12.1. Introduction to Threshold-Dependent Gene Drives

Gene drives have been proposed as valuable tools in the fight against a range of globally important issues, including vectors of disease, invasive species and agricultural pests. These approaches are classified primarily based on their persistence and/or invasiveness. Here we consider persistent (i.e. self-sustaining) and low-invasiveness (i.e. threshold-dependent) approaches using engineered underdominance as a case study.

Gene drive is a phenomenon whereby a particular gene (or suite of genes) can bias inheritance in its (their) own favour, thus allowing it to increase in frequency over successive generations, even when deleterious to carrier individuals (Sinkins and Gould, 2006; Alphey, 2014, 2020; Champer et al., 2016; NASEM, 2016; Leftwich et al., 2018;) (see Raban and Akbari, Chapter 8; Champer, Chapter 9, this volume). This may occur by a range of natural or synthetic mechanisms, most of which act in one of two ways: (i) conversion of progeny individuals into other genotypes; or (ii) reducing the fitness (or killing) of progeny individuals of certain genotypes.

The precise configuration of gene drive components can lead to systems with a wide range of different behavioural characteristics, and these are often used to classify the various gene drive systems. Perhaps the most common classifications are based on their intended purpose (usually population modification or suppression), invasiveness (ability to spread into non-target populations) and persistence (whether they remain in the population or diminish over time).

Owing mainly to coverage in popular media outlets, the term gene drive is often associated with only widely known systems such as some CRISPR-based homing approaches. These are expected to have relatively straightforward behaviour in that they are highly invasive (spreading from extremely small releases and so likely also to spread to all populations linked by any degree of gene flow), highly persistent (at least in absence of resistance) and able to be used flexibly for either population suppression or modification (see Bottino-Rojas and James, Chapter 11, this volume). As discussed previously (James, 2005; NASEM, 2016; Harvey-Samuel et al., 2019; Long et al., 2020; Lanzaro et al., 2021), the first gene drive trials are likely to be conducted in remote areas such as highly isolated islands to limit the probability of spill-over into non-target populations. It is debatable whether geographical containment of this type is adequate for non-localized (sometimes also referred to as ‘global’) gene drive approaches, as it would be difficult to guarantee perfect confinement within the trial area. Another potential drawback is that any gene drive releases would presumably need to gain regulatory approval from all affected countries, which for global systems could be argued to be all countries in which the target species (and any capable of forming fertile hybrids) are present (see Beech et al., Chapter 25, this volume). For the first gene drive trial releases, such widespread regulatory approval required for a global system would seem challenging to obtain, at least for widely distributed species. Localized gene drives may need approval only in the target territory and may be particularly suitable where homogenous modification of every population of the species is not desired.

Though not as widely discussed in the media as non-localized drives, several designs for localized drives have been proposed and subject to considerable analysis. Here we focus on two-locus engineered underdominance (UD) – an example of threshold-dependent gene drive that should be persistent, reversible and spatially restricted. This would appear to answer much of the concern around non-localized gene drive approaches, since the system is unlikely to spread significantly beyond any initial trial site and can be reversed easily in the event of unintended consequences – features likely to prove desirable to regulatory bodies and other stakeholders in the context of initial proof-of-concept gene drive field trials.

In this chapter, we outline a range of modelling approaches that have been used to demonstrate the key characteristics of this approach, including threshold introduction frequencies, reversibility, spatial limitation and robustness to mutation/resistance. We then go on to discuss alternative configurations based on the use of sex-specific components and their effect on introduction thresholds. We conclude with a discussion on the cycle of information between mathematical models and experimental data along with a range of areas for future modelling that will be important in providing information on the anticipated effects of these systems when released into target populations.

