An analytical model for genetic hitchhiking in the evolution of antimalarial drug resistance

Abstract

We analytically study a deterministic model for the spread of drug resistance among human malaria parasites. The model incorporates all major characteristics of the complex malaria transmission cycle and accounts for the fact that only a fraction alpha of infected hosts receive drug treatment. Furthermore, the model incorporates that hosts can be co-infected. The number m of parasites co-infecting a host is either a constant or, more generally, follows a given frequency distribution. Although the model is formulated in a multilocus setup, for our results we assume that drug resistance is caused by a single locus with two alleles - a sensitive one and a resistant one. We assume that the resistant allele has a selective advantage only in treated hosts and pays metabolic costs, which causes this allele to be deleterious in untreated hosts. We provide necessary and sufficient conditions for the fixation of the resistant allele. Moreover, provided the resistant allele will sweep through the population, we derive a formula for the time until it reaches a given frequency and in particular for the time until quasi-fixation. Furthermore, we establish an analytical solution for allele frequency changes at a linked neutral biallelic locus due to the rapid increase in frequency of the resistant allele. Our solution describes a local reduction in heterozygosity among parasite chromosomes around the resistant allele, the effect commonly referred to as the hitchhiking effect, as a function of alpha and m. The result therefore allows the investigation of selective sweep patterns under specific demographic settings. We find that the hitchhiking effect is similar but different from the standard model of genetic hitchhiking that assumes random mating and homogeneous selection. In particular, the process of recombination and selection cannot be decoupled. We further explain why standard hitchhiking theory cannot be applied to drug resistance in malaria. Furthermore, we will show that a genome-wide reduction in relative heterozygosity can occur provided a fraction of hosts is infected by a single parasite haplotype. Finally, we show how to incorporate host heterogeneity, and generalize our results to this biologically more realistic case.

Figure 1

Illustration of the transmission cycle.

Figure 2

Illustration of the model structure. Hosts acquire infections by multinomially sampling m sporozoites from the parasite population according to their distribution in generation t. Selection occurs in the host population. Parasites are selected independently in each host. Gametocytes are taken by the mosquitos during their blood meals, and recombination occurs only between gametocytes originating from the same host (only gametocytes further participate in the transmission cycle). After recombination the parasite population is pooled together yielding the frequency distribution of sporozoites in generation t + 1.

Figure 3

Density plots for the threshold value for α. If α is beyond this value the resistant allele will become fixed. The fitnesses are parameterized as in (5). In the white regions the resistant allele is deleterious and gets lost for all values of α. The parameter d_R is specified in the plot legends. The color code is shown in the panels. The white, dashed contour lines are at 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9, respectively.

Figure 4

Density plot of the time until the resistant allele has reached a frequency β as a function of d_R and d_S, which are according to (5). The parameters are specified in the plot label. The color code is shown on the right in numbers of generations. In the white region the resistant allele cannot be maintained in the population. The white, dashed contour lines are at t = 10, 20, 30, 40, 50, 100, 200, 300, 400 and 500 generations, respectively. The black line corresponds to d_S − d_R = 0.2. It is seen that the time to reach frequency β decreases along the black line if d_R and d_S increase.

Figure 5

Illustration of Example 2. (a) Fitnesses of the resistant and sensitive alleles as functions of the drug concentration give by (8). The parameters are specified in the label panel. (b) Contour plot of the time to quasi-fixation. The color code is shown on the right in numbers of generations. In the white region the resistant allele cannot be maintained in the population. The parameters are specified in the label panel. The fintesses are as in (a).

Figure 6

Average relative heterozygosity ℋ^(m)(r) as a function of r for different values of m. The fitnesses are parameterized according to (5). The parameters are indicated in the different label panels. Note the different scaling of the x-axes in the different plot panels.

Figure 7

Average relative heterozygosity ℋ^(m)(r) as a function of r for different values of m for the model accounting for host heterogeneity. The fitnesses are chosen as in Example 3, with N = 10. Hence, there are eleven host classes. The parameters are indicated in the different panels. In figures (a) and (b), the parameters are chosen as in Figures 6(a), and (b), respectively. In figures (c) and (d), the parameters are chosen such that λ and μ are equal to the corresponding values in Figures 6(c), and (d), respectively.

Figure 8

Average relative heterozygosity ℋ^(κ)(r) as a function of r for different distributions for κ as described in Example 4. The parameters are chooses as in Figure 6.