 and CH848, enrolled in the Center for HIV/AIDS Vaccine Immunology 001 acute infection cohort (Tomaras et al., 2008). CH505 developed the CD4 binding site-targeting bnAb CH103, which was first detectable 14 weeks after HIV-1 infection (Liao et al., 2013). CH103 maturation was found to be associated with viral escape from another antibody lineage, CH235, that ultimately developed significant breadth (Gao et al., 2014; Bonsignori et al., 2016). CH848 developed a bnAb, DH270, targeting a glycosylated site near the third variable loop (V3) of Env (Bonsignori et al., 2017). Similar to the bnAb development process in CH505, escape from ‘cooperating’ DH272 and DH475 lineage antibodies was observed to contribute to the maturation of DH270 (Bonsignori et al., 2017).

To quantify HIV-1 evolutionary dynamics, we sought to infer a fitness model that best explained the changes in the genetic composition of the viral population observed in each individual over time. In recent years, a wide variety of approaches have been developed to infer the fitness effects of mutations from temporal genetic data (Bollback et al., 2008; Illingworth and Mustonen, 2011; Illingworth and Mustonen, 2012; Malaspinas et al., 2012; Mathieson and McVean, 2013; Lacerda and Seoighe, 2014; Feder et al., 2014; Steinrücken et al., 2014; Foll et al., 2014; Terhorst et al., 2015; Schraiber et al., 2016; Tataru et al., 2017; Paris et al., 2019; Mathieson and Terhorst, 2022; He et al., 2023; Sohail et al., 2022; Sohail et al., 2021; Shimagaki and Barton, 2025b; Gao and Barton, 2025; Lee et al., 2025). The vast majority of these methods focus on a single locus at a time, ignoring correlations between genetic variants at different loci. While the recombination rate of HIV-1 is high (Neher and Leitner, 2010; Romero and Feder, 2024), the virus also evolves under strong selection, which can lead to interference between clones with different beneficial mutations (Rouzine and Weinberger, 2013; Pandit and de Boer, 2014; Garcia and Regoes, 2014; Garcia et al., 2016; Williams and Pennings, 2020; Sohail et al., 2021). Thus, we applied MPL, an inference method that systematically accounts for genetic correlations (Sohail et al., 2022; Sohail et al., 2021; Shimagaki and Barton, 2025b; Gao and Barton, 2025; Lee et al., 2025), to estimate the fitness effects of HIV-1 mutations.

Here, we provide a brief overview of the key steps in the MPL approach to inferring selection. Further details are available in Methods and in prior work (Sohail et al., 2022; Sohail et al., 2021; Shimagaki and Barton, 2025b; Gao and Barton, 2025; Lee et al., 2025). First, we assume that the effect on viral fitness of each individual mutation 𝑎\begin{document}$a$\end{document} at each site 𝑖\begin{document}$i$\end{document} is quantified by a selection coefficient 𝑠𝑖(𝑎)\begin{document}$s_i(a)$\end{document}, with positive coefficients 𝑠𝑖(𝑎)>0\begin{document}$s_i(a) \gt 0$\end{document} denoting mutations that are beneficial for the virus and 𝑠𝑖(𝑎)<0\begin{document}$s_i(a) \lt 0$\end{document} denoting deleterious ones. We further assume that the cumulative fitness effects of mutations are additive, such that the overall fitness 𝐹𝛼\begin{document}$F^\alpha$\end{document} of a viral sequence α is given by the sum of the selection coefficients for all the mutations that it bears. That is, (1)Fα=F(gα)=1+∑i=1L∑a=1qsi(a)gi,aα,\begin{document}$$\displaystyle F^\alpha= F(g^\alpha) = 1 + \sum_{i=1}^L\sum_{a=1}^q s_i(a)g_{i,a}^\alpha\,,$$\end{document}

where 𝑔𝛼=((𝑔𝑖,𝑎𝛼)𝑎=1𝑞)𝑖=1𝐿\begin{document}$g^\alpha= ((g^\alpha_{i,a})_{a=1}^q)_{i=1}^L$\end{document} represents the viral sequence, with 𝑔𝑖,𝑎𝛼\begin{document}$g_{i,a}^\alpha$\end{document} equal to one if genotype α has allele 𝑎\begin{document}$a$\end{document} at site 𝑖\begin{document}$i$\end{document} and zero otherwise. 𝐿\begin{document}$L$\end{document} is the length of the genetic sequence, and 𝑞\begin{document}$q$\end{document} represents the number of genetic states (i.e., 𝑞=5\begin{document}$q=5$\end{document} for nucleotide sequences and 21 for amino acids, including gaps/deletions).

