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# Novel combination of CRISPR-based gene drives eliminates resistance and localises spread

## Abstract

Invasive species are among the major driving forces behind biodiversity loss. Gene drive technology may offer a humane, efficient and cost-effective method of control. For safe and effective deployment it is vital that a gene drive is both self-limiting and can overcome evolutionary resistance. We present HD-ClvR in this modelling study, a novel combination of CRISPR-based gene drives that eliminates resistance and localises spread. As a case study, we model HD-ClvR in the grey squirrel (Sciurus carolinensis), which is an invasive pest in the UK and responsible for both biodiversity and economic losses. HD-ClvR combats resistance allele formation by combining a homing gene drive with a cleave-and-rescue gene drive. The inclusion of a self-limiting daisyfield gene drive allows for controllable localisation based on animal supplementation. We use both randomly mating and spatial models to simulate this strategy. Our findings show that HD-ClvR could effectively control a targeted grey squirrel population, with little risk to other populations. HD-ClvR offers an efficient, self-limiting and controllable gene drive for managing invasive pests.

## Introduction

CRISPR-based gene drives have the potential to address problems in public health, agriculture and conservation, including the control of invasive species1. Invasive species impact livelihoods, have severe economic consequences, and are among the major driving forces behind biodiversity loss2,3,4. Current control methods such as shooting, trapping, and poisoning are inhumane, labour-intensive, expensive, and ineffective in dealing with the scope of the problem in most situations5,6,7. Examples of damaging invasive species as a result of human mediated introduction include rabbits and cane toads in Australia, Asian carp in the US, and the grey squirrel and American mink in the UK.

In this study, we use the grey squirrel (Sciurus carolinensis) which is considered invasive in the UK as a case study for gene drive population control. First introduced in the 19th century, the grey squirrel is now widely distributed across the UK8. Since their introduction there has been a major decline in native red squirrels (Sciurus vulgaris). Grey squirrels are both larger and more aggressive than red squirrels and are passive carriers of Squirrelpox virus, which is lethal to red squirrels9. Without intervention, red squirrels could be lost from the UK mainland within the next few decades10. In addition to their impact on native red squirrels, grey squirrels also suppress natural forest regeneration through bark stripping of trees11 and likely have a negative impact on biodiversity of native woodland birds by preying on eggs and chicks12. As an invasive pest they are estimated to cost the UK economy more than £14 million per year by debarking trees, gnawing through electricity cables and other forms of property damage13. A manageable and robust grey squirrel control strategy remains to be established7.

Although there are still technical challenges14,15, CRISPR-based gene drives may offer a humane, efficient, species-specific and cost-effective method for controlling invasive species, including grey squirrels in the UK16,17; filling a distinct void in the conservation toolbox. Broadly, a gene drive skews the inheritance ratio of an allele towards a super-Mendelian rate and therefore drives itself to spread quickly through a population18. The CRISPR-Cas system that these gene drives are based on comprises two components: a guide RNA (gRNA) and a nonspecific Cas nuclease19. The gRNA directs the Cas nuclease to a specific sequence in the genome where it generates a double stranded break. Several synthetic CRISPR-based gene drives have been proposed with three major types suitable for population control: homing, driving-Y and cleave-and-rescue (Fig. 1)1,20,21,22. A homing gene drive works through a process called ‘homing’1. The system utilises germline-specific expression of CRISPR-Cas and subsequent cleavage in the germline, which leads to homology-directed repair (HDR) copying the gene drive element onto the homologous chromosome. By locating the homing gene drive cassette within the coding sequence of a haplosufficient female fertility gene, thereby disrupting the gene’s function, female somatic homozygotes will be infertile. As population growth is typically controlled by female reproductive performance18, the population will decline in size due to an increasing number of infertile females within the population. A driving-Y gene drive (also called X-shredder) specifically expresses CRISPR-Cas from the Y-chromosome during spermatogenesis to shred the X-chromosome at multiple locations beyond repair20,23. Therefore, only Y-bearing sperm mature and all or most offspring of a driving-Y father will inherit a gene drive harbouring Y-chromosome and be male. This eventually leads to a population decline due to the lack of breeding females. Cleave-and-rescue gene drive (also called toxin-antidote) uses CRISPR-Cas to cleave an essential gene while also supplying a recoded, uncleavable ‘rescue’ copy of this gene within the gene drive cassette21,24. In the case of a haploinsufficient essential gene, offspring must inherit the gene drive to be viable. For population suppression, the gene drive cassette is located in a haplosufficient female fertility gene like the homing gene drive, and thus this gene drive eventually also causes population decline due to female infertility24. In toxin-antidote terminology, this type of cleave-and-rescue gene drive is called Toxin-Antidote Dominant Embryo (TADE) suppression drive24.