12.2. Two-Locus Engineered Underdominance

Underdominance, also known as negative heterosis, is a natural phenomenon and the opposite of the better-known overdominance (positive heterosis or hybrid vigour). Thus, underdominance occurs where hybrids are of lower fitness than either of two different true-breeding parental strains; for practical purposes, one of these parental strains is wild-type, the other is the under-dominance-based (UD) gene drive strain. In a single locus scenario, modelling of such selection against hybrids has been shown to allow for the eventual fixation of one allele, with the other being eliminated (Wilson and Turelli, 1986; Altrock et al., 2010, 2011). More recently, transgenes displaying these properties have been developed and tested (Reeves et al., 2014; Maselko et al., 2020; Buchman et al., 2021). The UD concept can also be expanded beyond a single locus. UD gene drives can potentially be developed using transgenic constructs containing toxin and antidote components. The particular configuration considered here is two-locus UD as originally proposed by Davis et al. (2001). This approach requires two distinct transgenic constructs to be inserted at independently segregating (unlinked) genomic loci, each of which comprises a lethal effector (toxin) and a suppressor (antidote) for the toxin of the other transgenic construct (Fig. 12.1). One (or optionally both) of these transgenic constructs will also contain a genetic cargo aimed at producing a desirable phenotype within the target population, for example a reduced ability to transmit a given pathogen (e.g. Franz et al., 2006, 2014; Buchman et al., 2019) (see Franz, Chapter 22, this volume). This combination of transgenic components gives an underdominance-like effect by creating a negative selection (via a lethal effect) on individuals carrying just one of the transgenic constructs (since they contain a toxin but not the antidote from the other transgenic construct). This results in a positive selection pressure for individuals to carry either both or neither of the transgenic constructs (Fig. 12.1). The precise strength of this selection pressure is dependent on several factors, including the degree of lethality conferred on affected genotypes and the fitness cost caused by the presence of the gene drive constructs, for example imperfect suppression of the lethals by the suppressors, or insertional effects of the transgene on nearby genes.

Fig. 12.1

A schematic diagram illustrating the workings of a two-locus engineered underdominance gene drive system. (Left) This gene drive design requires the introduction of two distinct transgenic constructs at independently segregating genomic loci. Each transgenic (more...)

12.3. Mathematical Modelling Approaches

Building gene drives in the laboratory is an inherently time-consuming and expensive activity. Mathematical modelling, on the other hand, can be conducted relatively quickly and, by comparison, inexpensively and allows systematic exploration of parameter space far more readily than empirical methods. It is therefore extremely beneficial to model any proposed gene drive approach in advance of (or at least concurrently with) the laboratory development of transgenic components, and this has indeed been widespread practice. Models can be used to determine essential performance targets that must be met for engineered gene drives to achieve their intended function, particularly within laboratory-based experiments or field-based trial releases. The structure and complexity of such gene drive models can vary dramatically, with each having their own respective benefits and limitations. It is here that experienced mathematical modellers are key in determining the most appropriate model to use in any given scenario while understanding and being able to communicate to non-modellers how a given model structure may influence modelling outcomes. It seems intuitive that mathematical modelling has the potential to save vast amounts of experimental time, effort and money where gene drive designs and engineered components are not fit for purpose. Perhaps less obvious is the requirement for models, and model-based conclusions, to be used (or at least scrutinized) by experienced modellers who understand the impact of model assumptions, complexity and limitations.

A variety of mathematical modelling approaches have been used to provide insight into the predicted performance of UD gene drives; however, much of this work has focused on deterministic population genetics models. These can provide insight into the basic function and utility of UD systems. Such models typically consider an idealized scenario consisting of an infinitely large population (avoiding stochastic effects and the need for integer numbers of individuals) that is isolated (closed from any migration) and panmictic (randomly mating). For simplicity, it is also commonly assumed that females mate only once and produce a 1:1 (female to male) sex ratio in their offspring. Attention is typically restricted to the case whereby no resistance mechanisms can emerge and where each component of the introduced transgenes is immutable and perfectly linked (i.e., toxin, antidote and cargo genes are unable to separate from one another). Finally, it is commonly assumed that generations of offspring in modelled populations are synchronous (i.e., non-overlapping), which may not always be realistic but can apply to laboratory caged populations or wild populations that are synchronized by climatic factors (e.g., wet and dry seasons). This allows for the use of simple recurrence relations (i.e., difference equations) to model the population genetics resulting from the release of such a gene drive. This typical set of simplifying assumptions is also adopted in the example below.