We assume that viral replication is stochastic, where viruses with higher fitness are more likely to spread infection to new cells than ones with lower fitness. Let us write the number of viruses of each genotype in the population at time 𝑡\begin{document}$t$\end{document} as 𝑛(𝑡)=(𝑛1(𝑡),𝑛2(𝑡),…,𝑛𝑀(𝑡))\begin{document}$n(t) = (n_1(t), n_2(t), \ldots, n_M(t))$\end{document}, where 𝑀=𝑞𝐿\begin{document}$M = q^L$\end{document} is the total number of possible genotypes. In our model, the probability of obtaining a new distribution of genotypes 𝑛(𝑡+1)\begin{document}$n(t+1)$\end{document} in the next generation is multinomial,(2)P(n(t+1))=N!∏α=1Mpα(n(t))nα(t+1)nα(t+1)!,\begin{document}$$\displaystyle P\left(n(t+1)\right) = N! \prod\limits_{\alpha= 1}^{M}{\frac{p_\alpha(n(t))^{n_\alpha(t+1)}}{n_\alpha(t+1)!}}, $$\end{document}

with N=∑αnα\begin{document}$N = \sum_\alpha n_\alpha$\end{document} the total population size. The probabilities 𝑝𝛼(𝑛(𝑡))\begin{document}$p_\alpha(n(t))$\end{document} are influenced by fitness, as well as mutation, recombination, and the current frequency of each genotype (see Methods). Essentially, viruses that are fitter than the current population average are more likely to increase in frequency, while ones that are less fit are likely to decline. Mutations and recombination introduce genetic variation into the population. The population size 𝑁\begin{document}$N$\end{document} determines the relative stochasticity of the dynamics, with smaller populations having more random fluctuations than larger ones.

Following this model, we can then quantify the probability of any evolutionary history of the viral population (i.e., the distribution of viral genotypes over time) as a function of the selection coefficients (Methods).

Assuming that the population size 𝑁\begin{document}$N$\end{document} is large and the selection coefficients are small (such that |𝑠|≪1\begin{document}$|s|\ll1$\end{document}), we can write an analytical expression for the selection coefficients that best fit the viral dynamics observed in data (Sohail et al., 2022; Sohail et al., 2021):(3)s^=(Cint+γI)−1[Δxint−Δuint].\begin{document}$$\displaystyle \begin{equation*} \hat{s} = \left(C_{\mathrm{int}} + \gamma I \right)^{-1} \left[\Delta x_{\mathrm{int}} - \Delta u_{\mathrm{int}} \right]\,. \end{equation*}$$\end{document}

Here, 𝐶int\begin{document}$C_{\mathrm{int}}$\end{document} is the covariance matrix of allele frequencies (i.e. the linkage disequilibrium matrix) integrated over time, and γ is a regularization parameter. The terms Δxint\begin{document}$\Delta x_{\mathrm{int}}$\end{document} and Δuint\begin{document}$\Delta u_{\mathrm{int}}$\end{document} represent the net change in allele frequency and the total expected change in allele frequency due to spontaneous mutations alone, respectively (see Methods for details). Intuitively, this expression says that net allele frequency changes that cannot be explained by mutations are likely due to selection, either on that specific allele or the associated genetic background, which is quantified by Cint\begin{document}$C_{\mathrm{int}}$\end{document}. Alleles that have large, rapid changes in frequency are more likely to be under strong selection than those with smaller, slower frequency changes. Beyond HIV-1 (Sohail et al., 2021), this approach has been successfully applied to study the evolution of SARS-CoV-2 (Lee et al., 2025) and experimental evolution in bacteria (Li and Barton, 2023; Li and Barton, 2024).

As described above, we used MPL (Sohail et al., 2022; Sohail et al., 2021) to infer the selection coefficients that best fit the viral dynamics observed in data from CH505 and CH848. While the great majority of HIV-1 mutations were inferred to be neutral (𝑠𝑖(𝑎)∼0\begin{document}$s_i(a) \sim0$\end{document}), a few mutations substantially increase viral fitness (Figure 1). Strongly beneficial mutations occurred in clusters along the genome and preferentially appeared in specific regions of Env (Figure 2).

