- Research article
- Open Access
Dynamics of a combined medea-underdominant population transformation system
© Gokhale et al.; licensee BioMed Central Ltd. 2014
Received: 7 October 2013
Accepted: 28 April 2014
Published: 7 May 2014
Transgenic constructs intended to be stably established at high frequencies in wild populations have been demonstrated to “drive” from low frequencies in experimental insect populations. Linking such population transformation constructs to genes which render them unable to transmit pathogens could eventually be used to stop the spread of vector-borne diseases like malaria and dengue.
Generally, population transformation constructs with only a single transgenic drive mechanism have been envisioned. Using a theoretical modelling approach we describe the predicted properties of a construct combining autosomal Medea and underdominant population transformation systems. We show that when combined they can exhibit synergistic properties which in broad circumstances surpass those of the single systems.
With combined systems, intentional population transformation and its reversal can be achieved readily. Combined constructs also enhance the capacity to geographically restrict transgenic constructs to targeted populations. It is anticipated that these properties are likely to be of particular value in attracting regulatory approval and public acceptance of this novel technology.
Curbing the spread of vector borne diseases such as malaria or dengue is possible by eliminating the transmission capabilities of the insect vectors. One of the many approaches to achieve this is population transformation of vector species. In the most commonly discussed application of population transformation the aim is to introduce transgenes into insect populations which render them refractory to spreading diseases. Usually the technique seeks to use evolutionary principles to establish such transgenes at high frequency in populations through the release of genetically transformed stocks (also called population replacement, ). Synthetic disease refractory genes have already been developed for human malaria, dengue fever and avian malaria [2–5]. However, to stably transform insect populations with transgenes that are not selectively advantageous it will be necessary to link refractory transgenes to systems that drive them to high frequency in a population [1, 6–8]. Three transgenic population transformation systems have been shown to be effective in laboratory populations of insects. One is a homing endonuclease based system (HEG), which works by converting heterozygotes to homozygotes . The remaining two systems work by reducing the average fitness of heterozygotes and are: Medea [10, 11] and a bi-allelic form of underdominance . Here we explore theoretically a mono-allelic form of underdominance the implementation of which has to date not been published.
While most studies examine the theoretical properties of transgenic constructs embodying single drive mechanisms [7–10, 13], the observation that “most of them have specific characteristics that make them less than ideal” led Huang et al. 2007  to explore combinations. They demonstrated that certain combinations resulted in enhanced properties relative to single systems while others had the opposite effect. Here we take an analogous approach for autosomal Medea and mono-allelic underdominance constructs (not examined in Huang et al. 2007 ). We provide a rigorous and flexible analytical framework to explore salient properties across the entire parameter space. Intuitively, the inclusion within a single transgenic construct of more than one drive mechanism provides a degree of resilience to either mutations in the transgenic construct or to drive-resistance alleles which may exist in target population. While the value of this desirable functional redundancy is not analytically explored here, it does however provides an additional motivation for analyzing the properties of combined systems. Similar to Huang et al. the motivation for the analysis presented here comes from the realization that intuitive predictions about combined systems can be misleading and that identifying the parameter space where synergistic enhancements occur can motivate technical developments, including the development of mono-allelic form of underdominance.
We briefly summarise the previously known properties of Medea and mono-allelic underdominant systems separately. Then we look in turn at each of the properties of interest and determine if the combined model performs better than each of the techniques independently. The discussion focuses on the impact of a combined system and provides an assessment of its strengths and weaknesses.
When the heterozygote is less fit than both the possible homozygotes then we have a case of underdominance. However there are only a few examples where alleles at a given locus have been robustly inferred to exhibit underdominance . In a random mating, Hardy-Weinberg population, rarer alleles have larger sojourn times in the heterozygote state, consequently where an underdominant construct is rare it will mostly be in this unfit genotype. Due to the inherently unstable nature of underdominance, if the construct exceeds a threshold value through releases of sufficient homozygotes it is predicted to proceed to fixation within the population (Figure 2b). Intentional underdominant population transformation is inherently reversible where it is realistically possible to release sufficient wildtype individuals to traverse the unstable equilibrium in the lower frequency direction. However, underdominant constructs can be viewed as unappealing when transforming large populations due to the high release numbers required to initiate population transformation (Figure 2b) [20, 21]. The mono-allelic underdominance modeled here describes the situation where there is a transgenic allele at a single autosomal locus (the site of the transgenic construct integration). We have only examined situations where an insert is underdominant in both sexes. A recent publication  describing the development of a single locus bi-allelic form of underdominance where there are two functionally distinct transgenic alleles is not applicable to the mono-allelic underdominance analysis described here.
Medea and underdominance in a single transgenic construct
Here we explore the properties of combining both Medea and underdominance in a single transgenic construct on an autosome. As single locus transgenic underdominance effective in both sexes cannot by definition be configured on sex chromosomes we have modeled both systems on autosomes to permit the most direct comparison between single and combined systems. By combining systems, some properties will be discounted, remain the same or synergistically enhanced. We find a broad parameter space where the applied properties of single systems can be argued to be synergistically enhanced. The principle criteria being: (i) lower transformation threshold, (ii) faster population transformation and (iii) enhanced spatial stability of the transformed population.
