During early human immunodeficiency virus (HIV) infection multiple CD8+ T cell responses are elicited almost simultaneously. These responses exert strong selective pressures on different parts of HIV’s genome, and select for mutations that escape recognition and are thus beneficial to the virus. Some studies reveal that the later these escape mutations emerge, the more slowly they go to fixation. This pattern of escape rate decrease (ERD) can arise by distinct mechanisms. In particular, in large populations with high beneficial mutation rates interference among different escape strains –an effect that can emerge in evolution with asexual reproduction and results in delayed fixation times of beneficial mutations compared to sexual reproduction– could significantly impact the escape rates of mutations. In this paper, we investigated how interference between these concurrent escape mutations affects their escape rates in systems with multiple epitopes, and whether it could be a source of the ERD pattern. To address these issues, we developed a multilocus Wright-Fisher model of HIV dynamics with selection, mutation and recombination, serving as a null-model for interference. We also derived an interference-free null model assuming initial neutral evolution before immune response elicitation. We found that interference between several equally selectively advantageous mutations can generate the observed ERD pattern. We also found that the number of loci, as well as recombination rates substantially affect ERD. These effects can be explained by the underexponential decline of escape rates over time. Lastly, we found that the observed ERD pattern in HIV infected individuals is consistent with both independent, interference-free mutations as well as interference effects. Our results confirm that interference effects should be considered when analyzing HIV escape mutations. The challenge in estimating escape rates and mutation-associated selective coefficients posed by interference effects cannot simply be overcome by improved sampling frequencies or sizes. This problem is a consequence of the fundamental shortcomings of current estimation techniques under interference regimes. Hence, accounting for the stochastic nature of competition between mutations demands novel estimation methodologies based on the analysis of HIV strains, rather than mutation frequencies.
Strong immune responses both drive and shape the early within-host evolution of acute Human Immunodeficiency virus (HIV) during infection [1]. The earliest of these responses are mediated by CD8+ T cells, which exert selective pressures on the virus that are important to the clinical outcome of infections [2]. A precise quantification of their protective capacity for the host is of great relevance, and could inform HIV vaccine design [3–5]. The significance of CD8+ T cells for the control of viral replication is supported by ample experimental evidence [4,6–14]. One of the most important effector functions of CD8+ T cells is to recognize and kill viral target cells (CD4+ T cells) that present viral peptides, termed epitopes, on their surface, thereby signaling the presence of intra-cellular pathogens [15–17]. Mutant viral strains that code for epitopes that are sufficiently distinct from those recognized by CD8+ T cells, avoid clearance and so receive a substantial selective advantage [3,4, 18]. This phenomenon is termed escape. Mutations that escape from the effects of CD8+ T cells are a recurring phenomenon in HIV and can, in some individuals, be observed multiple times during acute infection [19,20]. Escape mutations occur at multiple sites in the viral genome [19–21]. The presence of several distinct escape mutations during early infection (which we define to last up to ∼4 months) implies that CD8+ T cell pressures act on multiple epitopes simultaneously [20,22]. The selective pressure exterted via the epitope-specific CD8+ T cells influences how fast escape mutations go to fixation. These escape rates are commonly estimated under the assumption of independence between escape mutations: Irrespective of the presence of other escape mutations, the escape rate of an individual escape mutation is assessed by fitting a logistic curve to its frequency time course [22–26]. The CD8+ T cell killing efficacy is then approximated by the estimated escape rate plus an offset arising from fitness costs [24]. Fitting such curves to sample points of the fixation trajectories of mutations in different epitopes shows a pattern of escape rate decrease: late-emerging escape mutations have lower escape rates (that is, they go to fixation more slowly) than early-emerging escape mutations [22,24,26]. Escape rate decrease (ERD) could be generated by different independent mechanisms. In particular, it has been hypothesized to partly stem from effects resulting from HIV’s large population size and high mutation rate [27–31]. In this scenario, beneficial mutations, such as escape mutations, are likely to arise simultaneously at distinct sites in HIV’s genome [32,33]. Since selection dominates the early HIV evolution within patients, beneficial mutations should thus start to rise in frequency [33]. Eventually, they will enter a state of competition, delaying each other’s fixation [34]. Such delays are generated by the mutually inflicted growth impairment among beneficial mutations, termed interference [35–37]. As a consequence, the fixation trajectories of individual escape mutations could become arranged in such way that they give rise to ERD [27]. Whether interference is prevalent enough during early HIV infection to cause or amplify phenomena like ERD remains an open question. On the one hand, given the low recombination rates [38,39] in relation to the selective pressures in early HIV infection, theoretical