To quantify patterns of selection in Env, we examined the top 2% of mutations inferred to be the most beneficial in CH505 and CH848. The fractions of nonsynonymous mutations within these subsets were 97% and 92% for CH505 and CH848, respectively. These fractions are significantly higher than chance expectations (𝑝=6.7×10−3\begin{document}$p = 6.7\times10^{-3}$\end{document} and 5.6×10−4\begin{document}$5.6\times10^{-4}$\end{document}, Fisher’s exact test; Methods), supporting the model’s ability to accurately infer fitness effects in this data. For CH505, we found 10.9-fold more strongly beneficial mutations in the first variable loop (V1) than expected by chance (𝑝=2.5×10−3\begin{document}$p = 2.5 \times10^{-3}$\end{document}). This is consistent with the presence of V1 mutations conferring resistance to autologous strain-specific antibodies (Gao et al., 2014). Mutations in V4, a region targeted by CD8 + T cells (Gao et al., 2014), were also 10.0-fold enriched in this subset (𝑝=9.0×10−5\begin{document}$p = 9.0\times10^{-5}$\end{document}). For CH848, mutations in V1, V3, and V5 were enriched by factors of 14.2, 6.3, and 19.1 among the top 2% most beneficial mutations (p=5.9×10−10\begin{document}$p = 5.9 \times 10^{-10}$\end{document}, 6.6×10−3\begin{document}$6.6\times10^{-3}$\end{document}, and 4.8×10−6\begin{document}$4.8\times10^{-6}$\end{document}). Mutations in these regions were shown to play a role in resistance to DH270 and DH475 lineage antibodies (Bonsignori et al., 2017). To test whether our results might be biased by overall sequence variability, we examined the relationship between our inferred selection coefficients and entropy, a common measure of sequence variability. Overall, we found only a modest correlation between selection and entropy, suggesting that the signs of selection that we observe are not due to increased sequence variability alone (Figure 1—figure supplement 1).

Reversions and mutations affecting N-linked glycosylation motifs were also likely to be beneficial. We define reversions as mutations where the transmitted/founder (TF) amino acid changes to match the subtype consensus sequence at the same site. Among the top 2% most beneficial mutations, reversions were enriched by factors of 19.9 and 17.8 for CH505 and CH848 viruses, respectively (𝑝=2.1×10−8\begin{document}$p=2.1\times10^{-8}$\end{document} and 8.5×10−13\begin{document}$8.5 \times 10^{-13}$\end{document}), consistent with past work finding strong selection for reversions (Zanini et al., 2015; Sohail et al., 2021). For CH848, this group also includes several strongly selected mutations observed in Nef (Figure 1B), a protein that plays multiple roles during HIV-1 infection (Das and Jameel, 2005; Barton et al., 2019). Mutations affecting N-linked glycosylation motifs (i.e., by adding, removing, or shifting a glycosylation motif) were enriched by factors of 4.6 (𝑝=7.0×10−3\begin{document}$p=7.0\times10^{-3}$\end{document}) and 8.7 (p=1.4×10−10\begin{document}$p=1.4 \times 10^{-10}$\end{document}). Changes in glycosylation patterns contributed to antibody escape for both CH505 and CH848 (Gao et al., 2014; Bonsignori et al., 2017).

To quantify levels of selection for antibody escape, we computed selection coefficients for mutations that were observed to contribute to resistance to bnAbs as well as bnAb precursors and autologous strain-specific antibodies (Supplementary file 1 and Supplementary file 3; Gao et al., 2014; McCurley et al., 2017; Kong et al., 2019; Saunders et al., 2022; Bauer et al., 2023). We mapped the inferred selection coefficients to the Env protein structure (Liao et al., 2013), highlighting the binding sites for bnAbs and resistance mutations for strain-specific antibodies, as well as important parts of Env (Figure 2, Figure 1—figure supplement 1). We also analyzed when these resistance mutations were first observed in each individual.

Overall, we observed stronger selection for escape from autologous strain-specific antibodies and/or changes in glycosylation during the first six months of infection (Supplementary file 4 and Supplementary file 5). This was then followed by more modest selection for escape from bnAb lineages (Figure 3). For CH505, mutations that conferred resistance to the intermediate-breadth CH235 lineage (Gao et al., 2014) were less beneficial than top mutations escaping from strain-specific antibodies (Figure 3A). In turn, resistance mutations for the broader CH103 lineage were less beneficial than CH235 resistance mutations (Figure 3A). CH848 is similar, with some highly beneficial mutations affecting glycosylation observed early in infection (Figure 3B). While a few mutations affecting DH270 appear strongly selected, these mutations appeared long before the DH270 lineage was detected (around 3.5 years after infection Bonsignori et al., 2017). Thus, these mutations may have initially been selected for other reasons.

Simian-human immunodeficiency viruses (SHIVs) have numerous applications in HIV/AIDS research (Hatziioannou and Evans, 2012; Li et al., 2016). Recently, Roark and collaborators studied SHIV-antibody coevolution in rhesus macaques (RMs), which they compared with patterns of HIV-1 evolution (Roark et al., 2021). Two of the SHIV constructs in this study included envelope sequences derived from CH505 and CH848 transmitted/founder (TF) viruses. There, it was found that 2 out of 10 RMs inoculated with SHIV.CH505 and 2 out of 6 RMs inoculated with SHIV.CH848 developed antibodies with substantial breadth.