Although all three population suppression gene drives are elegant and promising, they all face technical challenges. Homing gene drives face three major challenges. First, during in vivo testing, the formation of resistance alleles which block homing have been observed25,26. Resistance alleles can form through non-homologous end joining (NHEJ) instead of the desired homology-directed repair during homing. A potential solution is gRNA multiplexing16, but this is likely to reduce homing efficiency27,28. Second, another unresolved molecular challenge in mammals is the identification of a suitable promoter for sufficient, germline-specific expression of the gene drive. Ideally, the promoter should restrict Cas9 expression to a window when the endogenous meiotic recombination machinery is most active15. Third, a homing gene drive that was not hindered by resistant alleles could theoretically spread indefinitely, thereby compromising global ecosystem safety. To address this concern, approaches to make gene drives self-limiting have been divised, including versions called ‘daisy drives’ (see Fig. 4A)29,30,31,32. Of these daisy drives the ‘daisyfield’ drive is the most straightforward mechanism to limit spread. In a daisyfield gene drive, the gRNAs are scattered throughout the genome (forming a daisyfield)32. These daisy elements are inherited in a Mendelian fashion, and therefore, offspring inherits half of the daisy elements from each parent. Thus, the gene drive stops spreading as the daisyfield is diluted through matings with wildtype individuals. Once all daisy elements have disappeared, all elements of the gene drive will likely also disappear due to negative selection (as homozygotes are infertile) and drift. This is desirable in case gene drive individuals spread to a non-target population. In a population where further spread is required, gene drive individuals with a complete daisyfield can be supplemented to keep the gene drive spreading. The rate and extent of suppression can be controlled by the number of gene drive animals supplemented and how many daisy elements the introduced animals carry. In contrast to homing gene drive, driving-Y gene drives face problems with the formation of a population equilibrium depending on shredding efficiency33,34. Furthermore, a major challenge in developing driving-Y in mammals is the identification of a highly-specific spermatogenesis promoter to drive Cas-gRNA expression17. Cleave-and-rescue gene drives have the advantage that multiplexing does not reduce efficiency as there is no homing involved, and therefore, the formation of resistance alleles is limited. Furthermore, cleave-and-rescue gene drives also show frequency-dependent dynamics, which can be exploited to keep the gene drive contained24. This poses practical challenges however, as it requires an accurate estimate of population size and the release of a large number of animals simultaneously.

Population control gene drives still need improvements for optimal efficiency, self-limitation, and controllability. In this study, we present HD-ClvR, a novel combination of gene drives that eliminates resistance, is self-limiting, and can be controlled in a practical manner. HD-ClvR is composed of homing (H), daisyfield (D), and cleave-and-rescue (ClvR) gene drives. Our modelling in grey squirrel demonstrates the strategy is highly efficient and overcomes the ongoing issue of resistance allele formation of homing gene drives. The daisyfield gene drive ensures self-limitation and allows for controlled, localised spread. Therefore, HD-ClvR could effectively control a targeted grey squirrel population, with little risk to other populations. Our analysis includes a randomly mating population and a spatially distributed population, which mimics the UK grey squirrel, though it can be adapted to other species. This study provides the first promising steps towards the development and testing of HD-ClvR.

## Results

HD-ClvR is a combination of three gene drives: homing, daisyfield, and cleave-and-rescue. Our randomly mating and spatial modelling of this strategy in grey squirrel illustrates that HD-ClvR can effectively eliminate resistance allele formation, allows for optimised gRNA multiplexing, improves efficiency over standard cleave-and-rescue drives, and is both self-limiting and controllable. We find that the placement of supplemented animals significantly impacts the effectiveness of HD-ClvR, but that this is not prohibitive to the spread of the gene drive and that an effective placement strategy can achieve a rate of gene drive spread close to a randomly mating population.

