Phantom cutting by DCDs can bias type-1 resistance alleles

We simulate homing endonuclease gene drives in two populations with up to three unlinked diploid loci designated by a subscript: _A, _B, or _C. Briefly, a wild-type target allele (T) can be cut by the combined product of a nuclease allele (N) and a gRNA. The nuclease is target independent, whereas the gRNA always matches a specific wild-type target allele. The presence of a gRNA gene in an allele is indicated by a superscript of the locus to which it is targeted and can be together with a nuclease gene (N^A) or only a gRNA gene without the nuclease gene (G^A). A drive element can also lack a Cas9 or gRNA gene (E), being empty apart from the cargo gene/effector modification that is being spread. The nomenclature system we use is further expanded in Fig 1a.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Drive nomenclature system and overview of simulation parameters.

(a) Different alleles considered in our computational simulations: Wild-type target (T), type-1 resistance (1), type-2 resistance (2), and drive alleles (E, G, and N: collectively D). (b) When a target allele, a gRNA targeting that specific allele, and the nuclease are present in the same genotype, there is a certain likelihood that a cut occurs (by default 100%). The cut allele can be repaired by interhomolog HDR, or to a type-1 resistance allele or type-2 resistance allele. (c) Default nuclease and fitness parameters. The A drive allele has a fitness of 90% to account for general fitness costs (e.g., metabolic) and imperfect rescue of the haploinsufficient target gene by the drive element. This is the only fitness cost of the self-perpetuating gene drive. For DCDs, the B and C drive alleles have a fitness of 95%. Each fitness cost is dominant, with no additional recessive cost unless specifically listed for that simulation. Individuals carrying one or more of each different DCD allele are therefore assumed to have a fitness of 81% (90% ⋅ 95% ⋅ 95%). Type-2 resistance alleles at the A locus are assumed to be dominant lethal (fitness 0%). The B element is in a neutral locus; as such, its resistance alleles are all neutral and classified as type-1. The C locus is never cut, and resistance alleles are not generated at that locus. (d) Illustration of the two linked populations for which we report the allele frequency. Drive individuals are released into population one at the start of the simulation. Each generation, 1% of the surviving adults migrate to the neighbouring population. Population one receives 1% of population two, while population two receives 0.5% of population one and 0.5% of exclusively wild-types. (e) Default release frequency and simulation length parameters. DCD: Daisy-Chain Drive. SPD: Self-Perpetuating Drive.

https://doi.org/10.1371/journal.pgen.1010370.g001

Nuclease deposition into the embryo, differences in fitness, and nuclease expression in the germline can cause a change in the genotype frequency from one discrete generation to the next (Fig 1b and S1 Fig). The fitness and drive conversion rates for the many unique combinations of alleles are extrapolated from a small set of parameters listed in Fig 1c. The inheritance biasing parameters most closely resemble those of HEGs developed in the Anopheles mosquito [25, 30]. However, we assume that type-2 resistance alleles are dominant lethal at the A locus (haplolethal target gene) and the drive elements at that locus carry a rescue element [6, 25, 26]. The creation of a rescue element is generally achieved by including a recoded version (many synonymous codon substitutions) of the target site and downstream portions of the target gene. Essentially, engineering the drive allele to carry extensive type-1 resistance mutations. As such, we assume that a relatively high fraction of non-HDR repair results in type-1 resistance alleles (10% of non-HDR events, 0.5% overall) when compared to what has been achieved targeting a highly conserved locus (for which creating a viable rescue element may not be possible) [37].

For each simulation, we report the zygote genotype frequency before fitness, deposition, and expression-mediated changes in genotype frequency occur in that generation. We track genotype frequencies in two separate populations which have a 1% migration rate, with population two receiving a constant migration of wildtypes from the general population in addition to migration to and from population one (Fig 1d). Alleles for a particular population are indicated with an additional number in the subscript (e.g., _A1). Unless otherwise specified, our default release scenario for daisy-chain drives is such that at the start of the simulation, 25% of males (12.5% overall allele frequency) in population one are homozygous for all drive elements (Fig 1e). The self-perpetuating drive is initiated with a frequency of male drive carriers 100 times lower (0.25%) compared to the release scenario for DCDs.