To understand whether the patterns of HIV-1 selection observed in CH505 and CH848 are repeatable, and to search for viral factors that distinguish between individuals who develop bnAbs and those who do not, we analyzed SHIV.CH505 and SHIV.CH848 evolution in RMs (Roark et al., 2021). To prevent spurious inferences, we first omitted data from RMs with <3 sampling times or <4 sequences in total (Methods). After processing, we examined evolutionary data from seven RMs inoculated with SHIV.CH505 and six RMs inoculated with SHIV.CH848 (Supplementary file 6 and Supplementary file 7). We then computed selection coefficients for SHIV mutations within each RM. Reasoning that selective pressures across SHIV.CH505 and SHIV.CH848 viruses are likely to be similar, we also inferred two sets of joint selection coefficients that best describe SHIV evolution in SHIV.CH505- and SHIV.CH848-inoculated RMs, respectively (Methods).

As before, we examined the top 2% of SHIV.CH505 and SHIV.CH848 mutations that we inferred to be the most beneficial for the virus. Overall, we found consistent selection for reversions (17.9- and 14.2-fold enrichment, p=4.3×10−11\begin{document}$p = 4.3\times10^{-11}$\end{document} and 1.2×10−11\begin{document}$1.2 \times 10^{-11}$\end{document} for SHIV.CH505 and SHIV.CH848, respectively), with slightly attenuated enrichment in mutations that affect N-linked glycosylation (2.7- and 4.2-fold enrichment, 𝑝=5.4×10−2\begin{document}$p=5.4\times10^{-2}$\end{document} and 3.4×10−6\begin{document}$3.4\times10^{-6}$\end{document}). However, there is a small subset of mutations that shift glycosylation sites by simultaneously disrupting one N-linked glycosylation motif and completing another, where highly beneficial mutations occur far more often than expected by chance (158.3- and 191.1-fold enrichment, p=1.7×10−4\begin{document}$p=1.7\times10^{-4}$\end{document} and 7.9×10−11\begin{document}$7.9 \times 10^{-11}$\end{document} for CH505 and SHIV.CH505, respectively; 118.7-fold and 90.4-fold enrichment, 𝑝=2.0×10−4\begin{document}$p=2.0\times10^{-4}$\end{document} and 2.3×10−7\begin{document}$2.3\times10^{-7}$\end{document} for CH848 and SHIV.CH848).

Intuitively, one may expect that strongly beneficial SHIV mutations are more likely to be observed in samples from multiple RMs. However, we found that the number of RMs in which a mutation is observed is only weakly associated with the fitness effect of the mutation (Figure 4, Figure 4—figure supplement 1). While substantially deleterious SHIV mutations are rarely observed across multiple RMs, neutral and nearly neutral mutations are common. Thus, in this data set, it is not generally true that SHIV mutations observed in multiple hosts must significantly increase viral fitness.

We observed some differences between HIV-1 and SHIV in the precise locations of the most beneficial Env mutations. For example, mutations in V4 are highly enriched in CH505 due to a CD8 + T cell epitope in this region, but not in SHIV.CH505 (2.7-fold enrichment, p=0.13\begin{document}$p=0.13$\end{document}). For SHIV.CH848, beneficial mutations are modestly enriched in V1 and V5 (4.7- and 4.2-fold, 𝑝=4.0×10−3\begin{document}$p=4.0\times10^{-3}$\end{document} and 6.7×10−2\begin{document}$6.7\times10^{-2}$\end{document}), as in CH848, but not for V3 (0.9-fold enrichment, p=0.22\begin{document}$p=0.22$\end{document}).

Despite some differences in the top mutations, patterns of selection over time in SHIV were very similar to those found for HIV-1. As before, highly beneficial mutations, including ones affecting glycosylation, tended to appear earlier in infection. This was followed by modestly beneficial mutations at later times, including ones involved in resistance to bnAbs in the RMs who developed antibodies with significant breadth (examples in Figure 4).

A major goal of nonhuman primate studies with SHIV is to faithfully recover important aspects of HIV-1 infection in humans. However, due to the divergence of simian immunodeficiency viruses and HIV-1, SHIVs are not always well-adapted to replication in RMs (Bauer et al., 2023). To combat this problem, a recent study identified six SHIV.CH505 mutations that increase viral load (VL) in RMs (Bauer et al., 2023). These mutations result in viral kinetics that better mimic HIV-1 infection in humans.