### Eliminating resistance alleles

By combining a homing gene drive with a cleave-and-rescue gene drive, HD-ClvR eliminates resistance alleles which occasionally form during gene drive homing (Fig. 2a). This works as follows: as germline homing occurs, both copies of a haploinsufficient essential gene are cleaved, and their function is destroyed through erroneous NHEJ-based repair. However, the homing construct contains a recoded, uncleavable copy of this haploinsufficient gene as a ‘rescue’. For offspring to be viable, they must inherit the gene drive with the rescue to have sufficient expression of the haploinsufficient gene. Offspring that inherit a resistance allele instead of the gene drive will not develop as they lack the rescue gene to compensate for their broken copy of the haploinsufficient gene. This mechanism prevents the emergence of resistance alleles.

HD-ClvR also allows for independent optimising of gRNA multiplexing for both homing efficiency and resistant allele elimination. Multiplexing gRNAs can overcome resistance allele formation, allowing homing to take place even if some resistant gRNA sites are present. With a standard homing gene drive, the optimal number of gRNAs is a trade-off between homing efficiency and overcoming resistance allele formation. Two gRNAs has been proposed as optimal for homing, with efficiency decreasing when more than two gRNAs are used28. However, to also limit the formation of resistance alleles, the optimal number in the trade-off lies between 4 and 828. In contrast, with HD-ClvR it is possible to select the optimal number of gRNAs for homing, while multiplexing several gRNAs within the cleave-and-rescue to reduce the probability of resistance allele formation to effectively zero. Current data suggests four gRNAs is sufficient to prevent resistant allele formation28.

In grey squirrel, we have selected two genes through literature mining which are suitable for HD-ClvR: Progesterone Receptor (PGR) as a haplosufficient female fertility gene involved in ovulation35 and Delta-Like Canonical Notch Ligand 4 (DLL4) as a haploinsufficient essential gene involved in the Notch signalling pathway36. Both of these genes are conserved across many taxa (including vertebrates, arthropods, and nematodes among others) and if proven effective could also be used for other invasive species37. Figure 2b shows a candidate HD-ClvR contruct design for grey squirrel control, using 1 gRNA for homing and 4 gRNAs for cleave-and-rescue.

To demonstrate that combining a homing and cleave-and-rescue gene drive can eliminate the formation of resistance alleles, we model a standard homing gene drive, a standard cleave-and-rescue gene drive, and a homing-cleave-and-rescue gene drive in a randomly mating population of grey squirrels. We model different rates of NHEJ ($$P_n$$, Fig. 3), which is the probability that a gamete is repaired through NHEJ instead of HDR. When NHEJ occurs, we model a probability of 0.33 that the target is a functional resistant allele, which is the probability that a frameshift did not occur. Like Prowse et al.16, we model no fitness cost to heterozygote gene drive animals. Our model uses either 1 or 4 gRNAs to show multiplexing reduces resistance allele formation. For the standard cleave-and-rescue gene drive, we modelled the release of 1000 gene drive squirrels instead of 100 gene drive squirrels, as this form of drive is only effective at a large introduction frequency. A standard homing gene drive was effective at low rates of NHEJ ($$P_n$$ = 0.02 and 0.1) when multiplexing 4 gRNAs but is inhibited by resistant alleles when only 1 gRNA is used at the same rates of NHEJ. However, at a higher rate of NHEJ ($$P_n$$ = 0.5), squirrels with resistant alleles rescue the population from standard homing gene drive suppression despite multiplexing 4 gRNAs. In contrast, with a homing-cleave-and-rescue gene drive, resistant alleles are eliminated, and the squirrel population is completely suppressed across all rates of NHEJ when 4 gRNAs are used in the cleave-and-rescue component of the drive. When we compare the three gene drive types in a large population of carrying capacity 30,000 instead of 3000, we see the same dynamics (Supplementary Figure S1).

