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 (). 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 (). 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.
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:
where
is the sum of numerators in (
Equation 1)–(
Equation 4),
ε denotes the fitness (relative to wild-type) of an
individual carrying a transgenic construct and
t denotes the
generation from which the next allele frequency is computed. For simplicity, here we
assume that each transgenic construct (A and B) confers the same degree of fitness
cost on carrier individuals and that these are applied multiplicatively where
individuals carry more than one transgenic construct (up to a maximum of four, where
the relative fitness would be given by
ε4). Note
that the parameter
ε can take any value in the range from
zero (completely non-viable) to one (equally as fit as wild-type) and that the
consideration of multiplicative relative fitness ensures that the overall value for
any genotype also remains in the range from zero to one. Like much of the modelling
literature, here we assume that toxins are fully penetrant (i.e., no viable
offspring result) and similar for antidotes (i.e., a single antidote copy provides
full rescue).
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 .
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
() 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 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 ().
While useful in displaying the bistable (i.e. threshold-dependent) nature of a UD
gene drive, 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 ) 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.
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 .
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 (). 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 ), these begin to replace the mutated transgenes due to their
relative fitness advantage, eventually returning the population to a fully
wild-type state (). 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).
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 , 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.
12.5.2. UD reversal
Ideally, a gene drive, or any other intervention, should be simple to reverse in
the event of any unintended and undesirable consequences. For
threshold-dependent drives such as underdominance (UD), the simplest mechanism
for this is the release of wild-type individuals. Releasing a sufficiently large
number of wild-type individuals can push the UD system below the threshold
frequency, thus it undergoes negative selection and is driven out of the
population (consider, for example, ). To our knowledge, only one alternative reversal mechanism has been
proposed for reversing UD, namely releasing individuals carrying free
suppressors (i.e., individuals carrying just the antidote from one or both of
the original transgenic constructs) (Edgington and Alphey, 2019). This was proposed since modelling of
mutation in UD systems showed significantly greater accumulation of mutated
constructs that retained function of the antidote gene, which undergo positive
selection where a UD system is at high frequency. The same study also showed
that if free-suppressor elements conferred a non-zero fitness cost, then the
positive selection would be lost as the UD system became rare in the population,
thus allowing wild-type alleles to outcompete the free suppressor element,
returning to a fully wild-type state.
With there being two viable mechanisms for UD reversal, it is instructive to
compare the two. Each has benefits, but which would be more useful in real-world
scenarios? If a released UD system were to require reversal, then it would be
reasonable to assume that introducing large numbers of wild-type individuals
would be undesirable, as it could potentially increase the population above
pre-control levels, albeit transiently. Intensive suppression might reduce
numbers in the entire population such that the necessary wild-type releases
remain below pre-control levels, but that would have its own costs and issues.
In some cases, issues around release of wild-type individuals could be
alleviated somewhat by releasing individuals of a single sex (as discussed in
Leftwich et al.,
2018). For example, in many insect species (and certainly many of those
likely to be gene drive targets) females provide both the reproductive and
epidemiological potential of the population; thus, releasing wild-type males
should generally be quite benign.
The release of free-suppressor individuals provides an alternative reversal
mechanism and addresses some of the issues associated with wild-type release(s)
but also has some drawbacks of its own. For example, free-suppressor individuals
are theoretically able to function from an extremely small release. In practice,
it would be desirable to perform release(s) large enough to avoid stochastic
loss at low frequencies. Despite this, it should still be feasible to release
far fewer individuals than required for wild-type reversal and these may be of a
single sex. However, there are also some potential drawbacks that need to be
weighed against these benefits. Firstly, free-suppressor releases have been
shown to function much more slowly than wild-type reversal. For instance, Edgington and Alphey (2019) show that a
2:1 (reversal to wild) release of wild-type individuals can (approximately)
eliminate the UD gene drive in about 20 generations whereas an equal release of
free suppressor individuals took about 130 generations to reach the same point,
with about a further 150 generations required for the free-suppressor
individuals to be (approximately) eliminated. Another potential issue is whether
the appropriate regulatory body would approve the release of further transgenic
insects if the original system required reversal due to unintended negative
effects.