The three different HEG designs that we investigate are a single-element self-perpetuating drive (_AN^A) shown in Fig 2a and two different three-locus daisy-chain drives. For each drive, the A drive element is inserted into and rescues a haploinsufficient endogenous gene. For DCDs, the B and C elements are located at neutral sites. The first DCD design we call a direct DCD (_AE/_BN^A/_CG^B), shown in Fig 2b and is the most similar to the design modelled by Noble et al. [9]. The C element (_CG^B) carries a gRNA targeting _BT. The B element (_BN^A) carries both a nuclease and a gRNA targeting _AT. The drive element for A would, in practice, contain the desired effector modification, but does not contain any components (gRNA or nuclease) that are directly involved in inheritance biasing (denoted as _AE without a superscript).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Illustration of drive designs, default behaviour, and performance when subjected to different recessive and dominant _AD fitness costs.

Column 1. Self-Perpetuating Drive. Column 2. Direct Daisy-Chain Drive. Column 3. Indirect Daisy-Chain Drive. Row 1 (a-c). Illustration of the molecular designs of the three drives we simulate. Each design is shown as heterozygote, with on the right the product resulting from inheritance bias. Row 2 (d-f). A locus allele dynamics under the default conditions. The thin dashed line indicates a frequency of 90%. An allele at locus A in population one is indicated by _A1 (green lines), and a drive allele at locus A in population two is indicated by _A2 (gray line). Type 1 and 2 resistance alleles are not shown for population two in these panels. The allele dynamics for all loci are shown in S2 Fig. Row 3 (g-i). Parameter sweep for different recessive and dominant fitness costs applied to the _AD (_AN^A/_AE/_AG^A) allele. The shading of the heat map indicates the maximum _AD allele frequency in population one reached within the 100 generations simulated. The maximum frequency of _A1 alleles and outcomes for population two are shown in S7 Fig. An immediate decline after release results in the maximum frequency being the release frequency. Regions are boxed with coloured lines that indicate threshold values for the number of generations in which the _A1D allele frequency was greater than 90%. The black lines on each axis indicate the default value for the parameter being varied. Row 4 (j-l). Difference in the maximum _AD allele frequency between population one and two. Regions boxed with a red line are simulations in which the maximum allele frequency of population one or two was at generation 100 and the outcomes may change if additional generations are simulated.

https://doi.org/10.1371/journal.pgen.1010370.g002

The second DCD design we consider is an indirect DCD (_AG^A/_BN/_CG^B), shown in Fig 2c, which has the gRNA targeting the A element expressed from the A element itself (_AG^A). The chain structure is maintained as the A element will only function in the presence of the nuclease expressed from the B element (_BN). The C element is identical in both DCD designs. The indirect DCD has several practical benefits, including that A and B element transgenic lines can be developed independently, while for the direct DCD design, the A targeting gRNA(s) located on B limits interoperability. A DCD design functionally similar to this (using orthogonal nucleases) was proposed in Noble 2019 et al., but not explicitly investigated [9].

The allele dynamics of the self-perpetuating drive (_AN^A) shown in Fig 2d represents the maximally efficient implementation of an HEG, albeit with unlimited spread. The A drive element (the only drive element) in population one (_A1D) reaches a maximum allele frequency of 99.4% at generation 23. The fitter type-1 alleles slowly outcompete the drive allele, but this process will take many generations, since at generation 23 the type-1 resistance allele (_A11) frequency is only 0.6%. The drive element in population two reaches a maximum frequency of 98.8% at generation 24. While the initial spread of the drive element into population two lags a few generations behind that of population one, migration means that population one only reaches its maximum drive frequency when population two is also (almost) at its maximum drive frequency. The frequency of wild-type alleles in population two quickly drops and stabilises at just over 0.55%. This is due to the 0.5% migration of wild-type individuals from the general population each generation and the reduced fitness of drive-carrying individuals (90% of wildtype).

While the allele frequency dynamics of the B and C alleles are almost identical for the two DCD designs (S2 Fig), the A allele dynamics, especially the type-1 resistance allele dynamics, differs substantially (Fig 2e and 2f). The direct DCD reaches a maximum _A1E frequency of 93.8% at generation 25, which coincides with a type-1 resistance allele frequency of 2.1%. The maximum _A2E frequency in population two is 10.6% at generation 35. The indirect DCD reaches a maximum _A1G^A frequency of 95.3% at generation 27, which coincides with a type-1 resistance allele frequency of 0.6%. The maximum _A2G^A frequency in population two is 11% at generation 36.