Although we model the homing gene drive component of HD-ClvR targeting a haplosufficient female fertility gene in this study, HD-ClvR is adaptable and could target any desirable gene to generate a loss of function mutation through insertion disruption or propagate a genetic cargo of interest. The cleave-and-rescue component of the HD-ClvR targets a haploinsufficient developmental gene in this study but this could also be adjusted to a haploinsufficient both-sex infertility gene. Our results suggest it is marginally more efficient to target an embryonic lethal gene (Supplementary Figure S2), as this prevents infertile resistant individuals from competing with gene drive individuals for resources. From an ethical standpoint the reduction in efficiency when targeting a both-sex fertility gene, instead of an embryonic lethal gene, may be justified by the improved societal and political acceptance for a strategy that evades killing and suppresses through infertility. Additionally, we tested if overexpression of the cleave-and-rescue target gene should be biologically tolerable, as individuals could have more than 2 copies of this gene if cleaving of the wildtype gene fails (Supplementary Figure S2). We conclude that when multiplexing sufficiently for the cleave-and-rescue part of the gene drive, there is no difference. As can be seen from the dynamics when multiplexing less or not at all, allowing overexpression makes the gene drive initially faster to spread, but also allows resistance alleles to persist in the population.

### Self-limitation and control

A key benefit of HD-ClvR is that by including a daisyfield gene drive, it is self-limiting and can be controlled based on the number of supplemented gene drive animals and number of daisy elements each supplemented animal harbours (Fig. 4a). Unlike a standard homing gene drive, HD-ClvR can control the rate and extent of population suppression and, if required, suppression could be stopped by terminating further animal supplementation. Additionally, HD-ClvR does not require the large initial releases of standard cleave-and-rescue animals, which places pressure on the local ecosystem.

Using our randomly mating model, we show in Fig. 4b that by including a daisyfield system in a homing-cleave-and-rescue drive to form HD-ClvR, we can efficiently suppress a targeted population, while limiting risk to other populations, especially if those are bigger than the target population (Supplementary Figure S3). We modelled HD-ClvR with different daisyfield sizes in a population of 3000 grey squirrel over different rates of annual supplementation following an initial release of 100 HD-ClvR squirrels. The model shows that once the HD-ClvR runs out of daisy elements the population recovers. Therefore, HD-ClvR poses less risk to non-target populations than a standard homing gene drive. With 1% annual supplementation of HD-ClvR squirrels, the population size is reduced and maintained at an equilibrium, and with 10% annual supplementation the targeted population of grey squirrel is removed for all daisyfield sizes. Variability in the model with 1% supplementation and 60 daisyfield elements is caused by two factors: first, only a small number of individuals is supplemented due to the low population equilibrium, so randomness can have a big impact. Second, we round down the number of supplemented animals, to avoid perpetual maintenance of the population. Therefore with 1% supplementation, no animals are supplemented when the population is below 100 individuals. In Supplementary Figure S4, we show that it is possible to suppress a population without an accurate estimation of population size, which will be hard to obtain for most wild populations. To find the optimal combination of supplementation rate and daisyfield size, we ran a range of these two parameters and found that 5% supplementation would be sufficient to suppress a population, even with a small daisyfield (Supplementary Figure S5).

### Spatial dynamics and supplementation of HD-ClvR

To understand the spatial dynamics of homing-cleave-and-rescue drives, initially excluding daisyfield, we modelled this approach in a simple spatial model. Modelling a single release of 100 homing-cleave-and-rescue gene drive squirrels in populations of 3000 and 30,000 squirrels, the model demonstrated that the spatial life history of grey squirrel allows for the spread of the gene drive (Fig. 5). We also show that the removal of the target squirrel population is more delayed in the spatial model than in the randomly mating population model. This difference is approximately 5 years in a small population, and is increased to approximately 15–20 years in a big population. To test the sensitivity of our model to two crucial parameters, mating range and migration range, we performed a sensitivity analysis and conclude that the model is sensitive to a decreased mating range, but not to a decreased migration range (Supplementary Figure S6).

Using our spatial model, we then explored how the placement of supplemented HD-ClvR animals could impact population suppression. We show the impact of different supplementation placement schemes by modelling five strategies: mean of population location, mode of population location, randomly, randomly in 10 groups, and in a moving front (Fig. 6a). The moving front was implemented such that we start at the bottom and move upwards in ten steps, thereafter, supplementing at the topmost location. As can be seen in Fig. 6b, different placement schemes significantly affect the efficiency of the strategy. Placement at the mean population location was least effective and placement of squirrels randomly in 10 groups was most effective. Figure 6c shows three moments which represent key spatial dynamics of each placement scheme. For animations of the spatial dynamics over the whole timeline, see the animated GIFs (Supplementary Videos S7).