population genetics suggests that abundant beneficial mutations will generate multiple concurrent lineages [1,33,35–37,40–42]. Furthermore, the presence of interference effects in HIV is corroborated by some recent studies. In one instance, haplotype reconstruction techniques have revealed evidence for interference in one patient during early HIV infection [43]. The recently observed phenomenon of epitope shattering, is a further strong indication for interference at the intra-epitope level [2,22,44,45]. On the other hand, some studies suggest that HIV evolves by the sequential fixation of beneficial mutations [46], which is commonly associated with a lack of interference [33]. This is assumed to be the result of a severe bottleneck in population size during transmission in combination with strong selection [47]. Despite the uncertainty about its role in early HIV evolution, neglecting interference effects could presuppose an undervaluation of the protective role of CD8+ T cells. In interference regimes, the application of logistic curve fitting to escape data will produce biased estimates of killing efficacies, since beneficial mutations do not behave independently. In previous work [27], we have shown that this scenario can unfold in a realistic model of HIV dynamics that incorporates escapes at two loci, leading to severe underestimates of CD8+ T cell killing efficacies. In this study, we investigated whether features of the ERD pattern found in data are consistent with a simple model of interference [33] or alternatively, an interference-free model. To model interference, we developed a discrete-generation Wright-Fisher model with selection and multiple loci. Controlling population sizes, we then studied how increasing levels of interference affect ERD in simulations of early HIV infection under a variety of conditions. Selective advantages were assumed to be equal for all beneficial mutations. Besides different population sizes, simulations were run for different time spans of neutral evolution preceding immune-mediated selection, as well as different rates of recombination and numbers of loci. To assess the effects of alternative mechanisms that preclude interference, we derived predictions for the pattern of ERD under the assumption of free recombination and the presence of standing variation. We then assessed whether both models could replicate the ERD pattern features from data obtained in [26]. We found that interference consistently produces ERD patterns of similar magnitude and shape to those found in [26] in parameter regimes commonly associated with early HIV infection. How the number of loci, recombination rates and neutral evolution phases affect ERD in the multiple mutations regime is well explained by the underexponential decline of escape rates over time. Surprisingly, both the models with and without interference were able to replicate the underexponential decline pattern of patient ERD.
A beneficial mutation’s frequency time course is assumed to follow a logistic curve when going to fixation in the absence of interference [68]. This population genetic result has been rederived and used in the context of HIV dynamics [22–24,26]. In the notation introduced by [26], the frequency time course f (t) of a beneficial mutation is: f (t) = f 0 f 0 + (1 - f 0) e - ϵ t, (1) where f0 is the initial frequency of the mutant and ϵ is the average mutant fitness advantage. ϵ is termed the escape rate of the mutant, which is commonly used as a measure for the growth advantage that an escape mutation confers to its carrier. Besides the escape rate, another useful and intuitively accessible parameter is the time for the mutant frequency to reach fifty percent: τ 50 = 1 ϵ ln 1 - f 0 f 0. (2) This parameter is termed the escape time. Given ϵ and τ50, the logistic escape curve Eq (1) is fully determined. Both ϵ and τ50 must be inferred from experimental data. In simulations, we calculated the ERD by following methods that are commonly employed in experimental studies. Within each simulation run, we determined the trajectory of each mutated epitope’s frequency over time. Mimicking experimental procedures, we sampled these frequencies at predetermined time points similar to those chosen in Goonetilleke et al. [20]—densely sampling in the beginning, and sampling more sparsely late in infection. If not stated otherwise, the sampling times were 0,10,20,50,100,190 and 300 days after the onset of selection. To further approximate empirical settings, simulations were capped at 400 days. In each simulation, we then individually fitted Eq (1) to every epitope-mutation that had gone to fixation, as in [22,24,26]. Each fit for a beneficial mutation going to fixation provides a pair of estimates for the escape rate ϵ and the escape time τ50. Repeated fits lead to a set of pairs (τ50, l, ϵl) for each locus or epitope l ∈ {1, …, L}. These values were then used to calculate an association between successive escape rates and escape times (see S3 Fig for two simulation examples involving escapes, logistic model fits, and the escape rate and time association). This was achieved by fitting a linear regression formula log10 (ϵ) = a + b ⋅ τ50 to the data pairs (τ50, l, ϵl). This is mathematically equivalent to the regression log10 (ϵ/a′) = b ⋅ τ50, with a = log10 (a′) and a′ > 0. The slope of this regression is the value of the parameter b, termed the value of escape rate decrease (ERD value). The more negative this value, the more pronounced the ERD. If the ERD value is zero, there is no escape rate decrease between subsequent escape rates. The logarithm of a value having units, such as log10 (ϵ/a′), does not have a well-defined unit itself, and will thus be treated as unitless. Therefore escape rate decrease values b will be used with units of day−1.