The difference in performance between the two DCD designs occurs due to a phenomenon we term phantom cutting, illustrated in S3 Fig. This occurs when an upstream element, such as the direct DCD B element(_BN^A), attempts to bias the inheritance of the downstream A element, even when the downstream element is not present. This is possible in our simulation as with a homing rate less than 100% the B drive element is occasionally co-inherited with a resistance allele generated at the A locus. Phantom cutting is only possible for the direct DCD as both the A gRNA and the nuclease are on B (_BN^A) and it does not occur with the indirect DCD, as the nuclease on the B element (_BN) requires the gRNA expressed from the A element drive allele (_AG^A).

We have assumed an HDR rate of 95% and a type-1 to type-2 resistance allele repair ratio of 1:9 at the A locus, resulting in 0.5% of DNA breaks generating type-1 alleles. In the indirect design, these type-1 resistance alleles persist and slowly outcompete the drive alleles but are not otherwise biased in any way. In the direct DCD, these type-1 resistance alleles can be biased in the same way that the A drive allele can (e.g., _A1T/_BN^AT→ _A11/_BN^AT), and therefore reach much higher frequencies. In our simulations, type-2 resistance alleles at the A locus cannot be biased by phantom cutting as even one copy is dominant lethal. Phantom cutting can only occur when there are _AT alleles remaining in the population. As the A drive and resistance alleles replace the _AT alleles, the frequency of phantom cutting events fades away.

The default single-sex drive frequency at release of 25% allows the daisy-chains to spread to high frequencies in population one, but not in population two. As such, the release frequency can be considered to be roughly tuned to the drive’s inheritance biasing efficiency, fidelity, and fitness. The self-perpetuating drive cannot be tuned and will spread to high frequencies in both populations. The A locus drive and type-1 allele frequency dynamics are shown for different HDR and migration rates in S4 and S5 Figs, respectively.

It is important to note that the differences between the two DCD designs depend on the assumptions we have made compared to previous work. Using the parameters and assumptions of Noble 2019 et al. [9] (100% cut-rate, no type-1 alleles, and type-2 alleles at A are dominant lethal), the direct and indirect DCD designs behave identically to each other (S6 Fig, columns one and two). Phantom cutting cannot occur because no progeny survive in which the B element segregates away from the A element. Moreover, the DCD designs that we evaluate differ from other models of DCDs [9, 35] in that they only have a nuclease expressed by the B element, not by both the B and C element (i.e., _AE/_BN^A/_CN^B). With a nuclease on the C element, the direct design simulations match those reported by Noble et al. [9] (S6 Fig, columns three and four). For the indirect DCD design, the nuclease expressed from the C element helps increase the spread of _AG^A, but also generates many more resistance alleles at the B locus due to phantom cutting. All other simulations in this study use DCD designs without a nuclease on the C element.

Fig 2g–2i show the maximum frequency of the A drive element, and the number of generations it remains above a 90% allele frequency under different recessive and dominant _AD fitness costs. The recessive fitness costs apply to individuals with two copies of the drive, the dominant fitness costs apply equally to individuals with one or two copies. Obviously, minimisation of the fitness costs is the most conducive to DCD spread. However, without adjusting the release frequency, this also allows the drive allele to spread beyond population one and reach high frequencies in population two (S7 Fig). In Fig 2j–2l we contrast the two populations by subtracting the maximum drive frequency in population two from the maximum drive frequency in population one. This reveals a more narrow parameter space in which the DCDs can spread to high frequencies in the target population, yet do not substantially invade population two. We have highlighted cases where the maximum drive frequency in either population coincides with the end of the simulation, indicating that simulating more generations may change the outcomes.

With a fixed release frequency, dominant fitness costs appear to be more desirable when evaluating the differential spread in the two populations. Unlike recessive fitness costs, dominant fitness costs do not additionally penalise the drive for spreading to a high frequency in population one and immediately apply to the drive alleles that have migrated to population two, even when they are at low frequencies. Furthermore, dominant fitness costs can apply to heterozygous individuals carrying a type-1 resistance allele. Individuals with a type-1 allele are never affected by a drive with an exclusively recessive fitness cost, increasing the relative benefit for an individual carrying such a functional resistance allele.