## Discussion

This research presents HD-ClvR, which is a combination of three gene drives: homing, cleave-and-rescue and daisyfield. Our modelling indicates that HD-ClvR overcomes an important trade-off in current homing gene drive designs: the trade-off between resistance allele formation and gene drive efficiency. This strategy benefits from the efficiency of a homing gene drive and the evolutionary stability of cleave-and-rescue gene drive. Due to the inclusion of a daisyfield system, HD-ClvR is self-limiting and can be controlled by supplementation of gene drive animals.

### HD-ClvR compared to other gene drives

Over recent years, many different gene drives have been published and developments have been geared towards both efficiency and safety38. An ongoing issue has been the development of resistance alleles. For CRISPR-based homing gene drive there are two fundamental approaches to combat resistance allele formation: careful gRNA targeting and gRNA multiplexing. When a gRNA targets a conserved sequence in a gene, resistance alleles are likely to disrupt gene function through NHEJ repair and will therefore reduce fitness39. Recently, population suppression was already shown to work with a carefully targeted homing gene drive in contained mosquito populations39, however, current data suggests that homing might be less efficient in mammals than in insects14. A recent paper has proposed the concept of ‘tethered homing gene drive’, which combines a threshold-dependent underdominance gene drive with a homing gene drive for improved suppression capabilities40. We use this concept in a different manner in HD-ClvR, by relying on a daisyfield rather than threshold-dependence for self-limitation. Very recently, two new papers have proposed a gene drive similar to HD-ClvR, but intented for population modification instead of suppression41,42. These studies also combine homing and cleave-and-rescue principles to combat resistance alleles and their modifications are able to persist stably in cage experiments, which is promising for HD-ClvR.

In addition to targeting conserved sequences, when gRNA multiplexing, resistant allele allele formation is reduced because multiple sites are targeted simultaneously. For homing gene drives, multiplexing has been shown to reduce homing efficiency when more than two gRNAs are used28. In contrast, cleave-and-rescue gene drives do not have this problem, as they do not use homing and can therefore multiplex gRNAs without any efficiency costs. HD-ClvR separates the elimination of resistance alleles and homing efficiency, and therefore gRNAs can be optimised for both goals separately.

To date, most gene drive research has focused on improving the efficiency, however, equally important is the development of strategies that allow for containment, or even reversibility, of the gene drives29,43. For contained gene drives, density dependence is often used, which requires large numbers of gene drive individuals to be released into a target population to spread44. Therefore, non-target populations are unlikely to be affected by this type of gene drive. However, a large single release of gene drive individuals can put significant pressure on the local ecosystem, and if a population is already at carrying capacity, it may lead to starvation or mass migration of the population. In contrast, HD-ClvR uses ongoing input in the form of gene drive animals to control the extent of population suppression and contain spread, while the total amount of gene drive animals necessary for release is similar to threshold-dependent gene drives. Therefore, the use of HD-ClvR seems more feasible than threshold-dependent gene drives. Although self-limitation comes with increased cost and labour relative to unlimited gene drives, we believe this is justified by the control and safety of HD-ClvR.

As stated above, the initial introduction frequency for a standard cleave-and-rescue gene drive in our randomly mating model was increased 10-fold over the other homing-based strategies. This increase is necessary due to the significant cost to the reproduction rate that is incurred when using a standard cleave-and-rescue gene drive. On average, cleave-and-rescue animals will produce 50% less offspring than wild-type animals21,24. This significantly slows the spread of the gene drive and due to density dependent dynamics, requires large initial releases of cleave-and-rescue animals for population suppression. With a homing-cleave-and-rescue drive, more offspring inherit the drive and there is less cost to the reproduction rate. Effectively, for homing-cleave-and-rescue, the reproduction rate of gene drive individuals is equal to the homing efficiency (plus half of the homing failure rate, where the gene drive is inherited by chance), which so far has been shown to range from 0.7 to 1 in different organisms14,39,45.