Much of the modelling of UD gene drives uses genotype-based population genetics models. In the case of a two-locus approach such as UD, this results in a total of nine possible genotypes – homozygous, heterozygous or wild-type for the transgene at each of two loci – and therefore a set of nine difference equations. However, since UD is based entirely on Mendelian inheritance and lethality to certain offspring genotypes, the offspring in each generation are directly related to the proportions of each allele present in the parental generation rather than the precise parental genotypes. This allows the consideration of a simpler model that requires the tracking of only the four haplotype frequencies (ab, Ab, aB and AB, where a/b represent wild-type alleles and A/B their transgenic counterparts). This results in a set of four difference equations that may be solved recursively in a manner similar to that originally presented by Davis et al. (2001). Magori and Gould (2006) considered a similar model structure but additionally allowed for multiple insertions of each transgene, though this is not included here. This model is of the form:

The model presented here provides one of the simplest possible models of UD gene drive and is useful for determining various base-level characteristics of the system. As with all models, this is based on a range of simplifying assumptions (described above), each of which is likely to have its own implications. There is a wide range of other modelling approaches that can be (and have been) used to capture the anticipated effects of relaxing one or more of these model assumptions. Several of these are briefly discussed in the following sections, focusing primarily on results rather than extensive modelling detail. Modelling of UD systems has spanned a range of model structures, including difference equations, ordinary differential equations (ODEs), delay differential equations (DDEs), partial differential equations (PDEs) and stochastic models, each of which provides insights into different aspects of anticipated UD behaviour.

12.4. Introduction Thresholds

Underdominance acting at a single locus has been shown to produce bistable dynamics: either homozygotic state can be stable, depending on the initial frequencies and relative fitness of each type, as shown for underdominant alleles (Wilson and Turelli, 1986; Altrock et al., 2010, 2011) and chromo-some translocations (Curtis, 1968). UD gene drives seek to capture a similar effect synthetically, via the introduction of toxin and antidote elements. While the threshold dependence of UD has been demonstrated using numerical simulation under a range of release sizes, to our knowledge a full equilibrium analysis has yet to be conducted. This can be achieved either analytically or computationally and here we focus on the latter, using the numerical continuation software package XPPAUT (Ermentrout, 2002), producing results shown in Fig. 12.2.

Fig. 12.2

A bifurcation diagram showing the possible equilibria of a two-locus engineered underdominance gene drive and their associated stability properties. Here stable equilibria are shown by solid black lines while unstable equilibria are shown by red dashed (more...)

These results show two distinct regimes of behaviour separated by a particular relative fitness parameter ε* ≈ 0.725 (as these are applied multiplicatively, this gives UD double homozygotes a relative fitness of just ∼0.276). In the region ε < ε* there are two possible equilibrium states, with either the wild-type (stable) or gene drive (unstable) alleles at fixation. This can be interpreted as a scenario in which the gene drive is unable to establish itself, no matter how many gene drive-carrying individuals are introduced. The unstable equilibrium whereby gene drive alleles are at fixation is not biologically feasible, since it would imply there were no wild-type individuals present at the time of release – rendering the release of a gene drive unnecessary. The more interesting region (ε > ε*) displays four possible equilibrium states. The first is the unstable (and not biologically feasible) equilibrium state with gene drive alleles at fixation. The three remaining equilibria together constitute a bistable system (i.e. two stable equilibria separated by an unstable equilibrium). Focusing on the gene drive allele (Fig. 12.2c) below the unstable equilibrium, the elimination of the gene drive is the only stable equilibrium – representing negative selection against the gene drive when introduced at a sub-threshold frequency. Above the unstable equilibrium, the gene drive moves towards a stable equilibrium with high gene drive frequency – representing positive selection when introducing the gene drive above the threshold frequency. Theoretically it is possible that the system would attain the unstable equilibrium state and remain there; however practically this is exceedingly unlikely, due to the many and varied stochastic effects present in the real world.