The relatively low default fitness costs of the A drive rescue element make these simulations representative of low fitness cost population modification drives (e.g., spreading an antibody gene in a vector of human disease). Another class of drives is aimed at population suppression, with the goal of reducing a target species’s population density. We have performed simulations at different release frequencies for a recessive lethal (S8 Fig), and a sex-specific recessive lethal (S9 Fig) A element target gene. In both cases, the A element does not contain a rescue gene and imposes the same fitness cost as a type-2 resistance allele (i.e., disrupts the target gene as it homes). As may be expected, the self-perpetuating drive approaches fixation under the broadest set of conditions, largely independent of the release frequency. All drives rapidly decrease in frequency after reaching their maximum, with type-1 resistance alleles driven to near-fixation by the self-perpetuating drive (type-1 alleles would reach fixation were it not for the low levels of migration of wildtypes from the general population). The DCDs only approach a maximum drive frequencies similar to the self-perpetuating drive with upward of 70% of males carrying the DCD at the start of the simulation. In all cases, the indirect design is burdened by fewer type-1 mutations and spreads to higher frequencies.

Together, these results show that even low rates of type-1 resistance allele generation cause the two DCDs to perform differently from each other. Furthermore, the maximum frequency achieved by a DCD is highly dependent on its fitness profile, with dominant fitness costs providing the largest difference between the spread in population one and two.

DCDs are severely impacted by reductions in cut-rate

Like the generation of type-1 resistance alleles, Cas9:gRNA cut-rates that are less than saturating (<100%) have been reported for multiple HEGs [15, 18, 19, 27, 30, 33, 38]. In Fig 3 we show A allele dynamics of simulations with a cut-rate of 85%, with the allele dynamics of all loci shown in S10 Fig. The spread of the self-perpetuating drive is delayed, but otherwise largely unaffected by a lowering of the cut-rate as each element can simply try again in the next generation (Fig 3a). With a cut-rate of 85% the two DCDs reach maximum A element frequencies of 79.7% and 92% for the direct DCD and indirect DCD, respectively (Fig 3b and 3c). Interestingly, despite the lowered cut-rate, type-1 alleles are still produced in large numbers with the direct DCD design, even exceeding the drive allele frequency by the end of the simulation. Simulations of a range of cut-rates are presented in Fig 3d–3f and S11 Fig.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Daisy-chain gene drive element prematurely segregate with reduced cut-rates.

Column 1. Self-Perpetuating Drive. Column 2. Direct Daisy-Chain Drive. Column 3. Indirect Daisy-Chain Drive. Row 1 (a-c). Individual allele dynamics with a cut-rate of 85%. The thin dashed line indicates a frequency of 90%. An allele at locus A in population one is indicated by _A1 (green lines), and a drive allele at locus A in population two is indicated by _A2 (gray line). Type 1 and 2 resistance alleles are not shown for population two in these panels. The allele dynamics for all loci are shown in S10 Fig. Row 2 (d-f). A element drive allele dynamics for population one with different cut-rates. Allele dynamics of the A locus drive allele in population two, and type-1 resistance alleles for both populations are shown in S11 Fig. Row 3 (g-i). Parameter sweep for the male drive frequency at release and germline cut-rate. The shading of the heat map indicates the maximum _AD allele frequency in population one reached within the 100 generations simulated. The maximum frequency of _A1 alleles and outcomes for population two are shown in S13 Fig. An immediate decline after release results in the maximum frequency being the release frequency. Regions are boxed with coloured lines that indicate threshold values for the number of generations in which the _A1D allele frequency was greater than 90%. The black lines on each axis indicate the default value for the parameter being varied. Row 4 (j-l). Difference in the maximum _AD allele frequency between population one and two. Regions boxed with a red line are simulations in which the maximum allele frequency of population one or two was at generation 100 and the outcomes may change if additional generations are simulated.

https://doi.org/10.1371/journal.pgen.1010370.g003

With a reduction in the cut-rate the DCD B element can now be passed along together with an uncut _AT allele (as opposed to necessarily a drive element or a resistance allele when the cut-rate is 100%). This expands the possible genotypes under which phantom cutting can occur to include genotypes like _ATT/_BN^AT and creates the possibility of simultaneous cleavage of homozygous target alleles at the A locus. This complicates the nuclease conversion scheme as HDR could, in principle, recursively regenerate a T allele from the homologous T allele, opening up the opportunity of repeated and simultaneous cutting events. The simulation presented in Fig 3 were performed under the assumption that HDR does not occur when a TT genotype is cleaved, instead, additional resistance alleles are generated. In S12 Fig we present simulations with various cut-rates assuming HDR is possible, and individual T alleles in a TT genotype are cut sequentially. Regardless of the approach taken, the indirect DCD is more invasive than direct DCD at equivalent cut-rates and produces fewer type-1 resistance alleles.