Our analysis readily identifies the SHIV.CH505 mutations shown to increase VL. Five out of the top six SHIV.CH505 mutations with the largest average selection coefficients are associated with increased VL (Supplementary file 8). The final mutation identified by Bauer et al., N130D, is ranked tenth. We also find highly beneficial mutations in SHIV.CH848 that are distinct from those in SHIV.CH505 (Supplementary file 9). Highly ranked mutations identified here may be good experimental targets for future studies aimed at increasing SHIV.CH848 replication in vivo.

Next, we explored the similarity in the overall viral fitness landscapes inferred for HIV-1 and SHIV, beyond just the top mutations. First, we computed the fitness of each SHIV sequence using the joint SHIV.CH505 and SHIV.CH848 selection coefficients inferred from RM data. Then, we computed fitness values for SHIV sequences using selection coefficients inferred from HIV-1 evolution in CH505 and CH848.

We observed a remarkable agreement between SHIV fitness values computed from these two sources (Figure 5). For both SHIV.CH505 and SHIV.CH848, the correlation between viral fitness estimated using data from humans (i.e. CH505 and CH848) and RMs is strongly and linearly correlated (Pearson’s r=0.96\begin{document}$r=0.96$\end{document} and 0.95, p<10−20\begin{document}$p \lt 10^{-20}$\end{document}). This implies that evolutionary pressures on the envelope protein SHIV-infected RMs are highly similar to those on HIV-1 in humans with the same TF Env. In fact, this relationship holds even beyond the SHIV sequences observed during infection. Fitness estimates for sequences with randomly shuffled sets of mutations are also strongly correlated (Figure 5—figure supplement 1). In contrast, the inferred fitness landscapes of CH505 and CH848, which share few mutations in common, are poorly correlated (Figure 5—figure supplement 2). This suggests that the similarities between viral fitness values in humans and RMs are not artifacts of the model, but rather stem from similarities in underlying evolutionary drivers.

Given the similarity of HIV-1 and SHIV evolution, we sought to identify evolutionary features that distinguish between hosts who develop broad antibody responses and those who do not. Figure 5 shows that SHIV sequences from hosts with bnAbs often reach higher fitness values than those in hosts with only narrow-spectrum antibodies. We hypothesized that stronger selective pressures on the virus might drive viral diversification, stimulating the development of antibody breadth. Past studies have associated higher viral loads with bnAb development and observed viral diversification around the time of bnAb emergence (Liao et al., 2013; Moore et al., 2015; Landais and Moore, 2018). Computational studies and experiments have also shown that sequential exposure to diverse antigens can induce cross-reactive antibodies (Wang et al., 2015; Escolano et al., 2016; Sprenger et al., 2020).

To further quantify SHIV evolutionary dynamics, we computed the average fitness gain of viral populations in each RM over time. We observed a striking difference in SHIV fitness gains between RMs that developed broad antibody responses and those that did not (Figure 6). In particular, SHIV fitness increased rapidly before the development of antibody breadth. SHIV fitness gains in RM5695, which developed exceptionally broad and potent antibodies (Roark et al., 2021), were especially rapid and dramatic. These fitness differences were not attributable to bnAb resistance mutations, which were only moderately selected and generally appeared after bnAbs developed.

One outlier in this pattern is RM6072, infected with SHIV.CH505. Antibody development in RM6072 followed a path that was remarkably similar to CH505, including a lineage of antibodies, DH650, directed toward the CD4 binding site of Env (Roark et al., 2021). However, resistance to the DH650 lineage is conferred by a strongly selected mutation that adds a glycan at site 234 (T234N, with an inferred selection coefficient of 4.5%). Broadly neutralizing antibodies similar to DH650 are able to accommodate this glycan due to shorter and/or more flexible light chains (Zhou et al., 2013; Roark et al., 2021), but DH650 cannot. Antibody evolution in RM6072 thus proceeded along a clear pathway toward bnAb development but lacked critical mutations to achieve breadth.

Next, we quantified how different types of SHIV mutations contributed to viral fitness gains over time. We examined contributions from VL-enhancing mutations (Bauer et al., 2023), antibody escape mutations (Bauer et al., 2023; Roark et al., 2021), other mutations affecting Env glycosylation, and reversions to subtype consensus. We found increased fitness gains across all types of mutations in RMs that developed broad antibody responses, compared to those that did not (Figure 6—figure supplement 1). VL-enhancing mutations, known antibody resistance mutations, and reversions typically made the largest contributions to viral fitness.