When beneficial mutations are rare in a population of asexually reproducing individuals, they commonly will not overlap temporally, and will go to fixation in a sequential manner. However, if they are common, different beneficial mutations will be present simultaneously in the population, and the periods in which they grow in frequency will likely coincide. Assuming that selection dominates drift, the fixation trajectories of the mutant frequencies will thus be affected by the presence of other, concurrent mutations, i. e. they will interfere. The fixation time—the time at which a beneficial mutation is present in all individuals of a population– of one single mutation will be delayed with respect to the concurrence-free scenario. Mathematically, a beneficial mutation is defined to interfere with another whenever its establishment time –the average time for a mutation to become so prevalent that it is assumed to go to fixation deterministically– is smaller than the average fixation time [33]. In population genetics, this is called the “concurrent mutations regime” [33]. Desai and Fisher distinguish two aspects of the concurrent mutations regime: clonal interference [32] as well as multiple mutations [33] effects. Here, we will focus on the multiple mutations case only, where each mutation is assumed to confer the same selective advantage s. Due to its simplicity, this case serves as a useful null-model for interference. If a system is in the multiple mutations regime, it will also display interference. Whether interference occurs in the adaptive evolution of an asexual system is assessed by the measurement of three parameters: the population size N, the beneficial mutation rate per generation per individual (or sequence) μb and the fitness advantage s associated with such a beneficial mutation. The system is in the concurrent mutations regime if [32,33]: N μ b ≳ 1 ln (N s). (3) From this inequality it follows that increasing the population size N makes interference more probable if μb and s are held fixed. In fact, we can interpret Eq (3) as defining a threshold value for N, N0 (μb, s), above which interference should be the norm in a system, and below which sequential fixations should be prevalent. Fig 1 shows the difference between Nμb and 1 ln (N s) over a wide range of values of N and for different values of s. The curves have a zero at N0, which defines a threshold for the transition from the sequential fixations to the concurrent mutations regime at higher N [33]. For the parameters used to model interference in this study, μb = 10−4 and s = 0. 5, we obtain N0 ≈ 1. 5 ⋅ 103. Although this theory can only be applied indirectly to HIV –since HIV exhibits some recombination it breaks the prerequisite of asexual evolution– the above considerations can lead to useful insights. Assuming that due to strong selective pressures HIV is quasi-asexual, most estimates for N and s found in the literature should place HIV in the concurrent mutations regime during early infection, right of N0 part of Fig 1. The estimates for HIV’s effective population size lie between 104 [52,60,61] and 107 [28,29,69] during the asymptomatic phase. The values used in models that focus on early HIV infection tend to have a broader range, namely between 102 and 105 [46,52,70]. The selection coefficients range from s = 0. 81 [24,46] to s = 0. 01 [20,24,26]. Thus, these considerations raise the question whether interference effects are consistent with common estimates found the HIV literature, and ERD in particular. To investigate whether interference could cause ERD in escapes occurring simultaneously at multiple loci, we adopted a simple strategy: In simulation experiments, we tracked how ERD changed with increasing population size. According to the population genetic theory outlined above, a system must transition from a sequential fixations to a multiple mutations regime, where interference effects are present. If interference causes ERD, we would thus expect pronounced ERD to emerge as the system enters an interference regime. To track ERD, we utilized a Wright-Fisher model of early HIV infection that incorporates multiple epitopes or loci, selection, mutation and recombination (see Materials and Methods). We examined the regime transition under different recombination rates, different neutral phase lengths and different numbers of loci. Distinct effective recombination