The most straightforward way to compensate for a reduction in invasiveness may be to increase the drive frequency at release. However, for the self-perpetuating drive, this has a very limited effect when the reduction in invasiveness is due to an extremely low cut-rate. Under these conditions (cut-rates lower than ≈15%) the drive element remains unable to overcome its fitness costs independent of the drive frequency at release (Fig 3g). In contrast, any reduction in cut-rate affects the DCD much earlier through the loss of co-inheritance of the different drive elements (Fig 3h and 3i). For these conditions, an increased release frequency can allow DCDs to reach frequencies where orphaned drive elements are likely to be reacquainted by random mating. For DCDs, the frequency-dependent consequence of a reduction in cut-rate limits the DCDs ability to invade population two when the cut-rate is reduced (S13 Fig). This frequency-dependent effect does not apply to the self-perpetuating drive as only a single-element needs to be inherited to affect a cutting reaction at the target locus. Fig 3j–3l shows this consequently provides a broad range of cut-rate and release frequency parameter combinations under which the DCDs spread to high frequencies in population one, but not population two.

The production of type-1 resistance alleles through phantom cutting from isolated B elements causes the direct DCD to be more severely affected by reductions in cut-rates compared to indirect DCD. In addition to failing to spread with low cut-rates, the type-1 resistance alleles produced by DCDs may persist and theoretically impact a future release of an otherwise appropriately tuned drive release targeting the same locus. Next, we set out to determine how deposition would affect DCDs and potentially reveal additional differences between the drive designs.

Maternal deposition can aid daisy-chain containment

The inclusion of deposition greatly expands the genotypes that can be exposed to cutting by the drive. We assume exclusively maternal deposition without any paternal deposition, and deposition occurs uniformly into each progeny, with the deposition cut-rate determining the fraction of target alleles that are cut in each progeny. The most straightforward deposition scenario is the simultaneous deposition of the Cas9 protein and gRNA as a complex. However, gRNAs in CRISPR HEGs are generally expressed from (putatively) constitutive pol III promoters. In line with experimental data [27, 31, 38, 39], in our simulations embryonic cutting can also occur when the Cas9 protein is deposited into an individual that has inherited a gRNA expressing transgene from their father. In contrast, due to the reduced stability of the isolated gRNA [40–42], we assume that the pre-complexing of Cas9 and a gRNA in the mother is necessary for the gRNA to be deposited. Generally, cutting by the deposited nuclease is thought to occur much earlier in development, when HDR rates are much lower and HDR may in some cases be impossible due to physical isolation of homologous chromosomes [43–45]. As such, we assume that HDR repair does not occur in the early embryo and all cuts result in resistance alleles.

In Fig 4a–4c we show the A locus allele dynamics of simulations with a maternal deposition cut-rate of 25%. In contrast to the outcomes when the cut-rate is changed, maternal deposition affects the two DCDs very similarly. The two DCDs reach maximum _A1D element frequencies of 64.5% and 66.6% for the direct DCD and indirect DCD, respectively. For both DCDs, the B drive element carrying the nuclease drops in frequency rapidly due to its association with deposition-induced fitness costs at the A locus (S14 Fig).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Maternal deposition affects the drives when they are at low frequencies.

Column 1. Self-Perpetuating Drive. Column 2. Direct Daisy-Chain Drive. Column 3. Indirect Daisy-Chain Drive. Row 1 (a-c). Individual allele dynamics with a maternal deposition cut-rate of 25%. The thin dashed line indicates a frequency of 90%. An allele at locus A in population one is indicated by _A1 (green lines), and a drive allele at locus A in population two is indicated by _A2 (gray line). Type 1 and 2 resistance alleles are not shown for population two in these panels. The allele dynamics for all loci are shown in S14 Fig. Row 2 (d-f). A locus drive allele dynamics for population one with different maternal deposition cut-rates. Allele dynamics of the A locus drive allele in population two, and type-1 resistance alleles for both populations are shown in S15 Fig. Row 3 (g-i). Parameter sweep for the male drive frequency at release and maternal deposition rates. The shading of the heat map indicates the maximum _AD allele frequency in population one reached within the 100 generations simulated. The maximum frequency of _A1 alleles and outcomes for population two are shown in S18 Fig. An immediate decline after release results in the maximum frequency being the release frequency. Regions are boxed with coloured lines that indicate threshold values for the number of generations in which the _A1D allele frequency was greater than 90%. The black lines on each axis indicate the default value for the parameter being varied. Row 4 (j-l). Difference in the maximum _AD allele frequency between population one and two. Regions boxed with a red line are simulations in which the maximum allele frequency of population one or two was at generation 100 and the outcomes may change if additional generations are simulated.

