Estimating the stoichiometry of HIV-1 trimer neutralization, N

The topic of HIV-1 neutralization stoichiometry has previously been studied experimentally and computationally, with N = 1 being the most common conclusion [26, 29, 34]. However, ambiguity in these estimates prevailed, predominantly due to previously poorly defined parameters of HIV-1 entry stoichiometry, T, and mean virion trimer number, , which are essential for the analysis of N [27–29]. Here, we utilized recently determined values of T and [28] to conclusively estimate N across different HIV-1 strains and bnAbs identified in recent years, for which estimates of N are currently lacking. As precise information on N across nAbs is indispensable for an in-depth numerical understanding of HIV-1 nAb neutralization (Fig 1A), we set out to estimate N for a range of nAbs and HIV-1 strains.

To estimate N, we measured nAb neutralization of HIV-1 pseudoviruses carrying mixed trimers of nAb-sensitive and resistant Envs and analyzed the data with mathematical models (Fig 2A). Relative virus infectivities (RI) under saturating nAb concentrations were set in relation to the fraction of resistant Env (f_R), shown for nAb 2F5 with Envs JR-FL wt (2F5 sensitive) and mutant JR-FL D664N (2F5 resistant) (Fig 2B, S1 Fig, S2 Table). Taking the JR-FL entry stoichiometry (T = 2) and mean trimer number per virion () into account (S3 Table), our model predicts different RI profiles for different N (Fig 2C). Mathematical analysis of the experimental data indicated a neutralization stoichiometry of N = 1 for JR-FL by nAb 2F5 (Fig 2C). We confirmed the robustness of the N = 1 estimate against variation in virus entry stoichiometry (T) and mean virion trimer number () by sensitivity analyses (Fig 2D). To investigate the observed deviations of the model fit from the experimental data (Fig 2C), we performed a goodness-of-fit analysis (Fig 2E). This analysis indicated that lower values of T and/or could result in better curve fits. While we expect T to be constant for each viral strain, fluctuations in from experiment to experiment are conceivable and provide a potential explanation for the deviation between experimental data and model predictions. In addition, mean virion trimer numbers of a given virus preparation may decrease over time as spontaneous Env shedding can occur resulting in non-functional trimers [36].

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Fig 2. Estimating the stoichiometry of HIV-1 nAb neutralization (N) with mixed trimer assays.

(A) Scheme depicting the combined experimental-mathematical approach employed here to estimate N. (B) Neutralization by nAb 2F5 of mixed trimer HIV-1 pseudovirus stocks containing the indicated ratios of JR-FL wt (2F5 sensitive) and JR-FL D664N (2F5 resistant) Envs. The relative infectivity (RI) of each stock is given by the percentage of target cell infection (i.e., the inverse of % neutralization) under saturating 2F5 nAb concentrations. (C) Theoretical model predictions of the relation between N (N = 1, 2 or 3; colored lines) and RI of virus stocks with different fractions of neutralization-sensitive to resistant Env (f_R). The experimental JR-FL nAb 2F5 RI data from (B) are plotted as black dots, representing two independent experiments. Data fitting resulted in an estimate of N = 1, assuming T = 2 and trimers for each JR-FL virion (S3 Table). (D) Robustness analysis for the N = 1 estimate of HIV-1 strain JR-FL and nAb 2F5 against variation in T and . Blue areas show combinations of T and resulting in estimates of N = 1, green areas show combinations of T and that would result in estimates of N = 2. The actual values of T and for JR-FL (see above) are marked by the white dot, indicating that the N = 1 estimate is robust. (E) Since the model fit shown in (C) for JR-FL and nAb 2F5 is imperfect, we analyzed the goodness-of-fit. A slightly reduced or slightly increased T compared to those used for the analysis (white dot, S3 Table) could improve the model fit to the experimental data.

https://doi.org/10.1371/journal.ppat.1006313.g002

To further test variation in T and on predictions of N, we assessed N for nAb 2F5 against Env variants of HIV-1 strains JR-FL and NL4-3 that differ in T or (S2A–S2F Fig, S3 Table) [28, 37]. We indeed observed divergent RI profiles upon variation in T or as predicted by our model. However, for all Env variants tested we retrieved estimates of N = 1.

To explore if nAb avidity influences estimation of N, we compared nAb 2F5 IgG and Fab fragment. Both yielded overlapping RI profiles for strains JR-FL and NL4-3 and identical estimates of N = 1 (S2G–S2I Fig), confirming that estimation of N is independent of nAb valency.