rates were achieved by varying the genomic distances between loci coding for beneficial mutations on the genome. The number of loci considered is in accordance with the empirically observed number of CD8+ T cell clones simultaneously targeting distinct epitopes [20,22,71–73] (see Materials and Methods). For each combination of parameter values considered, 100 simulation repeats were run. To be able to compare the ERD values in the simulations with data, we adopted the empirical sampling scheme described in Materials and Methods. We observed that in all examined scenarios, as the system transitions into the interference regime, the ERD values become negative, indicating more pronounced ERD (see Fig 2). This corroborates the notion that interference does cause ERD in our null-model system. This effect persists across realistic parameter ranges. In particular, over the range of populations sizes examined, the higher the population size N, the more interference we expected, and the more negative the ERD values measured in simulations were. These results have to be interpreted in the context of empirical ERD values. The ERD values in the patients analyzed in [20], based on the data presented in the Supplementary Material of [26] are comparable in magnitude, as reported in [27]: For escapes with τ50 within the first two years, values are about −0. 006 day−1 (CH44), −0. 008 day−1 (CH77) and −0. 01 day−1 (CH58). These values are only reached at population sizes of about N = 104−105 in our simulations. We also tested whether the goodness of the logistic model fits was affected by the emergence of interference. As a goodness of fit statistic, we used the medians across simulation repeats of the medians of the residual sum of squares (RSS) of logistic fits across epitopes. The goodness of fit worsened as more interference was attained (see S4 Fig). As expected, recombination markedly improved fit quality. Higher levels of standing variation should lead to more interference (see below). Thus, shorter phases of neutral expansion also improved fit quality, which we hypothesize is explained by the reduction of standing variation. In the following, we examine and explain how sampling schemes, the number of loci, recombination rates and neutral phase lengths affect ERD values. We further analyze whether ERD values can be assumed to significantly differ from zero. The support for the notion that interference does entail ERD in the multiple mutations regime led us to analyze whether conversely, observed ERD contains information about the presence of interference during early HIV infection. Several mechanisms, such as different times of emergence of CD8+ T cell responses [70], or genetic standing variation in escape mutations segregating in the population might importantly affect ERD. Since in past studies the CD8+ T response emergence times [75] and strengths [26] were not predictive of the escape features during early infection (∼4 months), we focused on how standing variation might relate to ERD. To address this question, the effects of standing variation in selective advantages need to be disentangled from interference effects in the pattern of ERD. To this end, we derived a function for the pattern of ERD under the assumptions of free recombination and variation in selective advantages (variation in s, or equivalently, ϵ of mutations). More specifically, the function predicts what escape rate ϵ should be expected at a specific escape time τ50 under these assumptions. To obtain this function, we again employed a simple model of early infection. We followed the fate of a single escape mutation through two phases: A neutral phase of viral expansion, free of selective pressures [52] and a second phase with strong selection. In the neutral phase, the escape mutation is subject to expansion dynamics. We calculated the mutation’s frequency at the time of the onset of selection ps. For the selection phase we used the insight that given an ϵ of an escape mutation (corresponding to some selective advantage s), the frequency trajectory of that escape mutation is fully specified by ps when going to fixation in the absence of interfering mutations. More specifically, the trajectory follows a logistic curve Eq (1) with escape rate ϵ and f0 = ps. Considering a range of ϵ of an escape mutation, will thus account for standing variation of mutations with different ϵ.