https://doi.org/10.1371/journal.pgen.1010370.g004

In the previous simulations, the fitness cost of type-2 alleles were only experienced by those individuals that inherited it, not the individual in whose germline it was generated. To account for somatic genotype conversion by embryonic cutting, fitness costs are applied after maternal deposition-based gene conversion occurs (but before expression-based conversion in the germline). Any individual in which its _AT alleles are cut due to deposition has its fitness lowered in proportion to the combined fitness of the individual genotypes of the now mosaic individual. Although cutting of _B T alleles can occur due to deposition, resistance mutations are neutral at this locus. However, deposition still reduces the proportion of uncut target alleles available for subsequent homing in the germline. It is important to note that despite the A drive element being a rescue, the drive inheriting individual are fully affected by resistance alleles that emerge from deposition as the target gene is haplolethal.

Fig 4d–4f show the _A1D allele dynamics for different maternal deposition rates. The _A2D and _A1 allele dynamics are shown in S15 Fig. The same simulations were performed without the possibility of gRNA deposition. Restricting deposition in such a way may, on the face of it, be expected to aid the drives. However, it actually makes the fitness costs associated with deposition exclusively tied to inheritance of a gRNA expressing drive element, favouring individuals that do not inherit the gRNA expressing drive element. There may be more complex interactions with nuclease-only deposition for the two DCD designs; however, we find no substantial differences in the overall outcomes (S16 Fig).

There have been several reports of high rates of inheritance bias from maternally deposited nuclease (shadow drive) [27, 31, 38, 39]. This appears to be observed mainly in cases where Cas9 is provided by the mother and the gRNA gene from the father [8]. S17 Fig shows simulations of different maternal deposition rates with the possibility of shadow drive. We find that this specific scenario is so rare (and impossible for the self-perpetuating drive) that it has a negligible impact on the drive outcomes.

Finally, we tested how the drive frequency can be adapted to compensate for increasing rates of maternal deposition (Fig 4g–4i and S18 Fig). This shows that major changes in the deposition rate can be compensated for with relatively minor changes in the release frequency. This is in contrast to the same analysis described for changes to the germline-based cutting rate, which displayed a much more linear relationship. Strikingly, spread into population two was much more limited for DCD with maternal deposition, even at high release frequencies (S18 Fig). This provides a large parameter space with substantially higher drive spread into population one compared to population two (Fig 4j–4l) for the DCDs. The explanation for this is that once all target alleles have been cut, deposition ceases to have any consequence. However, at low frequencies, such as with the migration of drive-carrying individuals to a population of mostly wild-types, the deposition-induced fitness costs still occur and limit the drives ability to spread.

Overview

The model we present here is capable of tracking the frequency of different genotypes through discrete life stages and generations. To facilitate that, we devised a nomenclature system capable of defining the range of possible genotypes and parental deposition states that individuals can take on for daisy-chain gene drives (the most complex case we consider). This is required to model a range of possible different drive designs and the many different individual states that a single drive can give rise to during its spread. During the embryonic and germline life stages, drive-induced genotype conversion occurs, changing the relative frequency of the genotypes. We use non-overlapping discrete-generation recursion equations for genotype frequencies, treating males and females separately. This method is applied in a simple deterministic allele frequency simulation of a single release of gene drive individuals into a population of all wildtypes.

Stages and genotype nomenclature

The model tracks the frequency of different genotypes through different life stages. Each discrete generation (t), the same stages are repeated. The stages and their corresponding frequency vectors are: zygote (z), embryonic (e), germline (g), and gamete (p) stages. All vectors apart from the zygote vector are sex-specific, and male versions are indicated with a bar below the symbol (e.g., p). The overall genome frequency may vary through these stages; however, each generation the frequency is normalised to 1. For the particular drive system we are considering, d is the number of unique diploid genotypes, while h is the number of possible haploid genotypes. As such, z_i is the frequency of the i^th zygotic genotype, with i ranging from 1 to d.