NAb binding to a single Env subunit is sufficient for trimer neutralization

Having validated our approach to estimate N (Fig 2), we sought to obtain a comprehensive analysis of N for various HIV-1 nAbs including VRC01, NIH45-46, PGV04, b12, PGT121, PGT128, PGT135, 2G12, PG9, PGT145, and 2F5 (S4 Table). We therefore generated a panel of Env mutants in five divergent HIV-1 strains with single or combined nAb resistance mutations (S1 Fig). Several of these Env mutants showed a significant reduction in virus infectivity (S2 Table). This is critical to note, as matched virus entry parameters, notably T and , are a prerequisite for the analysis of N (S2 Fig). Indeed, we observed that strong infectivity differences between nAb-resistant and sensitive Envs in mixed trimer assays result in substantial deviations of the RI profiles that prohibit determination of N (S3 Fig). We thus restricted our analysis to mixed trimer assays with nAb sensitive-resistant Env pairs of comparable infectivity (≤ 2-fold infectivity difference). In a direct comparison of eight different nAbs across five Env mixed trimer combinations, we thereby obtained a consistent estimate of N = 1 irrespective of nAb epitope specificity, potency or HIV-1 strain (Fig 3A–3E, S4 Fig). The estimate of N = 1 was confirmed by bootstrap analyses (S5 Fig) and goodness-of-fit plots across all analyzed Env-nAb pairings (S6 Fig). As shown before (Fig 2E), the goodness-of-fit would in many cases be improved for lower values of T and/or , possibly indicating slight deviations in these parameters within the experimental setup.

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Fig 3. N = 1 is a general feature of broadly neutralizing antibodies and polyclonal HIV-1⁺ patient plasma IgG.

(A) to (E) Mixed trimer neutralization setups with the Envs and nAbs indicated in each panel, yielding estimates of N = 1 in all cases. (F) to (G) Envs with multiple nAb resistance mutations allowed parallel assessment of various nAbs on the same set of mixed trimer virus stocks. Mathematical analysis indicated N = 1 in all cases. (H) Mixed trimer setup with Envs ZA110 wt and ZA110-V1V2^1.7, the latter being sensitive to autologous HIV-1⁺ patient plasma. The model fit to the experimental data (black dots) yielded a plasma neutralization stoichiometry of N = 1. (I) Mixed trimer setup with Envs ZA110 wt and ZA110 ΔV1V2 and nAbs b6, 17B and 48D and three heterologous HIV-1⁺ patient plasma (Pat117, Pat118 and Pat122). The Envs in this setup are not infectivity-matched precluding mathematical analysis, but graphic comparison indicates equal N of nAbs and patient plasma. (J) Mixed trimer setup with Envs JR-FL L175P and JR-FL P369L M373R D664N and nAbs b6, 1.79, 2F5, b12 and heterologous patient plasma SP122, yielding equal estimates of N = 1. All data points depict two independent experiments. (K) N = 1 is a universal feature of all anti-HIV nAbs tested here. This defines decisive numerical requirements of HIV-1 virion neutralization by antibodies.

https://doi.org/10.1371/journal.ppat.1006313.g003

In the above analysis we employed a hard threshold model in which, according to our analyses, binding of one nAb to a trimer results in complete loss of function. Soft threshold models, which allow for partial loss of trimer functionality upon nAb binding, were introduced in earlier studies by us and others [26, 29]. To investigate how a soft threshold model would fit to our current data set, we re-analyzed our data accordingly [26]. Here, we allow the probability of a virion to infect a cell to scale with the number of functional trimers (a soft threshold for virus entry) and the functionality of a trimer to decrease by successive nAb binding (a soft threshold for neutralization). Our analysis revealed that the functionality loss of a trimer upon nAb binding to one subunit is dominant, that is, a soft threshold model is not supported (S7 Fig). This finding thereby underscores the assumption of a hard threshold for the stoichiometry of HIV-1 nAb neutralization.

To validate that N = 1 across all nAbs tested here, we tested a set of nAbs targeting different epitopes against Env variants with multiple nAb escape mutations, allowing parallel analysis of diverse nAbs on the same set of mixed trimer virus stocks (Fig 3F and 3G, S8 Fig, S2 Table). Two Env combinations (Fig 3F and 3G) were infectivity matched and allowed mathematical analysis of N, yielding N = 1 for all nAbs tested. To derive an estimate of N in the non-infectivity matched setups we included nAb 2F5 throughout, as we previously established that it neutralizes with N = 1 (Fig 2). Furthermore, nAb PG9, which is known to bind with only one nAb per trimer [38, 39], was included in two of these setups (S8 Fig). As 2F5 and PG9 yielded highly similar curves compared to the other nAbs, we conclude that our analysis univocally inferred N = 1 for all probed nAbs.

Regarding this consistent N = 1 estimate across all nAbs tested, we asked whether non-antibody Env inhibitors would show a similar neutralization behaviour. To test this, we generated Env mutants resistant against the peptide T-20, a clinically used HIV-1 entry inhibitor targeting the Env gp41 subunit [40]. Using mixed trimer assays, we found that T-20 neutralizes with an N = 1 stoichiometry (S9 Fig). This indicates that, regardless of inhibitor type, interference with one Env trimer subunit is sufficient for HIV-1 trimer neutralization.