12.5.3. Spatial effects
A key feature of UD gene drives is their threshold-dependent nature, since this
is often stated to be capable of preventing the system from establishing in
non-target neighbouring populations. It may even prevent the system reaching an
appreciable frequency, due to negative selection when present at sub-threshold
frequencies. The modelling of such spatial factors is therefore important in
assessing how well confined UD systems will remain and under what conditions
this confinement could potentially fail. These questions are expected to prove
important when seeking to attain regulatory approval for UD releases into the
field; with highly robust confinement, regulatory approval far beyond the
release site(s), for example regional or multi-national approval, may not be
required, in contrast to current thought regarding more invasive approaches.
Spatial effects can conceivably be studied in a variety of ways, including
n-deme population genetics (difference equations),
n-deme population dynamics (ordinary or delay differential
equations), reaction–diffusion (partial differential equations) or
individual-based models. Each of these has been used in the study of gene drive
approaches, although not all in the context of UD, and possesses its own
positive and negative features. Here we discuss a range of these approaches in
the context of UD gene drives, focusing primarily on findings rather than
technical details. While we cover a broad range of studies here, this is by no
means intended as a comprehensive review.
Perhaps the most commonly used technique for assessing spatial properties of gene
drives are n-deme population genetics models. Briefly, these
consider two or more demes (semi-isolated (sub)populations), each of which has
its own set of difference equations of the form outlined in section 12.4. For simplicity, the literature primarily
considers a scenario with just two demes. These models capture the migration of
individuals between demes via a simple exchange of a proportion of individuals
in each generation. It is commonly assumed that the two populations are of equal
size, such that the number of each migrant type is simply based on the haplotype
frequencies in each population – a reasonable assumption where large
(modelled as infinite) populations are considered. This approach has been used
to consider spatial aspects for a wide range of gene drive classes, including UD
(Marshall and Hay, 2012; Harvey-Samuel et al.,
2019; Edgington et
al., 2020b). One such model estimated that a UD system with a
homozygote fitness cost of 5% (applied additively) and a bidirectional migration
rate of 1% per generation would reach near-fixation in the target population
(e.g. results in ) but
reaches a frequency of just 0.032 in a non-target neighbouring population (Marshall and Hay, 2012). The same work
then went on to estimate that the same UD system would require a bidirectional
migration rate of 4.3% per generation to become established in both populations,
thus supporting the notion that UD is robustly confineable.
The approximation of equal population sizes has also been relaxed in a number of
studies focusing both on UD (Dhole
et al., 2018) and on other gene drive classes (Dhole et al., 2019,
2020). These studies include an
approximate scaling of migration rates to account for the differences in
respective population sizes that result from gene drive fitness cost and lethal
effects. While this captures an additional level of realism absent in the
non-scaled migration models, the results obtained do not suggest that this will
have a large impact on the ability of UD systems to remain confined. In fact, we
would suggest that this is likely to improve the confinement, since transgene
fitness costs and lethal effects will reduce the target population size, meaning
fewer migrants into the non-target population. However, this will likely
increase the influence of wild-type migrants from the non-target population,
potentially creating a necessity for slightly larger UD release(s) to ensure the
system remains above the introduction threshold.
Several studies have also considered extensions to these population genetics
models by taking account of various ecological factors affecting the life cycle
and size of the target and non-target insect populations when subject to the
release of a UD gene drive (e.g., Edgington and Alphey, 2018; Khamis et al., 2018, 2020). These each consider their own model structures to
capture density-dependent effects during the immature stages of the insect life
cycle, each with its respective advantages and disadvantages. One
density-dependence function used in such models is that of Maynard Smith and Slatkin (1973) and is of the form:
where
N denotes the size of the insect population,
a is a density parameter (1/
a relates to the
number of breeding sites) and
b defines the strength of
density-dependent competition. This function is known to be flexible in that it
can capture a range of density-dependence scenarios (
Bellows, 1981) and has been used in the study of UD
(
Edgington and Alphey, 2018) and
other gene drive classes (
Alphey and
Bonsall, 2014). This work outlines a variety of possibilities not
extensively discussed in the results of population genetics models. In
particular it identifies three possible outcomes of a UD release: (1) no
introgression in either population; (2) establishment in both populations; and
(3) introgression into the target population with extremely limited spread into
the non-target population, with the latter usually considered the most desirable
outcome for UD (
Edgington and Alphey,
2018).