In the analysis above, we used a simple model where the net fitness effect of multiple mutations is simply equal to the sum of their individual effects. Recently, methods have also been developed that can infer epistatic fitness effects from data, which include pairwise interactions between mutations (Sohail et al., 2022; Shimagaki and Barton, 2025b). We reanalyzed these data to examine how inferred fitness changes when epistasis is included in the model, using the approach of Shimagaki and Barton (2025; Methods). Overall, the inferred epistatic interactions were modest (Figure 6—figure supplement 2). In CH505, we found that the CD4 binding site, V1 (especially sites 136–146 in HXB2 numbering) and V5 regions were modestly but significantly enriched in the most beneficial (top 1%) of epistatic interactions (2.5-, 1.2-, and 1.8-fold enrichment with p=1.0×10−21,6.3×10−6\begin{document}$p=1.0 \times 10^{-21}, 6.3 \times 10^{-6}$\end{document} and 6.3 × 10^-5, respectively). Epistatic interactions between N280S/V281A and E275K/V281G, which confer resistance to CH235 (Gao et al., 2014), ranked in the top 6.5% and 13.0% of interactions. In CH848, we found 1.3-, 1.5-, and 2.3-fold enrichment in strong beneficial epistatic interactions in the CD4 binding site, V4, and V5 regions, respectively (p=4.0×10−6,p=2.5×10−14\begin{document}$p=4.0 \times 10^{-6}, p=2.5 \times 10^{-14}$\end{document} and 3.2 × 10^-19).

To compare the typical fitness effects of individual mutations in the model with epistasis to those in the additive model, we computed effective selection coefficients for the epistatic model. For each mutant allele 𝑎\begin{document}$a$\end{document} at each site 𝑖\begin{document}$i$\end{document}, we computed the average difference in fitness between sequences in the data set with the mutation and hypothetical sequences that are the same as those in the data, except with the mutant allele 𝑎\begin{document}$a$\end{document} reverted to the TF one. In this way, the effective selection coefficient measures the typical effect of each mutation in the data set, while also accounting for epistatic interactions with the sequence background. We found that the effective selection coefficients were highly correlated with the selection coefficients from the additive model (Figure 6—figure supplement 3). We also found strong agreement between the additive and epistatic model fitness values for each sequence in both HIV-1 and SHIV data (Figure 6—figure supplement 4).

Finite sampling of sequence data could also affect our analyses. To further test the robustness of our results, we inferred selection coefficients using bootstrap resampling, where we resample sequences from the original ensemble, maintaining the same number of sequences for each time point and subject. The selection coefficients from the bootstrap samples are consistent with the original data (see Figure 6—figure supplement 5 for a typical example), with Pearson’s 𝑟\begin{document}$r$\end{document} values of around 0.85 for HIV-1 data sets and 0.95 for SHIV data sets, respectively.

We retrieved HIV-1 sequences from CH505 (703010505) and CH848 (703010848) from the HIV sequence database at Los Alamos National Laboratory (LANL) (Los Alamos National Laboratory, 2023a). The rhesus macaque (RM) SHIV sequences (Roark et al., 2021) were obtained from GenBank (Benson et al., 2012). We then co-aligned SHIV.CH505 and SHIV.CH848 sequences with CH505 and CH848 HIV-1 sequences, respectively, using HIValign (Los Alamos National Laboratory, 2023b).

In this section, we describe the inference framework used to infer the fitness effects of mutations (selection coefficients) from temporal genetic data.

In addition to fitness models derived from SHIV data for individual RMs, we inferred a joint model under the assumption that virus evolution within each individual RM with the same TF virus is governed by a similar fitness landscape. This method improves inference accuracy from the WF process and deep mutational scanning data (Sohail et al., 2022; Shimagaki and Barton, 2025b; Hong et al., 2024). The joint path likelihood for allele frequency trajectories across RMs with the same TF virus is then(19)p(((xα(tk))k=0Kα)r=1R|s,μ,γ)=(∏r=1R∏k=0K−1pM(xr(tk+1)|xr(tk);s,μ,N))p(x(t0)) p(s|γ).\begin{document}$$\displaystyle \begin {aligned} p(&((x^\alpha(t_k))_{k=0}^{K_\alpha})_{r=1}^R|s, \mu, \gamma) = \left(\prod_{r=1}^R \prod_{k=0}^{K-1} p_{\rm M}(x^r(t_{k+1})|x^r(t_{k}); s, \mu, N) \right) \;p(x(t_0))\ p(s|\gamma)\,. $$\end{document}

Here, 𝑥𝑟\begin{document}$x^r$\end{document} is the allele frequency of the 𝑟\begin{document}$r$\end{document}-th individual and 𝑅\begin{document}$R$\end{document} is the number of replicate individuals (i.e. the number of RMs sharing the same TF virus). The initial state is 𝑝(𝑥𝑟(𝑡0))=𝑝(𝑥(𝑡0))\begin{document}$p(x^r(t_0)) = p(x(t_0))$\end{document} for all 𝑟\begin{document}$r$\end{document} for individuals with the same TF virus. The solution of the joint path likelihood is given by(20)s^=(C¯int+γI)−1[Δx¯int−Δaμ¯int].\begin{document}$$\displaystyle \hat{s} = (\overline{C}_{\rm int} + \gamma I)^{-1} [\overline{\Delta x}_{\rm int} - \overline{\Delta a\mu}_{\rm int}].$$\end{document}

Here, the overbar denotes the sum over the replicate RMs.