Antibodies in HIV-1⁺ patient plasma neutralize with a N = 1 stoichiometry

Antibody responses in the majority of HIV-1 infections are largely ineffective in neutralizing the virus within the patient, and bear only limited neutralization potency and breadth against other HIV-1 strains. In contrast, the bnAbs available to date were isolated from rare HIV-1⁺ patients with high HIV-1 neutralization potency and breadth [41–43]. In our stoichiometry analysis, these bnAbs uniformly showed N = 1 (Fig 3A–3G). This raises the question if weakly neutralizing Abs as most commonly elicited during HIV-1 infection require a higher N, and whether N = 1 is a distinguishing feature of bnAbs. To test this, we determined N of the polyclonal antibody mix in HIV-1⁺ patient plasma. We first tested plasma from an individual with typical HIV-1 neutralization escape [44] and corresponding plasma neutralization-resistant and sensitive Env variants (ZA110 wt and ZA110-V1V2^1.7, respectively; S2 Table). Mixed trimer assays yielded N = 1 for the plasma Abs of this individual (Fig 3H). We further tested plasma from three chronically HIV-1 infected individuals (Pat117, Pat118, Pat122) showing only weak HIV-1 neutralization activity. We tested these plasma on mixed trimer stocks of Envs ZA110 wt (resistant) and ZA110 ΔV1V2 (sensitive) and further included weakly neutralizing Abs b6, 17b and 48d in this analysis (Fig 3I). Comparison of the RI profiles indicated identical N for all probed plasma and nAbs. Additionally, plasma Pat122 neutralized HIV-1 strain JR-FL in an infectivity-matched setup with N = 1, as did several nAbs (Fig 3J).

Thus, irrespective of nAb potency, breadth or epitope specificity, neutralization of HIV-1 trimers requires only a single Env subunit to be bound by antibody (Fig 3K). This estimation of N defines numerical requirements for HIV-1 antibody neutralization [27, 29–31, 45], which we employed in subsequent modelling steps to assess HIV-1 virion population neutralization in vitro and in vivo.

In silico prediction of nAb inhibitory capacity

We previously established a mathematical framework that predicts the number of nAbs required to neutralize a given HIV-1 virion population based on the stoichiometry parameters N, T and [27]. The conclusive estimation of N (Figs 2 and 3), together with previously determined parameters T and [28], now enabled us to use our framework for quantitative predictions of nAb neutralization. We extended the framework by including the affinity of nAb binding to Env trimers (represented by the binding constant, K_D) [35, 46] to predict the fraction of neutralized HIV-1 virions for given nAb concentrations. In essence, this allowed us to simulate HIV-1 nAb neutralization curves in silico (Fig 4A).

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Fig 4. Predicting HIV-1 antibody neutralization curves and IC50.

(A) Scheme depicting the virus and nAb-specific parameters necessary to predict HIV-1 neutralization by nAbs in silico. (B) to (D) Predicted HIV-1 nAb neutralization curves in dependence of (B) nAb K_D, (C) HIV-1 entry stoichiometry, T, and (D) mean viron trimer number, . (E) Dependence of nAb IC50 on the number of Env trimers required for HIV-1 cell entry, T. (F) Relation between HIV-1 virion population size and the fraction of Env subunits that need to be bound by nAb to achieve virus population neutralization. (G) Comparison of experimental nAb IC50 values measured with HIV-1 strain BG505 [47, 48] (S5 Table) and IC50’s predicted by our model using BG505-specific T, and K_D [49] values (S3 and S5 Tables). For nAb PGT145 we assumed that only 1 nAb can bind per trimer; the PGT145 IC50 estimates with the 3-nAb-per-trimer model, used for all other nAbs, are shown in light grey (see also panel H). (H) For most nAbs, we assume that three nAbs can bind each trimer, while binding of one nAb is already sufficient for neutralization (N = 1). Binding of nAbs to a trimer that is already neutralized constitutes unproductive binding and may reduce antibody concentrations to a subcritical level [27]. To quantify this effect, we predicted neutralization by two nAbs with equal K_D, but assuming that either three nAbs can bind each trimer, or that nAb binding is anti-cooperative as observed for PG9/PGT145-like nAbs, which are known to occupy only one of three potential epitopes per trimer. We observed a slight neutralization advantage for PG9/PGT145-like nAb binding behavior, manifested as an approximately two-fold reduction in predicted nAb IC50.

https://doi.org/10.1371/journal.ppat.1006313.g004

For this analysis we utilized nAb binding constants, K_D, recently reported for the HIV-1 strain BG505 SOSIP trimer [49]. We employed these K_D values together with T and of BG505 (S3 Table) and N = 1 to model nAb neutralization of an HIV-1 BG505 virion population (Fig 4B–4D). As expected, we found that nAbs with high Env binding affinity (low K_D) are predicted to require lower concentrations to achieve virion population neutralization (Fig 4B). The required nAb concentrations increase slightly with fewer trimers needed for HIV-1 entry (low T) (Fig 4C) and higher virion trimer content (high ) (Fig 4D). The latter two trends can be rationalized as follows: in both cases (a small T or a high ), more nAbs will be needed to neutralize a given virion population, resulting in higher predicted nAb concentrations.

The relation between nAb trimer binding and HIV-1 virion population neutralization is influenced by all parameters included in our model (Fig 4A). Especially the influence of T and on nAb neutralization predictions should not be underestimated. This is highlighted in Fig 4E, depicting estimated values of nAb concentrations required for 50% virus population neutralization (IC50) in dependence on T and .