Methods and results
Genotype fitnesses and expected dynamics
The next generation offspring proportions
Population structure dynamics
We consider a simple two-deme model of population structure, where two populations of large and equal size are coupled by a symmetrical fraction of migrants m between the populations in each generation. Considering asymmetries in population sizes, migration needs to be dealt with separately, as in . Also migration dynamics with an explicitly set spatial system has been recently assessed  (albeit not for a combined system). In population i the expected genotype frequency of genotype k after migration is , where gk,i is the frequency of the k t h genotype in population i and gk,j is the k t h genotype frequency in population j. These adjusted genotype frequencies can then be substituted into Eqs. (1).
Discussion and conclusion
In the theoretical analysis of combined population transformation systems Huang et al. considered the combination of a transgenic two-locus form of underdominance (termed engineered underdominance  with two other natural phenomena (Wolbachia and sex-linked meiotic drive). Both Wolbachia and sex-linked meiotic drive were demonstrated to have the potential to significantly impact the feasibility and dynamics of population transformation in both positive and negative ways. It was clearly shown that intuitive expectations of combined systems could be misleading and that mathematical modeling was essential in identifying potentially useful combinations and parameter values (most notably those relating to genotypic fitness). An excellent example is the Huang et al. theoretical analysis of the two-locus form of engineered underdominance which has been only recently realised .
Here we have followed an analogous approach to explore the properties of combining two currently developed transgenic drive systems within a single autosomal construct. The underdominant and Medea systems are assumed to be physically interspersed in a manner that maximizes the probability that they remain linked (e.g. in a configuration analogous to that shown in Figure 2). The described modeling framework has allowed us to identify a broad parameter space where combined systems can in some circumstances outperform single systems in terms of (i) optimizing release thresholds (Figures 3, 4, 7) (ii) increasing the speed of population transformation and (Figure 5) (iii) enhancing the geographic stability of population transformation (Figure 6). In addition, the reliance on two distinct mechanisms for population transformation could reduce the probability that resistance to the transgenic construct arises in the target insects. If however, long term selective pressures within successfully transformed target populations would result in the loss of the underdominance mechanism, this essentially leaves a ‘Medea only’ construct at high frequency. This ‘Medea only’ construct would be impractical to remove (unless it was associated with a high fitness cost) and could spread to adjacent populations. Conversely, loss of the Medea mechanism from a combined construct has a considerably smaller impact on reversibility and stability (Figures 2 and 6). Recognizing that the loss of Medea is preferable to loss of underdominance, it would be prudent to engineer underdominance which is more mutationally stable than Medea (duplicating the underdominant mechanism would be one simple strategy). It is also noteworthy that many of the synergistic enhancements ascribed to combined systems are to a significant extent shared by Medea constructs inserted on sex-chromosomes . Consequently, depending on the empirical properties of autosomal versus sex-chromosome inserts the relative merits of both approaches would warrant evaluation within the specific objectives of a given program.
It has been assumed throughout that fitness costs are directly associated with the drive mechanism or mechanisms in a transgenic construct, however it is also likely that additional costs will also be associated with anti-Plasmodial or anti-viral genes included as part of a working construct. The analytical framework described here will permit the prediction of the properties of combined systems loaded with such disease refectory genes. The fitness cost of refractory genes has in some, but not all, circumstances been estimated to be quite high . Consequently the illustrative parameters values used in Figure 2 may represent plausible values for ‘loaded’ constructs (though the framework presented here allows exploration of the entire range of parameters). The immediate practical use of this method could help protect D. melanogaster from an unintended species wide Medea transformation if combined with underdominance for testing in the lab. The most likely application of population transformation is in species of the genera Anopheles and Aedes which act as devastating disease vectors . Within these genera there are significant differences in dispersal capacities estimated at various locations, in some instances individuals migrate hundreds of meters over their lifetime . Consequently, the capacity to restrict transgenic constructs to particular populations is likely to be considered of high value. Various configurations of underdominance have been proposed as representing the most likely system to maintain geographic stability [12, 29]. Geographic stability is generally achieved by maximizing the fitness of transgenic homozygotes fitness ν. However where this is not possible due to cost arising from the underdominant drive mechanism or of refractory genes, our analysis indicates that maximal geographic stability can be achieved by combining systems for intermediate values of ν (Figure 6). Exploitation of this phenomena, in addition to the value of functional redundancy in drive mechanisms, could provide a valuable practical incentive to explore combined drive systems experimentally.
Average genotype fitnesses and calculating the equilibria
where the time derivative of a variable is given by and so forth for y and z. From the form of these differential equations the equilibrial solutions are evident, either when the frequencies are zero (vertices of the simplexes in Figure 2) or when the bracketed terms are zero. Since the genotype frequencies sum up to 1, we can solve for just two frequencies. The solutions obtained though are complicated expressions with a possibility of imaginary roots.
We thank P. M. Altrock, J. F. Baines, J. Denton, and A. Traulsen, for discussions and V. L. Reed for comments on the manuscript. CSG is supported by the Emmy-Noether program of the Deutsche Forschungsgemeinschaft. RGR is supported by grant RE-3062/2-1 of the Deutsche Forschungsgemeinschaft. FAR is supported by the College of Natural Sciences, University of Hawai‘i at Mānoa, by the Max Planck Society, and by a grant from the Hawai‘i Community Foundation.
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