A feature evident in the results of Fig. 12.2 is that the equilibrium state for a UD system does not necessarily comprise only gene drive homozygotes. Where there are no fitness costs associated with the gene drive, the system can reach fixation (i.e. 100% gene drive homozygotes). However, where fitness costs are non-zero, wild-type alleles are expected to be present at a frequency that increases with the fitness costs of the system (Fig. 12.2b).

While useful in displaying the bistable (i.e. threshold-dependent) nature of a UD gene drive, Fig. 12.2 is not necessarily of direct use when planning a gene drive release. This is due to the combination of AB and Ab/aB haplotypes present at the unstable equilibrium (i.e. the introduction threshold). In practice it would be more convenient to know a single gene drive frequency above which the system must be introduced for it to increase in frequency within the target population. Fortunately, this can be calculated by summing to obtain the overall gene drive allele frequency for each point on the unstable equilibrium line. This results in a single threshold gene drive allele frequency (as shown in Fig. 12.3) that must be exceeded through any combination of heterozygote or homozygote individuals. Note that these results align with the pattern observed in several previous studies (e.g. Magori and Gould, 2006; Edgington and Alphey, 2017, 2018; Dhole et al., 2018, 2020; Leftwich et al., 2018), though precise thresholds may differ due to assumptions on the application of fitness costs to individuals carrying multiple transgenic constructs and the choice of presentation method.

Fig. 12.3

Threshold introduction frequencies required for an engineered underdominance system to spread in a target population over a range of relative fitness parameters. Note that this figure was generated by summing gene drive allele frequencies from results (more...)

12.5. Relaxing Model Assumptions

As discussed above, gene drive models are based on a range of simplifying assumptions, each of which has its own implications for the outcomes and applicability of models to different scenarios. The model outlined in section 12.3 represents one of the simplest useful representations of a UD gene drive system and, as shown in section 12.4, allows key characteristics of this gene drive design to be elucidated. Of course, this model structure can be altered to allow for the relaxation of any of the model assumptions outlined above, thereby allowing their implications to be explored. In the following sections we discuss studies exploring the relaxation of three such model assumptions: (1) the presence of resistance formation and mutation of transgenic constructs; (2) the reversal of UD gene drives; and (3) the incorporation of spatial effects.

12.5.1. Resistance formation and mutation

A common set of simplifying assumptions for gene drive modelling studies is that elements within a single transgenic construct are perfectly linked (i.e., unable to separate), do not undergo mutation (i.e., lose function of transgenic components) and that no other resistance mechanisms emerge. Edgington and Alphey (2019) relaxed this assumption by modelling a scenario whereby transgenic constructs accumulate loss-of-function mutations at a constant rate. To our knowledge, rates of mutation in insects likely to be targeted by gene drives are not well studied and could vary considerably, depending on the molecular biology of the gene drive system. Thus, mutation rates (per gene) are assumed to fall within the range 10^–4–10^–8 that should span rates relevant to a range of target insect species. This parameter range is based on measured mutation rates in Drosophila melanogaster (estimated as being of the order 10^–9 per nucleotide per generation) (Haag-Liautard et al., 2007; Keightley et al., 2014); the size of gene drive constructs in previous studies (∼1–10 kb) (Windbichler et al., 2011; Reeves et al., 2014; Champer et al., 2017) and an assumption that ∼1–10% of nucleotides in transgenic constructs are essential for gene function. Note that these mutation rates are assumed to be low enough that multiple mutations (i.e., in more than one gene) within a single generation may be neglected, leading to the pattern of mutation shown in Fig. 12.4.