In addition to predicting nAb neutralization curves and IC50 values, we determined nAb concentrations required to achieve sterilizing neutralization of HIV-1 virion populations. Due to unproductive nAb binding to Env trimers (Fig 4H) [27], the fraction of Env subunits required to be bound by nAb for sterilizing neutralization of a virion population increases with virion population size (Fig 4F). Likewise, the predicted nAb concentrations required for sterilizing neutralization of virion populations increase with virion population size (S10A–S10C Fig). As shown in Fig 4B–4D for nAb neutralization curves, these nAb concentrations are influenced by nAb K_D, T and (S10A–S10C Fig). Of note, only nAb K_D has a direct linear relationship with sterilizing nAb concentrations, while the influence of T and follows non-linear relations (S10D–S10F Fig).

We noted that our predicted neutralization curves show steeper slopes than commonly observed in HIV-1 neutralization assays (S11A Fig). We thus asked which parameters of our model could explain this deviation, and found that assuming broader virion trimer number distributions (i.e., higher variance in trimer numbers between virions) results in less steep predicted neutralization curves (S12 Fig). Of note, we also observed that our predicted curves are in closer agreement to in vitro neutralization data obtained with replication-competent virus and PBMC target cells (S11B Fig). While a detailed analysis of this relation and the underlying parameters is beyond the scope of this manuscript, this observation may be taken as indication that our model predicts neutralization curves better for replication-competent virus than for pseudovirus preparations.

To test the predictive power of our in silico approach, we compared experimentally derived IC50 values of HIV-1 strain BG505 and nAbs VRC01, PGV04, PGT121, PGT123, PGT145 and 2G12, originating from in vitro experiments by three independent laboratories [47–49], with nAb IC50 values estimated by our model (Fig 4G, S11 Fig, S5 Table). Our predicted IC50 values for nAbs VRC01 and PGT121 were in close agreement with the measured values. For nAbs PGT123 and 2G12, we observed wide variations in experimentally derived IC50 values; interestingly, the IC50 values predicted by our model lay in between these values. For nAbs PGV04 and PGT145, our estimated IC50s are slightly lower and higher than experimentally determined IC50s, respectively, though not far off (see below for a detailed discussion of the PGT145 IC50 estimation). This good agreement between experimental and predicted nAb IC50s highlights that the parameters included in our model (N, T, , nAb K_D and nAb concentration) capture relevant steps of HIV-1 virion neutralization. Of note, our model performs significantly better than a simpler model based on nAb K_D alone (S13 Fig).

We also derived predictions of nAb neutralization for scenarios of N = 1 but assuming that exclusively one nAb can bind one of the three epitopes displayed on each trimer (a binding behavior described for nAbs PG9, PG16 or PGT145 [38, 39, 49]). We compared these predictions to our standard model where we assume that all three trimer subunits can potentially be bound, although only one Env subunit needs to be bound to achieve neutralization. In this case, binding of the second and third nAb represents unproductive binding of nAbs, since the trimer is already neutralized by binding of the first nAb (Fig 3K). Importantly, we assume that also for PG9-like nAbs, initially the same number of epitopes is present as for “typical” nAbs (i.e., three per trimer), with the difference that binding of PG9-like nAbs shows negative cooperativity: after binding of the first antibody the remaining two epitopes are inaccessible. In our analysis, we assume two nAbs with the same K_D for their protomeric epitopes, but either a three-nAb-per-trimer or one-nAb-per-trimer binding behavior (Fig 4H). According to our predictions, binding of only one nAb per trimer results in a 2-fold decreased IC50, thereby demonstrating the effect of unproductive nAb binding (Fig 4H). Intriguingly, the PGT145 IC50 predictions using this model are closer to experimental PGT145 IC50 values than the predictions obtained with the standard model (Fig 4G). This indicates that nAbs with a one-nAb-per-trimer binding behavior should have a slight neutralization advantage, especially under conditions of low nAb concentrations.

Estimating the in vivo efficacy of passive nAb immunization and single-virion infectivity

We next predicted nAb neutralization efficacy and host infection probability in vivo by re-assessing data from four studies of rhesus macaque vaginal virus challenge following passive immunization with nAbs b12 [8], 2G12 [7], PGT121 [10] and PGT126 [16]. Our approach consists of two parts: first, we utilized the nAb neutralization prediction model developed above (Fig 4) to estimate how many virions of the challenge dose are neutralized in vivo in dependence on the nAb immunization regime. Secondly, we connected this number of non-neutralized virions to the number of animals protected or infected for a given immunization and challenge regime. Ultimately, this allows us to derive the probability for a single infectious virion to establish a host infection (Fig 5A).

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Fig 5. Antibody neutralization and host infection in macaque passive immunization challenge studies.