Throughout each stage, the genomes maintain their genotype unless they are exposed to a gRNA:nuclease complex and contain a target allele (T) that can be cut. The other alleles we consider are type-1 (1), functional, and type-2 (2) non-functional resistance mutations and drive alleles. The symbol for the drive alleles is distinguished by whether that allele contains a nuclease (N), no nuclease but a gRNA (G), or neither (E). The N allele can carry a gRNA, but does not necessarily have to. None of the alleles except T can be cut but, if present, can influence the repair outcome of the T allele on the homologous chromosome. The drive elements can, depending on the specifics of the drive, provide the components necessary for a cut to occur at another locus but do not influence the repair outcomes at that other locus.

To model daisy-chain drives, we must further distinguish genotypes at different genomic loci. The defining feature of the persistence of inheritance bias for a daisy-chain element is its priority in the chain [9]. Sequentially going from A, B, C, etc. To aid in interpretation, each element’s genotype (diploid combinations of: T/N/G/1/2) has a letter subscript that denotes its order in the daisy-chain (e.g., _ATT). Any genotype with a different letter subscript will be at a different genomic locus. For drive elements that carry a gRNA, a superscript indicates which T allele locus they target (e.g., _AN^A or _CG^B). Table 1 lists the genotype notation of the drives studied in this publication and the two element split-drive.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Genotype specification of drive designs.

Drives are listed in their heterozygous state.

https://doi.org/10.1371/journal.pgen.1010370.t001

Haploid to diploid stage

Each discrete generation, the current genotype frequencies can change due to the action of the drive and fitness effects (S1(a) Fig). For generations 2–100, this process starts by the combining the haploid genotype frequencies of the previous generation from the male (p^t−1) and female (p^t−1) individuals to form the frequency vector of the zygotes (z^t) (Eq 1). For generations 2–100, there is a single vector z, since, in our model, sex effects manifest only at the embryonic stage. However, for the first generation, the embryonic frequency is calculated separately for males and females according to the release frequencies. As such, released individuals are reduced according to their fitness before their opportunity to mate.

The model tracks the deposition of drive components from one generation to another. For Cas9 deposition, we differentiate each allele received by the zygote by a sex-specific Cas9 deposition factor (₀ or ₁). We do not differentiate whether the deposition comes from a nuclease element heterozygote, or homozygote parent. For example, a haploid T}₀ allele has come from a non-Cas9 carrying diploid parent (e.g., TT), while a T}₁ must have come from a nuclease carrying parent (e.g., TN). In the diploid genomes these notations are combined, with the drive status of the male parent listed first (e.g., }_0*) and then female parent (e.g., }_*0). Eq 2 illustrates the combining of p^t−1 and p^t−1 to form z^t. Note, that many genotypes are functionally equivalent and will be summed when reporting the outcomes of the drive (e.g., {T1}₀₀ = {1T}₀₀, {p_i/p_j}₀₀ = {p_j/p_i}₀₀, {p_i/p_j}₁₁ = {p_j/p_i}₁₁, {p_i/p_j}₁₀ ≠ {p_j/p_i}₀₁). Deposition of the gRNA is tracked in a similar way. The haploid genotypes are differentiated by the gRNAs that have been expressed in the diploid parent, and this is used in the calculation of deposition-based cutting in the diploid embryos. (1) (2)

Fitness costs, and genotype inter-conversion by deposition and expression

In the transition from the zygotic to the embryonic state, the model takes into account deposition and fitness effects. Through deposition, each individual zygotic genotype (z_i) can give rise to multiple embryonic genotypes (e). Genotype conversion is mediated by matrix K, which converts genotypes according to the deposition cut and repair rates. Furthermore, based on the fitness parameters (Fig 1c), we can, for each genotype, define a particular fitness cost (0–1) in the vector θ. The fitness cost vector has a female (θ) and male (θ) specific version because for some simulations sex-specific fitness costs are applied. The fitness of a particular genotype (θ_i) is determined by multiplying the fitness costs of individual alleles. The dominant fitness cost imposed by a particular allele is applied only once, even if two copies are present. Some simulations impose an additional recessive fitness cost on the A locus, which only occurs if two loss of function alleles are present at A (e.g., _AEE, _AE2, _A22). As the multiple genotypes produced by deposition are within the same mosaic organism, the fitness cost contribution of individual genotypes is summed and applied uniformly over all genotypes arising from the same zygotic genotype. The deposition conversion outcomes and fitness costs of any one genotype do not vary throughout a simulation, only the relative frequency of the input genotypes in the form of z. As such, conversion matrix D/D can be constructed that mediates both deposition conversion (K), and fitness costs (θ_i and θ_i). Matrix D is the product of the row-wise multiplication of matrix K_i,* by θ and normalisation. When z is then subjected to conversion matrix D it produces the embryonic frequency vector e (Eq 3). Vector e represents the genotype frequency at the end of the embryonic stage, after deposition and fitness costs have been applied. Vector e/e is subsequently multiplied by the expression-based gene conversion matrix C/C to produce the gamete frequency vector g/g (Eq 4). The construction of matrices C and K is discussed after gamete production. (3) (4)