We emphasize that the joint selection coefficients in Equation 20 are not the same as selection coefficients that are simply averaged across RMs with the same TF virus. The joint selection coefficients are more robust, as they are guided by the level of evidence within each individual rather than naive averaging.

The Pearson values we utilized to compare the fitness landscapes denoted as 𝐹𝑠(𝑔)∶=𝐹(𝑔|𝑠)\begin{document}$F_s(g):=F(g|s)$\end{document} and 𝐹ℎ(𝑔)∶=𝐹(𝑔|ℎ)\begin{document}$F_h(g):=F(g|h)$\end{document} can be expressed by the following simple relation:(21)r=Cov(Fs,Fh)Var(Fs)Var(Fh)=s⊤Chs⊤Csh⊤Ch.\begin{document}$$\displaystyle r = \frac{\mathrm{Cov}(F_s, F_h)}{\sqrt{\mathrm{Var}(F_s)} \sqrt{\mathrm{Var}(F_h)} } = \frac{s^\top C h }{\sqrt{s^\top C s} \sqrt{h^\top C h}}\,. $$\end{document}

Here, Cov(𝐹𝑠,𝐹ℎ)\begin{document}$\mathrm{Cov}(F_s, F_h)$\end{document} and Var(𝐹𝑠)\begin{document}$\mathrm{Var}(F_s)$\end{document} represent the covariance and variance values estimated from the samples being compared, (𝐹𝑠(𝑔𝑛),𝐹ℎ(𝑔𝑛))𝑛\begin{document}$(F_s(g^n), F_h(g^n))_{n}$\end{document}. 𝐶\begin{document}$C$\end{document} is the covariance matrix defined between arbitrary loci. The last equation can be interpreted as an angle between two vectors, 𝑠\begin{document}$s$\end{document} and ℎ\begin{document}$h$\end{document}, with a metric matrix 𝐶\begin{document}$C$\end{document}; if 𝑠=ℎ\begin{document}$s=h$\end{document}, the Pearson value clearly becomes 1. However, the ‘similarity’ also depends on how these vectors are projected by 𝐶\begin{document}$C$\end{document}; eigenmodes associated with larger variance of statistics will be more emphasized. The last expression readily implies an interpretation for the case of shuffled sequences; shuffling the sequences equates to diluting the covariance between loci, resulting in the metric matrix becoming a diagonal matrix. Removing the off-diagonal elements corresponds to lifting the constraints on the fitness landscape.

We consider the following pairwise epistatic fitness function, which depends on epistatic interactions 𝑠𝑖𝑗\begin{document}$s_{ij}$\end{document} across all possible pairs of loci:(22)F(g)=1+∑i=1L∑a=1qsi(a)gi,a+∑i<j∑a,b=1sij(a,b)gi,agj,b.\begin{document}$$\displaystyle F(g) = 1 + \sum_{i=1}^L\sum_{a=1}^q s_i(a) g_{i,a} + \sum_{i \lt j}\sum_{a,b=1} s_{ij}(a,b) g_{i,a} g_{j,b}\,.$$\end{document}

Our goal is to obtain the epistatic interactions 𝑠𝑖𝑗(𝑎,𝑏)\begin{document}$s_{ij}(a,b)$\end{document} as well as the selection coefficients 𝑠𝑖(𝑎)\begin{document}$s_i(a)$\end{document} from temporal genetic sequences.

The basic logic for inferring these fitness parameters parallels the additive case. The only practical difference is that epistatic interactions can influence the dynamics of additive and pairwise mutation frequencies. Under the diffusion limit, we obtain the following drift terms, which align with (12) in the additive model,(23)di(a)=∑k=1L∑c=1qCik(a,c)sk(c)+∑k<lL∑c,d=1qCikl(a,c,d)skl(c,d)+ui(a)dij(a,b)=∑kL∑c=1LCijk(a,b,c)sk(c)+∑k<lL∑c,d=1qCijkl(a,b,c,d)skl(c,d)+uij(a,b)+vij(a,b).\begin{document}$$\displaystyle \begin{array}{ll}d_i(a) = \sum\limits_{k=1}^{L}{\sum\limits_{c=1}^{q}{C_{ik}(a,c)s_k(c)}} + \sum\limits_{k \lt l}^{L}{\sum\limits_{c,d=1}^{q}{C_{ikl}(a,c,d)s_{kl}(c,d) + u_i(a)}} \\ d_{ij}(a,b) = \sum\limits_{k}^{L}{\sum\limits_{c=1}^{L}{C_{ijk}(a,b,c)s_k(c)}} + \sum\limits_{k \lt l}^{L}{\sum\limits_{c,d=1}^{q}{C_{ijkl}(a,b,c,d)s_{kl}(c,d) + u_{ij}(a,b) + v_{ij}(a,b)}}.\end{array}$$\end{document}