(A) Scheme depicting our approach of post-hoc analysis of macaque passive nAb immunization vaginal virus challenge studies. The approach delivers both estimates of in vivo virus inoculum neutralization and the probability of each infectious virion to start a host infection. (B) Total number of SHIV-P3 virions remaining infectious in the four analysed studies, obtained by multiplying the study-specific SHIV-P3 virus inoculum sizes with the fraction of virions remaining non-neutralized by nAbs. Bars depict the lowest, mean and highest number of potentially infectious virions, based on different assumptions of nAb K_D and mucosal nAb concentration (see S14 Fig). (C) Estimates for the probability that a single infectious virion starts a host infection, ψ, shown for each of the four individual macaque studies. The different symbols and bars depict the lowest, mean and highest values of ψ, based on the different estimates for the number of potentially infectious virions as shown in (B). For this analysis, only immunization and challenge regimes resulting in infection of test animals were considered. The average value of ψ (1.65x10^-5) across all studies is indicated.

https://doi.org/10.1371/journal.ppat.1006313.g005

This modelling of HIV-1 neutralization in vivo following vaginal challenge requires data on vaginal nAb concentrations and nAb K_D as well as the virus-specific parameters of inoculum size, , T and N. Importantly, we also require the probability of virions to penetrate mucosal epithelial layers to come in contact with target cells (p_pen); this parameter was recently estimated elegantly by Carias et al. [50]. The four selected macaque studies reported the majority of nAb and virus-specific parameters (S6 and S7 Tables), allowing us to employ our modelling approach in a post-hoc analysis. Essential data of the four analyzed studies are listed below:

1. (i). In the b12 study [8], 5 animals were immunized weekly with 1 mg/kg nAb b12, giving average serum b12 concentrations of 41.8 μg/ml and extrapolated vaginal mucosal concentrations of 0.47 μg/ml (S6 and S7 Tables). Animals were challenged bi-weekly with virus SHIV-P3, first 11 times with 3 TCID50 (397.500 virions) and subsequently 40 times with 10 TCID50 (1.325.000 virions). One animal was infected after 6 challenges with 3 TCID50, while 3 animals were infected after 6, 23 and 38 additional challenges with 10 TCID50, and one animal remained uninfected throughout the challenge regime.
2. (ii). In the 2G12 study [7], 5 animals were immunized once with 40 mg/kg nAb 2G12, giving average serum 2G12 concentrations of 1053 μg/ml and extrapolated vaginal mucosal concentrations of 12 μg/ml (S6 and S7 Tables). Animals were challenged once with 500 TCID50 (66.250.000 virions) of virus SHIV-P3. Three animals were protected, while 2 became infected.
3. (iii). In the PGT121 study [10], 5 animals per group were immunized once with either 5, 1 or 0.2 mg/kg of nAb PGT121, giving average serum PGT121 concentrations of 95, 15 and 1.8 μg/ml, respectively. Vaginal mucosal concentrations were measured at 0.9 and 0.2 μg/ml for the 5 and 1 mg/kg immunizations and extrapolated at 0.02 μg/ml for the 0.2 mg/kg immunization (S6 and S7 Tables). Animals were challenged once with 300 TCID50 (39.750.000 virions) of virus SHIV-P3. All animals with 5 and 1 mg/kg PGT121 immunizations were protected, while 3 animals were protected and 2 became infected for the 0.2 mg/kg immunization.
4. (iv). In the PGT126 study [16], 5 animals per group were immunized once with either 10, 2 or 0.4 mg/kg of nAb PGT126, giving average serum PGT126 concentrations of 98, 20 and 3.6 μg/ml, respectively. Vaginal mucosal concentrations were extrapolated at 1.1, 0.22 and 0.04 μg/ml for the 10, 2 and 0.4 mg/kg immunizations, respectively (S6 and S7 Tables). Animals were challenged once with 300 TCID50 (39.750.000 virions) of virus SHIV-P3. All animals with 10 mg/kg PGT126 immunizations were protected, while 2 animals were protected and 3 became infected for the 2 mg/kg immunization, and 1 animal was protected and 4 became infected for the 0.4 mg/kg immunization.

To derive nAb K_D data for the challenge virus Env P3 used in all four studies, we utilized the known IC50 for 2G12, b12, PGT121 and PGT126 against P3 and inferred the corresponding K_D values by linear regression (S6 and S7 Tables). We further assumed that virus strain P3 has T = 2 and as previously determined for the closely related strain P3N (S3 Table). These data enabled us to predict neutralization curves for the four nAbs against SHIV-P3 (S14 Fig). We then superimposed these nAb neutralization curves with the estimated nAb concentrations in the vaginal mucosa at the time of challenge. In this analysis, we allowed both nAb K_D and mucosal nAb concentrations to vary 2-fold around the estimated values to account for potential inaccuracies in the extrapolation procedures. This analysis yielded windows for the extent of SHIV-P3 inoculum neutralization by the four nAbs in vivo (S14A Fig, colored parts of the neutralization curves). We predicted almost complete SHIV-P3 inoculum neutralization for the highest PGT121 and PGT126 immunization regimes, while the b12 and 2G12 immunizations and the low doses of PGT121 and PGT126 immunizations yielded intermediate SHIV-P3 neutralization levels, mirroring the protective effects seen in the respective challenge studies [7, 8, 10, 16].

Importantly, our model also allowed us to determine the probability that a single infectious virion starts a host infection. To this end, we first calculated the fraction of SHIV-P3 virions that remained potentially infectious in the four macaque challenge studies, i.e. virions with at least T non-neutralized trimers (S14B Fig). Multiplying these fractions of non-neutralized virions with the total number of virions in the respective challenge inocula provided an estimate of the number of virions that remained potentially able to infect each animal (Fig 5B).