Fig. 12.4

Transgenic constructs are assumed to mutate at a rate of m per gene. This is assumed to be low enough that multiple mutations per generation may be neglected. For example, the initial transgenic construct (say, A) mutates at a rate of 3m, producing mutations (more...)

The original study describing this scheme of mutation (Edgington and Alphey, 2019) considered a genotype-based formulation, resulting in a set of 2025 genotypes, of which 819 were non-viable. This could be reduced to a haplotype-based formulation with 81 haplotypes (81 difference equations), making the model simpler and faster to formulate, code and simulate.

This study found that UD displays an increase in frequency that is almost completely unaffected by such mutation where mutated transgenic constructs conferred a greater fitness cost than their non-mutated counterparts – such cases will not be discussed further. Loss-of-function mutations are therefore only of concern where mutated transgenic constructs are of higher fitness than non-mutated versions. Here, the introduced UD system would initially increase in frequency if introduced above the required threshold. Over time each type of mutated transgenic construct will begin to accumulate, with the rate of accumulation and maximum frequencies varying depending on which loss-of-function mutations are present. Results in Edgington and Alphey (2019) show that, for a range of mutation rates and fitness costs, it is transgenic constructs with a single loss-of-function mutation in either the lethal or cargo gene that achieve the greatest maximal frequencies (Fig. 12.5). Constructs with loss-of-function in two genes achieve lower frequencies and are dominated by those where the antidote (suppressor) gene is unaffected. Combined, the mutated transgenic constructs reach high overall frequencies, with a concurrent decrease in the frequency of non-mutated constructs. Since UD approaches typically achieve an equilibrium in which wild-type alleles remain present (see Fig. 12.2), these begin to replace the mutated transgenes due to their relative fitness advantage, eventually returning the population to a fully wild-type state (Fig. 12.5). It is yet to be studied in depth whether the stable co-existence of mutated and non-mutated transgenic constructs is possible, but it was not observed under any parameter set or initial condition considered in Edgington and Alphey (2019).

Fig. 12.5

An example numerical simulation showing the effects of mutation within transgenic constructs of an engineered underdominance gene drive system. Results here are presented for a 1:1 (introduced to wild) introduction of non-mutated double homozygote (AABB) (more...)

For different fitness costs and mutation rates, the observed dynamics remained broadly similar to those in Fig. 12.5, though the timescales and maximal frequencies of each mutation vary. Higher mutation rates reduce the period over which the UD system persists at high frequency – essentially the period in which the gene drive would maintain efficacy. Even though UD systems can be eliminated by mutations, they are predicted to remain at high frequency for hundreds of generations – likely long enough for the system to have produced its desired effect. As an example, Aedes aegypti mosquitoes (vectors of dengue, Zika, yellow fever and chikungunya viruses) undergo approximately ten generations per year, meaning that the introduced UD system should persist at high frequency for at least ten years, even in the face of accumulating loss-of-function trans-gene mutations.

12.6. Linking Theory and Experimentation

The development of gene drive technologies, including UD, has generally followed a design–build–test cycle (Fig. 12.6). At each stage, modelling can play an important role in designing, understanding and analysing experimental work. Thus, in addition to providing insights, modelling can save a significant amount of research time, effort and money. To date there has been relatively little published experimental work on UD and so this section focuses on how one could link theory and experiment when experimental data becomes available. This will partially be informed by studies on alternative gene drive classes and will describe each phase of the design–build–test cycle; however, since the ‘design–build’ aspect of this cycle has largely been covered in previous sections, it will not be discussed further here.

Fig. 12.6

A cartoon showing the design–build–test cycle commonly followed in the development of gene drive technologies and that mathematical modelling is an important tool within each phase.