### Supplementation

As animal supplementation is a critical component of HD-ClvR, our modelling investigated how daisyfield size and the level and placement of supplemented HD-ClvR animals effects efficiency and safety of population suppression. Optimisation of these parameters can significantly reduce cost and labour, as well as reduce the risk of unwanted impacts on non-target populations. We modelled our supplementation as a percentage of the total population size, therefore the number of individuals needed for supplementation increases linearly with population size. We also want to minimise the risk of non-target populations being impacted by the gene drive, and therefore, there is a trade-off between safety (size of the daisyfield) and cost and labour (level of supplementation required).

The least number of daisy elements that can suppress the population with a realistic level of supplementation, but does not cause any serious issues in non-target populations, should be objectively established through an in-depth risk assessment process. In a larger population however, the spread is slower than in a small one. Therefore, for improved safety and efficiency, gene drives are best applied in small sub-populations separately. The impact of a single introduction, such as a rogue deployment or migration, depends on the population size. The smaller the population, the bigger the impact. This it is a concern when the target population is much larger than the non-target population, but this is not the case for invasive UK grey squirrels and many other invasive species.

The appropriate daisyfield size also depends on the rate of NHEJ ($$P_n$$) of the gene drive system; the higher the ($$P_n$$), the more embryonic lethal offspring will arise and the sooner daisyfield burns out. To choose a safe number of daisy elements, we also need an estimate of how many animals a rogue party could obtain, potential breed and add into a non-target population for their own benefit. Overall, each target population and prospective gene drive strategy needs to be considered on a case-by-case basis and include an in-depth multidisciplinary risk assessment process.

When we consider the spatial aspects of a HD-ClvR supplementation programme, the picture becomes more complex. A key factor is the supplementation location of individuals. Obviously, supplementing individuals in a location where the population has already been suppressed will be ineffective. Therefore, different placement strategies can be adopted to keep placing individuals in a relevant area. A monitoring system where not only the size of the population is known, but also the location can significantly help HD-ClvR continue spreading and suppress a targeted population.

In this study, we modelled HD-ClvR using five different supplementation placement strategies in grey squirrel. These were: supplementation at the mean of population location, the mode of population location, randomly, randomly in 10 groups, and in a moving front (Fig. 6a). With supplementation at the mean of the population location, supplementation started in the middle of the population. After a few generations, a gap appears in the middle due to local suppression. The mean of the populations location still lies in the middle, as can be seen in Fig. 6c at 20 generations. Therefore, supplementation is not effective until the population is also suppressed in another location, thereby shifting the mean. Additionally, when there is a single large patch of the population left and additional smaller clusters, supplementation in the middle of the large patch allows the smaller clusters to recover, as can be seen in Fig. 6c after 64 generations.

With supplementation at the mode of the population location, we supplement in a location where there are many individuals. This placement strategy avoids the problem of supplementing in a location without individuals, either in a doughnut-like spatial population structure or in a multi-patch population. However, this placement strategy still allows small patches to form and recover. Supplementation at a random location theoretically means that supplementation happens uniformly, but in reality, this is not the case. Initially HD-ClvR spreads in multiple locations, but after the population is suppressed in certain regions, supplementation in those regions becomes ineffective. Therefore, at a later stage of population suppression this placement scheme becomes increasingly ineffective.

Supplementation at random locations is more effective when they are broken up into multiple groups (ten in our model). The gene drive spreads in many locations initially like the random single location placement scheme. After significant suppression of the population some but not all of the 10 groups supplemented are at ineffective locations. The groups that are placed at relevant locations are enough to keep the gene drive spreading. In our model supplementation in groups at random locations gets close to the speed at which a gene drive spreads in a non-spatial model.

The moving front placement scheme is very effective initially, as the gene drive spreads uniformly across the front. In this case, supplementation keeps ahead of where the populations is being suppressed. This placement strategy allows the population to recover behind the moving front after effective initial spread and near-complete suppression. To improve efficiency of the moving front strategy, it may be beneficial to include random supplementation behind the moving front to prevent animals from re-establishing.

Finally, in our spatial model, it was evident that there is more uncertainty in levels of population suppression than a randomly mating model leads us to believe. As can be seen in Fig. 6b, the 95% quantiles are broader than the quantiles in Fig. 3. Therefore, we conclude that to tailor the amount of supplementation, it is vital to closely monitor a population where a gene drive is used.

