Clonal interference can cause wavelet-like oscillations of multilocus linkage disequilibrium

Abstract

Within-host adaptation of pathogens such as human immunodeficiency virus (HIV) often occurs at more than two loci. Multiple beneficial mutations may arise simultaneously on different genetic backgrounds and interfere, affecting each other's fixation trajectories. Here, we explore how these evolutionary dynamics are mirrored in multilocus linkage disequilibrium (MLD), a measure of multi-way associations between alleles. In the parameter regime corresponding to HIV, we show that deterministic early infection models induce MLD to oscillate over time in a wavelet-like fashion. We find that the frequency of these oscillations is proportional to the rate of adaptation. This signature is robust to drift, but can be eroded by high variation in fitness effects of beneficial mutations. Our findings suggest that MLD oscillations could be used as a signature of interference among multiple equally advantageous mutations and may aid the interpretation of MLD in data.

Keywords: genetic interference; human immunodeficiency virus; multilocus linkage disequilibrium.

Figure 1.

Origin of oscillations in multilocus linkage disequilibrium (MLD). (a) The space of all possible haplotypes, starting from the wild-type (no mutations: all zeros) for a system with L = 3 loci. (b) As evolution pushes the fitness distribution to higher Hamming distances, it generates a signature of over-representation of haplotypes with equal Hamming distance. This is reflected by the sequential rise and fall of k-mutant waves. The time shift between k-mutant waves here is 80 days. (c) Pairwise and three-locus Geiringer–Bennett linkage disequilibria, measured with the wild-type 000 as reference, over the course of the simulation (all the pairwise disequilibria overlap). When taken as a reference haplotype, all haplotypes with the same number of adaptive mutations produce an MLD of equal sign.

Figure 2.

Haplotype dynamics of DFMMI and FSMMI simulations. (a) Haplotype frequencies over the course of a DFMMI simulation with L = 4 loci. Beneficial mutations arise every τ_inter = 100 days (see electronic supplementary material, §SIII) and begin to sweep at a rate ε = 0.095 (see electronic supplementary material, §SIII, equation (S11)). Colours indicate haplotypes with an equal number of mutations k. (b) Haplotype frequencies over the course of a simulation of the FSMMI model with L = 4 selected loci, selection coefficients per mutation s = 0.1, population size N = 10⁵ and beneficial mutation rate μ_b = 10⁻⁴ per locus per generation.

Figure 3.

MMI-induced MLD oscillations are still detectable under drift. (a) Oscillation of the fifth-order MLD in a symmetric full escape graph. The dark blue line is the median of a set of 200 runs, and the upper and lower bounds of the light blue area represent the 2.5 and 97.5 percentiles of all measured MLDs, respectively. The MLD was calculated every 10 days using a sample size of 20 haplotypes. The red trajectory represents the measured MLD from one particular repeat. (b) The wavelet power spectrum in the time-period domain of the fifth-order MLD values obtained with the sampling points of the red line in (a) [48,49]. The horizontal grey line is the true oscillation period of the red time series in (a). The white contour lines indicate regions where the power spectrum values are significantly (less than 5%) non-random. The black lines indicate local power spectrum maxima. The half-transparent region demarcates a low-confidence wavelet power region. (c) The time-averaged wavelet power spectrum. The red and blue dots indicate whether the null-hypothesis that the time-averaged wavelet power may have been generated by white noise is rejected at below 0.05 and 0.1 significance levels, respectively. The maximum spectral density is attained close to the simulated period of T = 200 days (horizontal thick grey line) of the oscillations. (d) The analogous situation to (a) for 100 simulation runs of the FSMMI model with selection, run with parameters L = 4, N = 10⁵, μ_b = 10⁻⁴ and s = 0.1. Samples are taken every five generations or 10 days. (e) Wavelet power spectrum of one randomly chosen MLD trajectory (red line in (d)). (f) Analogous to (c), but without the horizontal line indicating expected value.

Figure 4.

Estimates of speed of evolution based on MLD oscillations versus MMI theory [16]. (a) Estimates of the speed of evolution obtained by wavelet analysis for FSMMI model simulations run for L = 3, 4, 5 and 6 loci, and selection coefficients s∈{0.01, 0.05, 0.1, 0.3} with population size N = 10⁵. v is estimated with equation (2.8) (s is known), where for f we use the MLD-based oscillation frequency estimate (figure 3). The inter-sampling period was Δt = 2 days. Coloured open circles and filled circles correspond to medians and non-parametric confidence intervals (95%), respectively, from those simulations among 100 runs that displayed significant (less than 0.05 level for wavelet-based test) oscillations. (b) Analogous figure for narrow FSMMI model. Here, to compute the speed of evolution v from the inferred oscillation frequency f, we use the mean of the gamma distribution from which the selection coefficients were sampled (s̄ values equal to s values in FSMMI). (c) Analgous to (b), but for the broad FSMMI model.