In the ‘build–test’ phase, experimentation commonly focuses on discrete generation laboratory cage-based experiments. Models of the form shown in section 12.4 assume discrete generations and so they are ideal for predicting and analysing experiments of that form. In the first instance, specific test crosses between transgenic insect strains can be performed and the number of offspring of each resulting genotype counted/screened (as in Hammond et al., 2016, 2017, 2020; Kyrou et al., 2018; Adolfi et al., 2020; Champer et al., 2020b,c; Simoni et al., 2020), with fluorescent markers commonly used to distinguish between types. This allows a first approximation of various system parameters (for example, relative fitness, toxin penetrance and strength of the antidote effect (Buchman et al., 2018b; Webster et al., 2020)) to be generated by calculating ratios between the mean number of each genotype produced. These can then be used to parameterize models and predict the expected outcomes of gene drive cage trials. Note that other gene drive classes may approximate fitness costs by fitting models to data from cage trials with only a single transgenic construct present (e.g. Webster et al., 2020), but this is not feasible for UD where a single transgenic construct produces a lethal phenotype.

Following the ‘test’ phase, i.e., after laboratory cage trial experiments, prior predictions of gene drive behaviour can be compared with the actual cage trial data. Using the same mathematical models (and potentially stochastic versions also, especially given the relatively small numbers of individual mosquitoes typical of a laboratory cage experiment), one can then assess whether model predictions are consistent with the observed outcomes (as in Hammond et al., 2016, 2020; Buchman et al., 2018b; Kyrou et al., 2018; Pham et al., 2019; Adolfi et al., 2020; Champer et al., 2020c; Simoni et al., 2020; Webster et al., 2020). If model and experimental results are consistent, then parameters can be varied to identify potential areas for improvement in the gene drive design. Conversely, if model and experimentation are not consistent, then further modelling may be required to identify sources of this mismatch, potentially informing future models and experimental designs (as in Hammond et al., 2017, 2020, for CRISPR-based gene drives).

12.7. Alternative Configurations of UD

Previous sections focused on a specific configuration of UD, namely that based on two mutually repressing bi-sex toxin genes, inserted into two unlinked and independently segregating genomic loci. Similarly, we have primarily focused on the release of both transgenic males and females. However, there is a wide range of alternative methods for engineering/releasing UD systems, a variety of which have been studied previously.

For the UD design considered thus far, a potential variation is in the time at which the toxin takes effect. This has been studied for a UD system with toxins and transgene fitness costs acting either in early (before density-dependent competition, for example eggs or early instar larvae for mosquitoes) or late (following density-dependent competition, for example pupae or pharate adults in mosquitoes) developmental stages (Edgington and Alphey, 2018; Khamis et al., 2018). While early- or late-acting lethal/fitness effects do not have any impact on threshold introduction frequencies, they can have more impact on other traits. For instance, Khamis et al. (2018) found that, for a system spreading a cargo gene conferring refractoriness to a pathogen, late-acting lethality produced a slightly larger reduction in disease burden. This is likely due to the greater reduction in both equilibrium and (transiently attained) minimum population sizes observed with late-acting lethality (as seen in Edgington and Alphey, 2018). Despite this potential epidemiological benefit, a greater reduction in population size may not always be good news. For example, Ae. aegypti mosquitoes are known to compete with Aedes albopictus (Edgerly et al., 1993; Juliano et al., 2002; Armistead et al., 2008). Therefore, an Ae. aegypti population may be displaced during the period in which the population is reduced by a late-acting UD – potentially reducing the epidemiological benefit as Ae. albopictus are also competent vectors of a similar set of pathogens, including dengue viruses (WHO, 2011). This necessitates some knowledge of ecological factors in the vicinity of gene drive target areas, and could be addressed by modelling approaches similar to those used for sterile insect technique (SIT) and release of insects carrying a dominant lethal (RIDL)-based control (Bonsall et al., 2010).