In this study, we investigated whether multiple interfering concurrent mutations of equal strength can generate patterns of ERD and conversely, whether the presence of ERD in data is indicative of interference effects. To do this, we utilized a computational model of HIV based on a multi-locus Wright-Fisher process with selection, mutation and recombination. The model implements much biological detail in recombination processes as well as in the neutral phase preceding the onset of immune responses. We found that when utilizing sampling procedures commonly employed in experimental settings, interference causes ERD in multi-epitope systems. However, the presence of ERD is not indicative of interference effects, and could be caused by alternative mechanisms. One caveat of this study might involve its reliance on the abundance of escapes to study CD8+ T cell mediated pressures. A recent study by Roberts et al. following 125 HIV-infected adults for three years, found that a third of patients did not exhibit any escapes, suggesting that escapes might occur less frequently than previously thought [78]. Our study focused on effects produced by beneficial mutations of the same fitness advantage; thus we neglected clonal interference interactions in the specific sense of [33]. According to this narrow notion of clonal interference, slightly beneficial mutations are outcompeted by more advantageous mutations that appeared in another background before going to fixation. Crucially, the possible acquisition of an additional rescuing beneficial mutation before extinction is neglected. In contrast, a system generating interference effects under the conditions simulated in this paper is said to evolve in the multiple mutations regime [33]. The combination of both of these processes is encompassed in the concurrent mutations regime [33,79]. Whether quasi-asexual systems evolve predominantly by multiple mutations or clonal interference depends critically on the shape of the density distribution of fitness effects [1,33,79–81]. Density distributions, which go to zero very slowly for larger positive values of s (as 1/s3), should show clonal interference effects that resemble sequential fixations [33]. Fitness distributions with a long positive tail, i. e. , which are broader than exponential, are expected to display clonal interference [82,83]. Short-tailed fitness distributions lead to multiple mutations, and the borderline case of an exponential fitness decay combines characteristics of both [80,81]. Theoretical work suggests that the fitness distribution should be exponential [84–88]. Since the distribution of fitness effects in HIV is not known, our analysis may have ignored important effects of clonal interference. The distribution of beneficial point mutations seems to depend on the selective environment [89], and the fraction of mutations that are beneficial varies among studies [89–91]. However, Desai and Fisher note that systems that satisfy the conditions for the emergence of clonal interference also satisfy the conditions for multiple mutations [33]. This is supported by theoretical results showing that even broad fitness distributions of beneficial mutations can result in a narrow fitness distribution of mutations ultimately going to fixation, averaging out most of the variance in fitness effects [92,93]. The assessment of the effects of the fitness distribution of mutations in HIV is further complicated by different abundances of distinct epitope-specific CD8+ T cell clones. On the one hand, this immunodominance hierarchy could effectively generate fitness distributions with long tails: More abundant CD8+ T cell clones could result in stronger selective pressures, and thus higher selective advantages for mutations evading them [94]. The relative CD8+ T cell clone abundances would then reflect the differences in selective advantages of each escape mutation. This seems to be suggested by the correlation found in Henn et al. between the escape rate and CD8+ T cell response frequencies [22], albeit over long durations of HIV infection of about 1600 days. On the other hand, whether the killing efficacy of CD8+ T cells is linearly dependent on their abundance or density is questionable. Models of CD8+ T cell action in the spleen show that killing efficacies saturate as a consequence of the mutual impairment of cell recognition and clearance induced by the presence of other CD8+ T cells [95,96]. This saturation effect could level the selective advantages of individual escape mutations, making them very similar. In fact, no correlation between average CD8+ T cell abundances and escape rates was found in three other patients [20,26]. In light of these considerations, this study only elucidates the minimal expectation of interference effects in the concurrent mutations regime, namely the emergence of multiple mutations. The inclusion of clonal interference effects should exacerbate the bias induced by logistic model fitting. In this study, we ignored fitness costs of escape mutations for several reasons. First, by proposing a very simple model of early HIV dynamics we were able to more clearly identify the sources of some of the effects observed, such as ERD. Second, there seem to exist few high-cost escape mutations [97] and the selective pressures during early HIV are high compared to the fitness costs [24]. This suggests a small role for fitness costs during the phase of HIV infection studied here. Third, compensatory mutations might also be neglected if they go to fixation faster than the escape mutations. Another caveat stems from CD8+ T responses decreasing in strength over time, as suggested by [98]. This behavior could produce the pattern of ERD if all CD8+ T cell responses were decreasing simultaneously. However, interference and decreasing CD8+ T responses are not mutually exclusive. Hence, a substantial fraction of the ERD pattern could still be produced by interference effects, despite declining immune responses. In a recent, paper, van Deutekom et al. have offered an