To investigate how this number of non-neutralized, infectious virions translates into systemic host infection we utilized a further parameter. During vaginal challenge not all virions of the inoculum will penetrate the vaginal epithelial layers to come in contact with mucosal CD4⁺ target cells; the probability of epithelial penetration is given by p_pen. We assumed that only 0.235% of virions will penetrate the genital tract epithelium as experimentally estimated [50]. Based on this, we derived the number of infectious virions in the four challenge studies that can potentially contact mucosal target cells and set these number in relation to the observed infection outcomes (Fig 5A). This delivered the probability for a single infectious virion to establish a systemic host infection, denoted ψ (Fig 5C). Intriguingly, we obtained closely matching ψ-estimates for the four independent macaque studies, with an average value of .

Next, we asked whether a similar analysis would be possible using in vitro nAb neutralization data (i.e., nAb IC50 and Hill parameter). For this analysis we utilized in vitro neutralization data of SHIV-P3 with nAbs PGT121, PGT126 and b12 (S11B Fig) and analyzed the respective macaque challenge studies. We obtained a closely matching estimate of ψ, i.e. (S15 Fig) as with our mechanistic population neutralization model (Fig 5C). Based on these two complementary analyses, we conclude that only one in ~30.000 to 60.000 infectious virions that have penetrated the vaginal epithelial layers will succeed in systemically infecting the host. This estimate thereby provides a quantitative evaluation of the bottlenecks encountered by HIV-1 during transmission in the genital mucosa.

Predicting human infection probabilities and nAb levels required to protect from HIV-1 acquisition

We next used the estimate of HIV-1 virion host infection probability () to predict male to female HIV-1 transmission risk per sex act and nAb protection efficacy. In this analysis, we define the per-act HIV-1 inoculum as the number of HIV-1 virions per ejaculate; this number is given by per-ejaculate semen volume and semen viral load. We then assumed that only a fraction of virions in the inoculum will penetrate the vaginal epithelial layers, defined by p_pen [50]. We next determined the number of virions that are potentially infectious, i.e. have at least T trimers, and multiplied the fraction of penetrating infectious virions with the probability for each infectious virion to initiate host infection, ψ. Thus, we obtained the probability of host infection in dependence on HIV-1 inoculum size. In addition, we modelled protective effects of nAbs present in the vaginal mucosa; here, the extent of HIV-1 inoculum neutralization (and hence the magnitude of protection) depends on mucosal nAb concentration and nAb binding affinity (K_D) for the Env trimer.

In a first step, we calculated infection probabilities of women during a single penile-vaginal intercourse in dependence on HIV-1 inoculum size, in absence of nAbs. We performed this analysis for HIV-1 virions with the entry characteristics of the transmitted-founder strain BG505 ( trimers per virion, T = 2 entry stoichiometry; S3 Table). The resulting relation between HIV-1 inoculum size and per-act female host infection probabilities is shown in Fig 6A.

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Fig 6. Predicting protective antibody levels in vaginal HIV-1 transmission.

(A) Model predictions for male to female per-act HIV-1 transmission risk in absence of nAbs and in dependence of HIV-1 inoculum size. The range of typical HIV-1 inoculum sizes in semen (defined by semen viral load and semen volume) observed in chronic and acute stages of HIV-1 infection and the corresponding predicted per-act infection risks are indicated in grey shadings. (B) Model predictions for male to female per-act HIV-1 transmission risk in dependence on vaginal mucosal nAb concentrations. A relatively high HIV-1 inoculum of 100.000 virions was assumed. NAbs with four different K_D are modelled, spanning the range from less potent to very potent nAbs. (C) Model predictions for male to female per-act HIV-1 transmission risk in dependence on vaginal mucosal nAb concentrations, assuming four different HIV-1 inoculum sizes and a medium-potent nAb.

https://doi.org/10.1371/journal.ppat.1006313.g006

This relation needs to be interpreted with regard to empirical data of human HIV-1 semen viral loads and per-act transmission risk. In chronic HIV-1 infection, semen viral load typically ranges between ≤ 200 and 20.000 HIV-1 RNA genome copies per mL seminal plasma (i.e., ≤ 100 to 10.000 virions / mL, assuming two viral genomic RNA copies per virion). During acute infection, semen viral load typically ranges between 20.000 and 200.000 RNA copies / mL (10.000 to 100.000 virions / mL) [33]. In rare cases, semen viral loads of several million RNA copies / mL (>1.000.000 virions / mL) were reported [51]. Assuming an average per-ejaculate semen volume of 3 mL [51, 52], this results in typical HIV-1 inoculum sizes of ≤ 300 to 30.000 virions during chronic HIV-1 infection, 30.000 to 300.000 virions during acute infection, and > 3.000.000 virions in rare cases. The corresponding per-act female infection probabilities predicted by our model are ≤ 0.001 to 0.11% during chronic infection (one in ≥ 87.000 to one in 876 sexual contacts), 0.11 to 1.14% during acute infection (one in 876 to one in 88 sexual contacts), and maximum values exceeding 11% (one in 9 sexual contacts) (Fig 6A).

