A formal description of the population genetics model to study the spread of drug resistant malaria

Transcription

1 A formal description of the population genetics model to study the spread of drug resistant malaria October 22, 2010 Simulations were run varying drug usage (i.e., proportion of infections treated) and fitness penalty between 0.0 and 1.0 in increments of 0.02 for both parameters. The models were run with constant MOIs of 1 (100 % selfing), 2, 3 and 4. The application to perform these simulations is made available in In order to facilitate understanding of the resistance profile of each clone we will use, whenever pertinent, a binary representation per locus where 0 represents a locus that is sensitive and 1 when a locus might confer resistance. As an example, in a case where we model two drugs and 1 resistance locus per drug, the totally sensitive clone can be represented by c 0,0 ; clone c 0,1 is sensitive to drug 1 and resistant to drug 2. For clarity and unless otherwise stated, all textual examples presented below will assume 2 drugs with 1 locus per drug. The following symbols are also employed: 1

2 l is the total number of loci involved in drug resistance; i is the concurrent number of infection on a human host or Multiplicity of infection (MOI); F k is the frequency of resistance profile k in the whole population in the current generation; F k if the frequency resistance profile k in the whole population in the next generation; f k is the frequency of resistance profile k inside a single human individual in the current generation; s is the strength of natural selection acting against each mutation; m is the number of mutated loci in each clone, e.g c 01,11 has m = 3; d is the drug treatment rate, defined as the proportion of infected individuals treated. If more than one drug is used, then all are used in equal proportions. n d is the number of drugs used. The calculations will be performed in a computer because, as will become apparent, the number of permutations of genotypes becomes huge. Here we provide the algebraic basis of the calculations; obviously we cannot describe each case so we will use illustrative examples. Selection pressure is assumed to be mediated through competition between all clones in the asexual blood phase in humans. The proportion of transmissions contributed by a resistance profile k in any single host is: 2

3 t k = f k(1 s) m (1) i f c (1 s) mc c=1 When no drug is present; the denominator is the familiar mean fitness averaged across the c clones of the infection. If a drug is used, only infections that are drug resistant are able to transmit (i.e., t k = 0 for sensitive forms). Being drug resistant depends on epistasis for models with more than 1 locus per drug: for full epistasis all loci are required to be resistant, for duplicate gene function a single locus in enough (and having more than one is slightly deleterious) and for asymmetrical epistasis, the first locus is required. After successful transmission to a mosquito via a blood meal, P. falciparum reproduces sexually (with the possibility of selfing), and as such is subject to recombination. This means that new genotypes can originate inside the mosquito, as an example, if a clone resistant to drug 1 (c 0,1 ) mates with a clone resistant to drug 2 (c 1,0 ), 4 types of offspring are possible: the initial versions plus profiles c 0,0 and c 1,1, i.e., totally sensitive parasites and multidrug-resistant ones. In the case of two drugs and one locus per drug and assuming that the recombination rate among loci is 0.5 (i.e., they are not physically linked) the frequency of transmission of sensitive parasites from a certain individual is given by: f 0,0 = t 2 0, t 0,0(t 1,0 + t 0,1 ) t 0,0t 1, t 1,0t 0,1 (2) 3

4 A similar reasoning applies for all other resistance profiles. In the general case, the probability of a certain infected human host being infected with MOI i composed of a certain combination of clonal genotypes is given by: p(i)p(c 0, c 1,..., c l ) (3) p(c 0, c 1,..., c l ) is calculated from a multinomial distribution using current frequencies of clonal genotypes in the population. In our example, this can be read as the probability of having i infections times the probability of having a combination of infections composed by c 0,0 sensitives, c 0,1 resistant to drug 1, c 1,0 resistant to drug 2 and c 1,1 multidrugresistant. The frequency of each resistance profile k transmitted to the next generation will be: F k = i i c 0 i=1c 0 =0c 1 =0... i c l 1... c 0 c l =0 n d f(e)p(i)p(c 0,..., c l )p(d)t ek e=0d=0 W (4) The structure of the equation can be explained as follows: summation over i allows the investigation of all MOI classes in the population (though in the examples researched, the value is fixed for the whole population, i.e., everyone has the same MOI). Summation over c 0, c 1,..., c l allows inclusion of 4

5 all possible combinations of genotypes within the MOI class. Summation of e allows to investigate the contribution of all environments (untreated individuals plus one or more different epistasis cases). f(e) is the fraction of the host population providing a certain environment (e.g., untreated, treated with duplicate gene function, etc.). p(d) is the probability of receiving a certain drug regimen where d = 0 means no drug. For the untreated environment p(0) = 1. For other environments p(0) = 0 and p(d > 0) is dependent on drug policy (e.g. 1 n d when using multiple therapies simultaneously in the same proportion). For each environment e there is a different transmission proportion for each profile (t ek ). W is a normalization coefficient equal to the sum of all the numerators (in order to assure that the proportions of each type of transmission 2l 1 sum to 1). F k k=0 A similar approach has been used before (Hastings, 2006), and is now extended to allow different environments, more than one drug and several models of epistasis. This more complex model has consequences on the computational cost of the simulation. The number of different genotypes that must be tracked is equal to 2 l, where l is number of drugs times the number of loci per drug. As per the formula above all possible combinations of genotypes for each MOI have to be considered. The number of permutations is then 2 li. This makes the calculation above computationally very intensive. As an example studying a clonal multiplicity of 4 with 64 different genotypes (3 drugs with 2 loci per 5

6 drug) requires considering 16 million cases. The most extreme case theoretically allowed, with a MOI of 7 would require dealing cases (which is not feasible in practice). Furthermore, this computation has to be done for each environment (untreated and each epistasis mode per drug), for every generation and for every simulation (the number of simulations being dependent on the ranges of both fitness penalty and drug usage). Linkage (gametic) disequilibrium is expected to be influenced by epistasis and is a necessary parameter to understand the relationship between frequency and prevalence of resistance. In order to understand and compare the impact of epistasis on linkage, we studied linkage disequilibrium using the correlation coefficient r which, while not completely removing the effects of allele frequency (Hedrick, 1987), still allows for a standardised qualitative comparison required for this study. Furthermore, unlike D, it preserves the original signal of D which is important to understand epistasis effects. Epidemiologically realistic MOIs While the main manuscript only discusses scenarios with constant MOIs (i.e. all humans have the same number of infections), epidemiologically realistic settings have mixed MOIs (Arnot, 1998). The model presented is capable of simulating mixed MOIs but, from a qualitative point of view the results are similar to constant MOIs. Therefore, for the sake of simplicity, the main text manuscript presents results with constant MOIs. 6

7 Below we present a chart of LD patterns using a more realistic truncated Poisson distribution of MOI (truncated at 7 clones with a conditional mean of 2.5). As it can be observed, the chart is qualitatively similar to Figure 2 on the main manuscript (constant MOI of 2). References Arnot, D Unstable malaria in sudan: the influence of the dry season. clone multiplicity of Plasmodium falciparum infections in individuals exposed to variable levels of disease transmission. Trans R Soc Trop Med Hyg, 92, Hastings, I. M Complex dynamics and stability of resistance to antimalarial drugs. Parasitology, 132, (DOI /S ). Hedrick, P. W Gametic disequilibrium measures: proceed with caution. Genetics, 117,

8 8