Other possible UD configurations revolve around the use of sex-specific toxins or insect releases, rather than the bi-sex versions considered above. Such considerations have been studied in terms of their effect on release thresholds and degrees of tolerable transgene fitness costs (Edgington and Alphey, 2017). These results showed that considering either male-only release(s) of gene drive-carrying individuals or female-specific toxins results in a lesser ability to tolerate fitness costs and higher introduction thresholds.

In a two-locus UD configuration, it is possible that the suppressor element from one transgenic construct is not sufficient to inactivate two copies of the toxin gene from the other transgenic construct. In the context of UD, this has been referred to as ‘weak suppression’ (Edgington and Alphey, 2017, 2018); however, the mathematical models and predicted dynamics are equally applicable to systems based on reciprocal chromosome translocations (Buchman et al., 2018b). These studies showed that reciprocal chromosome translocations (or weakly suppressed UD) generally have a higher introduction threshold than the UD systems discussed here. As discussed previously, this represents a trade-off between the increased cost/difficulty of gene drive introgression and the increased reliability of gene drive confinement to the target population.

Several gene drive concepts are based on toxin–antidote systems. The UD system considered thus far assumes two mutually suppressing lethals, each of which comprise a ‘toxin’ gene and an antidote that suppresses its effect, perhaps RNAi targeting the toxin gene. Other toxin–antidote concepts can also provide threshold-dependent gene drives. For example, a synthetic Medea drive was constructed in Drosophila using a maternally contributed (RNAi) toxin with zygotic expression of the antidote only in those offspring inheriting the Medea element (Buchman et al., 2018a). Medea is a low-threshold drive, zero-threshold in the absence of fitness costs, but a mutually repressing pair of such elements can provide a threshold-dependent drive known as UD^MEL (Akbari et al., 2013) or double Medea (Wimmer, 2013). For a rather different molecular basis, a CRISPR/Cas9 system (toxin) can be used to disrupt an essential endogenous gene, which can then be rescued with a recoded – and therefore toxin-resistant – antidote. A range of threshold-dependent gene drives using this technology have previously been modelled (Champer et al., 2021) and are based on the use of cleave and rescue (Oberhofer et al., 2019) or CRISPR toxin–antidote (Champer et al., 2021; Champer et al., 2020a; Champer et al., 2020b) elements. These have been discussed extensively in the original sources and produce broadly similar behaviour to the approach(es) discussed here. Additionally, the mathematical modelling frameworks considered in the studies listed above are similar to those explored throughout this chapter and so we do not discuss results of these studies any further.

12.8. Areas of Future Interest

Despite all the modelling work discussed above, there remain several areas in which further modelling could elucidate various characteristics of UD gene drives. Some have briefly been mentioned in the relevant sections above and so we focus predominantly on areas not yet discussed.

Above, we discussed the use of laboratory cage trial experiments for inferring parameters of the UD system. While these are useful for predicting system performance, these estimates are inherently flawed when moving into the field since they assume that laboratory wild-type strains – and environments – are a good approximation of insects in the wild. In practice, laboratory wild-type strains are recognized as having lower fitness than their wild counterparts (Leftwich et al., 2021), likely due to many generations of laboratory adaptation (Leftwich et al., 2016; Ross et al., 2019). Models can capture the impact of this to a certain degree by considering variation of relative fitness parameters about laboratory-derived estimates. However, the transition toward field-based experiments will potentially necessitate more detailed models capturing a range of ecological, behavioural and fitness effects (some of which have been discussed above). This enhanced modelling can then help to inform the design of UD releases as they progress from small-scale field cage trials right up to the eventual release in full large-scale control programmes.