How do these estimates compare to empirical data on female per-act infection probabilities? A frequently stated number for male-to-female penile-vaginal transmission risk is 1 in 1000 sexual contacts (0.1%). However, as previously discussed [32, 53, 54], this likely represents a lower bound of per-act infection risk and may be dangerously misleading. Our model predicts that an HIV-1 inoculum of ~26.000 virions (i.e., a semen viral load of ~17.000 RNA copies per mL, assuming 3 mL semen volume) would result in a female per-act infection risk of 0.1%. This semen viral load lies within the range typically observed during chronic HIV-1 infection. In contrast, several studies reported per-act penile-vaginal female infection risks between 0.5 and 10%, presumably linked to acute or late-stage HIV-1 infection of the male partner and correspondingly higher semen viral loads ([32, 53, 54] and references therein). Indeed, our model predicts HIV-1 inoculums of 130.000 to 2.770.000 virions (semen viral loads of 87.000 to 1.850.000 HIV-1 RNA copies / mL) to result in per-act infection risks of 0.5 to 10%. These values represent the range of semen viral loads typically observed during acute HIV-1 infection and rare cases of exceedingly high semen viral load. Thus, our model appears to precisely capture and recapitulate the interplay between HIV-1 inoculum size and infection probability during HIV-1 transmission in the female genital tract suggested by empirical studies.

Having thus obtained good indications that our model and the decisive parameter of ψ reliably mirror empirical data of human mucosal HIV-1 transmission, we next aimed to model the protective effects of nAbs present in the vaginal mucosa. First, we assumed an HIV-1 inoculum of 100.000 virions and modelled neutralization by four nAbs with a broad range of K_D (Fig 6B). Not surprisingly, this showed that with increasing nAb K_D (decreased Env trimer binding affinity), higher mucosal nAb concentrations are required to provide protection from infection. Specifically, we found that nAbs with a high Env binding affinity, such as PGT121, would afford protection from infection well below mucosal nAb concentrations of 0.1 μg/mL. This suggests that nAbs with high Env binding affinity could exert protective effects in vivo at relatively low doses.

We next looked more closely at the protective effects of a nAb with intermediate Env affinity and reduced potency, reasoning that such nAbs may be more readily inducible by vaccination. We chose nAb b12 as an example and modelled its protective effects in the vaginal mucosa against various HIV-1 inoculum sizes (Fig 6C). This analysis showed that such medium-potent nAbs may provide considerable protective effects starting in the range of 1 μg/mL mucosal nAb concentration.

In summary, the analyses shown in Fig 6 suggest that our model has the potential to retrieve information relevant to human mucosal HIV-1 transmission, offering opportunities for further model extension and application in HIV-1 vaccine research (Fig 7).

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Fig 7. Modelling HIV-1 mucosal transmission and antibody neutralization.

Summary of the parameters incorporated into our model (left), allowing in silico analysis of host HIV-1 infection probabilities and the in vivo protective effects of neutralizing antibodies (right). Parameters shown in red were estimated in this study, while parameters shown in black were retrieved from the literature. The model framework can be expanded by a range of parameters to investigate additional factors involved in HIV-1 transmission and prevention (bottom), and may serve as blueprint for similar efforts in other viral disease settings or mucosal infection processes.

https://doi.org/10.1371/journal.ppat.1006313.g007

Ethics statement

Plasma samples from chronically infected individuals (ZA110, Pat117, Pat118, Pat122) were obtained from biobank samples previously collected during two approved clinical trials, the Swiss treatment interruption trial [104–108] and the Zurich Primary HIV-infection (ZPHI) study (ClinicalTrials.gov identifier NCT00537966) [109]. Written informed consent was obtained from all individuals in the respective studies according to the guidelines of the ethics committee of the canton Zurich.

Cells and Env variants

293-T cells (obtained from the American Type Culture Collection) and TZM-bl cells [110] (obtained from the NIH AIDS Reagent Program) were cultured as described [111]. The origin of HIV Env plasmids is listed in S3 Table. Env point mutations were generated by site-directed mutagenesis (Agilent QuikChange II XL kit). All Env mutants were confirmed by sequencing. V1V2-deleted Envs were previously described [44]. The Luciferase reporter HIV-1 pseudotyping vector pNLLuc-AM was previously described [111].

Reagents

MAbs (see S4 Table) were kindly provided by: PG9, PGT121, PGT128, PGT135, PGT145, b12 and b6 by Dr. Dennis Burton, The Scripps Research Institute, La Jolla, USA. 2F5 and 2G12 by Dr. Dietmar Katinger, Polymun Scientific, Vienna, Austria. 17b and 48D by Dr. James Robinson, Tulane University, New Orleans, USA. 447-52D was purchased from Polymun Scientific. Expression plasmids for VRC01 and PGV04 were provided by Dr. John Mascola, National Institutes of Allergy and Infectious Diseases, Bethesda, USA. Expression plasmids for NIH45.46 and 1.79 were provided by Dr. Michel Nussenzweig, The Rockefeller University, New York, USA. T-20 was purchased from Roche Pharmaceuticals.