and diffusion matrices,(24)𝐶𝑖𝑘𝑙(𝑎,𝑐,𝑑)=𝑥𝑖𝑘𝑙(𝑎,𝑐,𝑑)−𝑥𝑖(𝑎)𝑥𝑘𝑙(𝑐,𝑑)𝐶𝑗𝑖𝑘𝑙(𝑎,𝑏,𝑐,𝑑)=𝑥𝑖𝑗𝑘𝑙(𝑎,𝑏,𝑐,𝑑)−𝑥𝑖𝑗(𝑎,𝑏)𝑥𝑘𝑙(𝑐,𝑑).\begin{document}$$\displaystyle \begin{equation*} C_{ikl}(a,c,d) &= x_{ikl}(a,c,d) - x_i(a) x_{kl}(c,d)\\ C_{jikl}(a,b,c,d) &= x_{ijkl}(a,b,c,d) - x_{ij}(a,b) x_{kl}(c,d)\,. \end{equation*}$$\end{document}

Here, 𝑢𝑖\begin{document}$u_i$\end{document} and 𝑢𝑖𝑗\begin{document}$u_{ij}$\end{document} represent the expected frequency changes due to mutations for additive and pairwise terms, while 𝑣𝑖\begin{document}$v_i$\end{document} represents the changes in pairwise frequencies due to recombination. These explicit expressions indicate that the 𝑢𝑖\begin{document}$u_i$\end{document} remains the same as in the additive fitness case; therefore, Equation 17 holds. The pairwise term is given as(25)uij(a,b)=∑c=1q([μbcxij(a,c)+μacxij(c,d)]−[μcd+μca]xij(a,b)).\begin{document}$$\displaystyle \begin{array}{ll} u_{ij}(a,b) &= \sum\limits_{c=1}^{q}{\Big([\mu_{bc}x_{ij}(a,c)+ \mu_{ac}x_{ij}(c,d)] - [\mu_{cd} + \mu_{ca}]x_{ij}(a,b) \Big)}.\end{array}$$\end{document}

and the 𝑣𝑖𝑗\begin{document}$v_{ij}$\end{document} is expressed as(26)vij(a,b)=−r|i−j|Cij(a,b),\begin{document}$$\displaystyle v_{ij}(a,b) = -r|i-j| C_{ij}(a,b),$$\end{document}

where 𝑟\begin{document}$r$\end{document} denotes the recombination rate. In this study, we set 𝑟=10−5\begin{document}$r=10^{-5}$\end{document} . More detailed derivations are provided in previous studies (Sohail et al., 2022; Shimagaki and Barton, 2025b).

The technical challenge of the epistasis inference is that the diffusion matrix 𝐶\begin{document}$C$\end{document} involves third- and fourth-order interactions, and the number of matrix elements scales as 𝑂((𝑞𝐿)4)\begin{document}$\mathcal{O}((qL)^4)$\end{document}, while the computational cost to invert it scales as 𝑂((𝑞𝐿)6)\begin{document}$\mathcal{O}((qL)^6)$\end{document}. Recently, a more efficient computational method was proposed, reducing both the necessary memory usage and computational times by 𝑂((𝑞𝐿)2)\begin{document}$\mathcal{O}((qL)^2)$\end{document} (Shimagaki and Barton, 2025b). The essential idea involves factorizing the higher-order covariance matrix using the rectangular matrix Ξ∈ℝ𝐷×𝑑\begin{document}$\Xi\in\mathbb{R}^{D\timesd}$\end{document} such that 𝐶=ΞΞ⊤\begin{document}$C=\Xi\Xi^\top$\end{document}, where 𝐷\begin{document}$D$\end{document} scales as 𝑂((𝑞𝐿)2)\begin{document}$\mathcal{O}((qL)^2)$\end{document} while 𝑑≪𝐷\begin{document}$d\llD$\end{document}. This method resolves the linear equation without obtaining an explicit expression of 𝐶\begin{document}$C$\end{document} and avoids any computations involving more than 𝑂((𝑞𝐿)2)\begin{document}$\mathcal{O}((qL)^2)$\end{document} operations (Shimagaki and Barton, 2025b).