Sánchez et
al. (2020a) considered similar effects for UD
MEL and
reciprocal chromosome translocations using a computational framework called
MGDriveE (
Sánchez et
al., 2020b;
Wu et
al., 2021).
The above approaches consider spatial effects via an exchange of individuals
between two (sub)populations, which are assumed to be well mixed. A possible
extension to this work is to consider spatial effects explicitly, using a model
defined over a continuous spatial domain (e.g., Champer et al., 2020d). This work takes
a fully computational approach, using an individual-based model implemented in
the open-source software package SLiM (Haller and Messer, 2016, 2019). Here, two circular regions (sub-populations) are linked by a
narrow migration corridor, with movement assumed to result from the birth of new
offspring (Champer et
al., 2020d). This showed that UD is robust against re-invasion
by wild-type but may display a greater degree of invasiveness into neighbouring
populations than predicted with the spatially implicit model structures
discussed above. However, the narrow migration corridor essentially forces
migrating individuals to encounter those moving in the opposite direction,
creating an approximately linear boundary between the two populations – a
scenario shown to facilitate easier gene drive invasion (Champer et al., 2020d).
An alternative scenario would allow migration to occur over a wider space,
meaning migrants encounter those moving in the opposite direction far less
frequently, thus eliminating the linear boundary within the migration corridor.
Here migrants would first encounter individuals when arriving at the boundary of
the opposite population, meaning they would encounter either a very high or very
low local gene drive frequency, likely producing results closer to those from
spatially implicit model structures. Such variation highlights the importance of
understanding a wide range of species, location and ecological traits when
predicting the outcome of a real-world UD release.
While the above approaches assess the likelihood of UD invading non-target
populations, spatial effects are also important in determining the ability of UD
to spread in a given target population. This has also been addressed using a
variety of different modelling approaches.
One possibility is to consider a lattice-based model in which the target region
is discretized into a collection of cells, each containing a well-mixed pool of
individuals (Huang et
al., 2011). Individuals move between cells according to a
dispersal kernel, defining the probability of an individual moving between any
two cells in the lattice on any given day. This model structure was used to
compare the relative efficacy of two UD release methods: (i) release into one
large area; and (ii) release into many smaller, equally distributed areas.
Interestingly, this work showed that either release method could be more
effective, depending on the degree of mobility exhibited by individuals and the
fitness costs associated with the UD system.
The individual-based model of Champer
et al. (2020d) has also been used to explore the ability
of UD to spread within a single population. This considered two spatial
scenarios based on the shape of the UD release area, namely a scenario with
either a straight-line (linear) scenario or a circle dividing regions of high
and low/zero gene drive frequency. Interestingly, the UD system was able to
spread or persist more readily in the linear scenario than the circular one.
This was proposed to be a result of the local gene drive frequency being lower
for the circular case, since the wild-type partially wraps around the high UD
region. However, we would expect this effect to diminish rapidly as the circular
region increases in size (reducing the curvature of the boundary).
Finally, another potential approach for considering spatial effects in either
single or linked populations is the use of reaction–diffusion equations
(i.e., partial differential equation models). These have been explored in the
context of highly invasive CRISPR-based gene drives (Beaghton et al., 2016; Tanaka et al., 2017),
but to our knowledge have yet to be widely applied to threshold-dependent
systems. Such models should enable a wide range of spatial scenarios to be
considered, while allowing various sources of heterogeneity to be considered
across the spatial domain. These models can also allow an explicit
representation of insect migration to be incorporated into population dynamics
model structures, such as those in Edgington and Alphey (2018) and Khamis et al. (2018, 2020).
12.6. Linking Theory and Experimentation
The development of gene drive technologies, including UD, has generally followed a
design–build–test cycle (). 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.
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 UDMEL (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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