Estimating the stoichiometry of neutralization, N, with mixed trimer assays

To produce HIV-1 pseudovirus stocks expressing mixed trimers with varying ratios of nAb-sensitive to nAb-resistant Env, 293-T cells in 6-well plates (250.000 cells per well in 2 ml complete DMEM, seeded 24 h pre-transfection) were transfected with 3 μg pNLLuc-AM and 1 μg Env expression plasmids, using polyethyleneimine (PEI) as transfection reagent. The ratio of sensitive to resistant Env was varied to yield combinations with 100, 90, 70, 50, 30, 10 and 0% of resistant Env. After overnight incubation the transfection medium was replaced with 2.5 ml fresh complete DMEM and virus-containing supernatants were harvested 48 h post transfection. To determine virus infectivity, serial dilutions of virus stocks were added to TZM-bl cells in 96-well plates (10.000 cells per well) in DMEM supplemented with 10 μg/ml DEAE-Dextran. TZM-bl infection was quantified 48 h post-infection by measuring activity of the firefly luciferase reporter (in arbitrary relative light units, RLU). The neutralization activity of mAbs and patient plasma against the mixed trimer virus stocks was evaluated on TZM-bl cells as described [111]. Sufficiently high starting concentrations of inhibitors were chosen to yield clear neutralization plateaus, and data were fitted in GraphPad Prism version 7.0 using the sigmoidal dose-response (variable slope) function. In cases where no clear neutralization plateaus were obtained (less than two consecutive data points giving the same level of neutralization), the position of the expected plateau was provided by the curve fit. Subsequently, the relative infectivity (RI) of each virus stock under saturating inhibitor concentrations was calculated. The resulting RI values were plotted over the fraction of resistant Env (f_R) of each virus stock and the data were analyzed with mathematical models.

Env expression analysis by flow cytometry

293-T cells were transfected with Env and Rev plasmids and processed for flow cytometry as described [37]. Env on the cell surface was detected with biotinylated mAb 2G12 (5 μg/ml) and Streptavidin-APC (BioLegend, San Diego, USA; 1:400 dilution) or Abs 1.79 and PG9 (5 μg/ml) and Cy5-conjugated F(ab′)₂ goat anti–human IgG (Jackson ImmunoResearch, West Grove, USA; 1:500 dilution) followed by cell analysis on a CyAN ADP flow cytometer (Beckman Coulter, Brea, USA).

Mathematical modeling to estimate N and HIV-1 population neutralization

Population neutralization model

1. (i). NAb concentrations that confer sterilizing immunity

To tackle the question how high nAb concentrations must be to neutralize a given virion population, we first study the number of nAbs required to perform this task. We start with a virus population with n_v virions. Each virion has a random trimer number S_i that follows a discretized Beta distribution with mean and variance (see above). We let nAbs bind to these virions until all virions are neutralized. Neutralization of virion i is achieved when at least (S_i − T + 1) trimers are bound to at least N nAbs. This simulation is repeated n_r = 1000 times and the mean of the nAb numbers to reach neutralization is calculated. We introduced this procedure in Magnus and Regoes, 2011 [27].

To transition from nAb numbers required for virion neutralization to nAb concentrations, we model the binding of a nAb, Ab, to an envelope protein, E, with a chemical binding equation [35]: where k_d is the off-rate constant and k_a the on-rate constant. Assuming a first order reaction, the quotient of the product of the reactant concentrations divided by the product concentration follows:

The fraction of bound envelope proteins, f_b, when the equilibrium is reached can then be calculated by

This equation can be transformed to

With this equation it is possible to determine the nAb concentrations needed for sterilizing neutralization by determining the fraction of bound envelope proteins with the simulation tool described above.

1. (ii). Calculating the neutralization levels for different nAb concentrations

We start with calculating the percentage of neutralized virions when n_Ab nAbs are bound to n_v virions. Each virion has a discretized Beta distributed trimer number, S_i, with mean and variance . We then distribute the n_Ab nAbs to the virions such that each envelope protein has the same probability of being bound. Thus A_i nAbs bind to virion i.

The probability that a virion with s trimers is neutralized when a nAbs bind to the complete virion can be calculated as follows: and where P(Y₁ = y₁, Y₂ = y₂, Y₃ = y₃) is the probability that y_j trimers are bound to j nAbs and ℇ_{a, s} is the set of all combinations of a nAbs to s trimers such that at least (s − T + 1) trimers are bound to at least N nAbs. For N = 1 this set is

The fraction of neutralized virions, , can thus be calculated by:

To calculate the mean fraction of neutralized virions, the above described procedure is performed n_r = 1000 times.

Several nAbs have been shown to bind with only one antibody per trimer, including nAbs PG9, PG16 and PGT145 [38, 47, 49]. To account for this binding behavior, we extended the above described model. We still assume that each trimer has three epitopes. However, as soon as one nAb binds to a trimer, no additional nAbs can bind. The fraction of bound epitopes is then f_{b, epitopes} = 1/3f_{b, trimers} and a virion with s trimers is neutralized when (s − T + 1) trimers are bound by one nAb.