Gamete production

The last step in each generation is the conversion of diploid genomes to haploid ones. This is done by multiplying the diploid genome frequency vector g by matrix H to form the haploid genome frequency vector p (Eq 5). Matrix H simply splits each diploid genome into the possible haploid genomes it can produce, an example of which is shown in Eq 6. An important step in this process is to keep track of nuclease and gRNA deposition. The presence of at least one N allele in the diploid germline genome causes the haploid genome to have an associated deposition marker 1 (*}₁). If no N allele is present in the diploid genome, the deposition marker is 0 (*}₀). Separately, any gRNA expressing elements in the diploid parent causes each haploid genome to carry an indicator for deposition of that gRNA (as long as Cas9 is deposited simultaneously). Like Cas9 deposition, this occurs even if the gRNA-expressing element is not inherited. We assume that alleles at separate loci segregate fully independently, as is shown in Eq 7 which shows H if two elements are considered. (5) (6) (7)

Deposition and drive induced conversion

To model drive activity within an individual, we approach gene editing as the conversion of genomes from one genotypic state to another genotypic state. The probabilities underlying this will depend on the probability that a DNA cut will occur (κ) and the HDR rate (α), together with the ratio between type-1 and type-2 resistance mutations (β). From this, DNA repair can result in three outcomes: interchromosomal-homology-directed repair (α), type-1 resistance mutations ((1− α) ⋅ β = μ), and type-2 resistance mutations ((1− α) ⋅ (1 − β) = ν). The distinction between type-1 and type-2 mutations is dependent on the target locus, and in our model only relevant for accounting for fitness effects. The probability of all repair outcomes sums to 1. For each simulation, the value of each of these parameters is fixed and the default values are listed in Fig 1c. Each of the previous parameters is defined separately for maternal deposition-based conversion, which is indicated by ₀₁ (κ₀₁, α₀₁, and β₀₁).

In our model, there are two stages where the drive can induce a genotype conversion: embryonic and germline. For each genotype and its associated deposition state, we define a rate at which it converts to the other possible genotypes or remains the same (the overall rate is always 1). These conversion probabilities are mediated by matrix C for the drive expression in the germline, and K for drive deposition in the embryo. For the majority of genotypes, the conversion rate maintaining the original genotype will be 1, and the conversion rate to all other genotypes will be 0. This is because only a select few genotypes can convert. The requirements for conversion are: first, the genotype must have a genomically expressed gRNA or deposited gRNA (in complex with Cas9). Second, that gRNA must target a locus that in that particular individual has one or more T alleles present. Lastly, there must be genomically expressed or deposited nuclease. If these conditions are met, we use the cut and repair probabilities to determine the conversion rate. If a T allele can be cut in the germline, κ ⋅ α of the T alleles will convert to whatever allele is on the homologous chromosome. κ ⋅ μ of the T alleles will convert to a type-1 resistance allele and κ ⋅ ν of the T alleles will convert to a type-2 resistance allele. Finally, 1-κ of the T alleles will remain uncut. These conversion rates populate matrix C, as shown in Eq 8. The exact same relationships hold for matrix K shown in Eq 9, with the κ₀₁, α₀₁, μ₀₁, and ν₀₁ parameters (with α₀₁ being 0, except in the case of the shadow drive simulations). The conversion matrix for genotypes with TT is more complex. For a TT genotype to be cut, the nuclease and gRNA must be expressed from a separate locus, or alternatively the nuclease, or nuclease and gRNA can be provided by maternal deposition. For TT genotypes, we assume that α = 0, or in a separate set of supplemental simulations that alleles are cut one by one, resulting in the conversion probabilities shown in Eq 10. (8) (9) (10)