alternative explanation for ERD employing an ODE-based virus dynamics model of HIV infection [70]. A mathematical analysis yields that the contribution to killing of each individual CD8+ T cell clone must be inversely associated with the breadth of the response. Thus, a broadening response will result in decreasing escape mutation advantages, which should in turn lead to ERD. In our study, we have neglected this effect of immune response broadening. However, interference and CD8+ T cell-broadening induced ERD are not mutually exclusive. Also, unlike van Deutekom et al. we have neglected the early fluctuations of the viral load. This simplification was motivated by the results of our previous work [27], where such fluctuations had little impact on the emergence of interference. We simulated population sizes up to 105, whereas the number of infected target cells lies between 107 and 108 [28,29,69]. Increasing the population size reduces the effects of linkage between beneficial mutations [1,99]. However, the virus does not necessarily replicate in all of the infected cells, since a large part of the integrated viral DNA could be non-functional [1]. Effective population sizes have been estimated to be lower, namely around 105 [60,61]. Our results have to be interpreted in the light of the mounting evidence for the role of interference and linked inheritance in a broad range of organisms replicating at large population sizes, ranging from viruses [100], to bacteria [92] to eukaryotes [101]. Our results stress the need for further investigation of the multilocus evolution of HIV during early infection. Accurate assessment of CD8+ T cell killing efficacies depends crucially on information about the dynamics of haplotypes constituting the population. If interference effects are important, neglecting linkage will bias estimates of CD8+ T cell killing efficacy towards smaller values and underestimate their potential for vaccine development. That interference effects in early HIV infection could bias the estimation of escape rates has only been recognized in a few publications [27,34,49]. ERD has been identified in simulations of stochastic multi-epitope models of HIV integrating an immune response [102]. However, as in [34,49], interference was not discussed as a possible causal mechanism for the ERD pattern. In a recent paper, Ganusov et al. presented a mathematical model of HIV within-host evolution which included recombination, but no explicit immune responses [49]. To simulate HIV adaptation following infection of the host, they utilized a Wright-Fisher process based on FFPopSim [48]. The simulations showed that delays increased between the emergence of mutations when compared to fully deterministic dynamics as the effective recombination rate was decreased. The bias in escape rate estimation was particularly pronounced at low sample sizes of around 20. In a similar paper, Kessinger et al. attempted to estimate escape rates from [20] with a multi-locus model partially allowing for interference effects [34]. The estimation was performed by modeling only a reduced set of haplotypes, assumed to go to fixation sequentially. In line with the predictions under interference, the corrected selective advantages of beneficial mutations were much higher than previously estimated, and showed no evident decrease over time. Evidence for the coexistence of multiple mutations is emerging especially within single epitopes—called epitope shattering. Mutational pathways found in epitope shattering were studied by Leviyang [103] in data from [104]. This shattering phenomenon suggests that if recombination rates are low, interference effects should also appear in inter-epitope mutations. Failure to identify such effects between epitopes would be highly indicative of either a long tailed fitness distribution of escape mutation advantages due to immunodominance or larger recombination rates than previously expected. Irrespective of the outcome of further research into the matter, the current practice of treating several mutations within an epitope as identical is problematic. The frequencies of these mutations are commonly added together to form the frequency of one overall escape mutation at that same epitope [20,22,26]. Interference effects within epitopes could spill over to between-epitope interactions, thereby producing considerable estimation biases, as can be seen when analyzing the data from [22,43].
Within-host evolution of human immunodeficiency virus (HIV) is shaped by strong immune responses mounted against the virus. Multiple CD8+ T cell populations, each recognizing a specific part of an HIV protein, simultaneously suppress HIV growth. Escape mutations that arise in HIV genome regions coding for these virion protein parts, impair CD8+ T cell recognition and are consequently strongly selected. The emergence and rise of these escape mutations exhibits an intriguing temporal pattern: the earlier an escape mutation arises, the faster it goes to fixation. This pattern is termed escape rate decrease (ERD). In this paper, we tested computationally whether interference, i. e. the coexistence of multiple genetically distinct HIV strains engaged in competitive interaction within the host, could be a possible source of ERD. As an alternative, we also mathematically derived the temporal pattern of escapes under interference-free conditions, and compared this with data. We found that interference between multiple beneficial mutations could generate ERD. However, ERD does not imply the presence of interference. Thus, more detailed data is required to unambiguously determine whether interference effects influence ERD generation. Nevertheless, interference should be considered when studying the within-host evolution of HIV. Ignoring its effects on population dynamics can severely underestimate the protective capacity of CD8+ T cells.
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Investigating the Consequences of Interference between Multiple CD8+ T Cell Escape Mutations in Early HIV Infection