A feature of most of the modelling discussed here is that it is deterministic and so does not account for the stochasticity inherent in the real world. In the context of UD, this will be important when the release of transgenic insects results in a gene drive frequency close to (or even below) the introduction threshold calculated from deterministic mathematical models (i.e., the unstable equilibrium discussed above). Here stochastic models can provide insight into the expected likelihood of success or failure (i.e., the probability that a UD system increases or decreases in frequency) of a given release strategy. To date, stochastic modelling of UD systems has been limited, to our knowledge, to only Marshall and Hay (2012) for this UD configuration. However, some stochastic modelling frameworks have been used to study other gene drive classes, from which such work could take a lead (for example: Magori et al., 2009; Champer et al., 2020a; Edgington et al., 2020a,b; Sánchez et al., 2020a; Wu et al., 2021).

A common feature in the modelling of many gene drive classes, and, in particular, toxin–antidote-based approaches, is an assumption that toxins and antidotes are fully penetrant (i.e., that toxins kill 100% of target genotypes and antidotes rescue 100% of carriers to full fitness). However, gene drive components engineered in the laboratory may not give this degree of efficacy. Laboratory experiments, for example life history analysis of different genotypes, can provide initial estimates of such incomplete penetrance. These data could be incorporated into models similar to those in section 12.4. In the absence of working gene drive components to test in the laboratory, one can use the same model to explore the expected behaviour for a range of toxin and antidote penetrance parameters, thus setting performance targets for laboratory-engineered gene drive components. This could be used to assess performance metrics including threshold frequencies, the speed of spread, the system invasiveness (with n-deme versions of the models) and tolerable fitness costs – all of which are likely to be vital when transitioning from laboratory to field-based testing.

The motivation for genetic control of mosquitoes is to reduce or prevent morbidity and mortality from mosquito-borne diseases. Thus, it is important to explore the anticipated epidemiological impact(s) expected from a given gene drive and release strategy. This can be explored by incorporating a gene drive model into a standard epidemiological modelling framework, for example susceptible–exposed–infectious–recovered (S-E-I-R) or a variety of extensions/modifications as previously considered for Wolbachia (Ndii et al., 2015, 2016a,b; Zhang and Lui, 2020) or RIDL (Atkinson et al., 2007) control approaches in Ae. aegypti mosquitoes. This will likely require the use of a population dynamics model similar to those of Khamis et al. (2018) and Edgington and Alphey (2018) for two main reasons: (i) models must produce results for gene drive and epidemiological dynamics at all time points; and (ii) the respective sizes of human and insect (vector) populations are important in determining a pathogen’s force of infection. Such models can provide important insights into potential epidemiological impacts. However, various factors required to formulate these models (such as infection numbers, exact population sizes, transmissibility of pathogen(s) and biting frequency) can be extremely difficult to measure, meaning that the consideration of model uncertainty will be important when interpreting results.

This chapter has focused on the use of mathematical modelling to predict the efficacy of UD gene drives from molecular design and laboratory testing right through to field testing and final applications. Such studies will likely be important in providing an evidential basis upon which regulatory decisions can be made. As further laboratory-based testing provides more and higher-quality data, we would anticipate that more detailed and species-specific models will be developed, providing greater insight into the anticipated efficacy of UD systems. Likewise, as more field studies into the ecology of potential gene drive target species and field-trial releases (of this or other technologies) become available, more detailed ecological, epidemiological and behavioural factors can be studied and incorporated into models, enabling the best possible predictions of gene drive function following release of transgenic insects. The previous literature and future focus areas discussed here demonstrate the key role that modelling plays in the development of gene drive technologies and emphasizes the necessity for gene drive research and development to follow an interdisciplinary approach. This will ensure that any future gene drive release has the greatest opportunity to function as intended, thus providing the maximum possible beneficial impact.

Acknowledgements

MPE is funded through a Wellcome Trust Investigator Award [110117/Z/15/Z] made to LA. LA is supported by core funding from the UK Biotechnology and Biological Sciences Research Council (BBSRC) to The Pirbright Institute [BBS/E/I/00007033 and BBS/E/I/00007038].

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Corresponding author, email: ku.ca.thgirbrip@yehpla.ekul