### Assumptions and future work

Our model works under the following six assumptions. First, our model excludes some complexities of the optimal number of gRNAs for homing. Although our model suggests that multiplexing gRNAs for both the homing and cleave-and-rescue gene drives is most effective, a recent study using a more complex model and in vivo data shows that the optimal number of gRNAs to use for homing in Drosphilia melanogaster is two. They report a decrease in homing efficiency with more than two gRNAs due to reduced homology and Cas nuclease saturation28. Therefore, our gene drive with four gRNAs for both homing and cleave-and-rescue will likely be less efficient in such a complex model. We suggest using two homing gRNAs and four cleave- and-rescue gRNAs is likely most efficient, while still eliminating all resistance alleles28. It would be prudent to analyse our gene drive in this complex model as well to get a definitive estimate, as Cas saturation is thought to have an influence on gene drive efficiency when multiplexing is used28.

Second, we assumed there was no embryonic Cas-gRNA expression. Embryonic Cas-gRNA expression might be problematic as it leads to resistance allele formation and can interfere with the cleave-and-rescue mechanism by cleaving alleles from the wildtype parent. As our gene drive eliminates resistance alleles, embryonic Cas-gRNA expression may not inhibit spread, depending on the rate. Additionally, if the embryonic Cas-gRNA expression turns out to be more common in grey squirrel or other species, the cleave-and-rescue part of the gene drive can be harnessed with a double rescue mechanism to overcome this issue, as reported by Champer et al.24.

Third, we did not take other types of resistance alleles into account such as mutations rendering the CRISPR-Cas non-functional. As this is a universal assumption in gene drive research, we will have to await multigenerational studies to see if this is problematic.

Fourth, HD-ClvR has not been tested in vivo, which is our next step. The two recent papers testing a gene drive similar to HD-ClvR for population modification have performed in vivo tests in Drosophila melanogaster which showed very efficient conversion rates41. Proof-of-concept testing of HD-ClvR would likely initially occur in D. melanogaster and mouse models before progressing to squirrel studies. Recent reports have shown that the VASA promoter for Cas expression in homing gene drives is not optimal and further investigation to identify a meiosis-specific germline promoter is needed15. Furthermore, the integration of many daisies in a squirrel genome will be a molecular challenge and is a feat which has not yet been reported on in any species. This task could be achieved using either a random integration strategy, such as lentiviruses46 or a targeted integration strategy that exploits neutral repetitive sequences in the genome as target sites32. Also, non-model species might be difficult to genetically engineer, although grey squirrel embryology will likely follow the extensive knowledge on rodent and farmed animal embryology, and similar reagents and equipment could be used. An important consideration when engineering gene drive is that the modified animals maintain enough wild vigour to survive and breed in a wild population. Promising technologies for generating gene drive harbouring mammals with as little intervention as possible include in situ delivery of CRISPR reagents to the oviduct47.

Fifth, for our spatial modelling, we assumed that an estimation of population size could be made every year, although there is a significant amount of room for error in this estimate. Additionally, for some of our placement schemes, we assumed an accurate estimate of population location. As the random placement in groups scheme turned out most effective, this is not a problem so much as further potential for improvement. Another direction for future spatial work is the modelling of real landscapes, which are more complex than what we modelled in this study48. In complex landscapes, it might be that gene drive spread is slower or even regionally confined in some situations. Additionally, there might be spatial dynamics to gene drives in general such as ’chasing’, which is the perpetual escaping and chasing of wildtype and gene drive animals34. Further efforts are necessary to create a more realistic spatial model before we can consider using a gene drive.

A final consideration is that the ecological services the grey squirrel and other invasive species provide are largely unchartered. Ecologists need to investigate the ecological services that an invasive species performs and how an abrupt suppression of this invasive population might impact the ecosystem as a whole. We need to consider other restorative measures such as reintroducing native species to fragmented habitats, amongst other ecological interventions49. From a regulatory perspective, there is no tested legislative framework for the release of gene drive organisms; and with regard to our test animal it is currently illegal to breed grey squirrels in the UK. Developing these legislative frameworks alongside gene drive research is important. More importantly, the UK needs to continue to broaden public engagement and see whether the public is receptive to the deployment of gene drive technology in parallel to a financial overview of how much it would cost to apply gene drives reflecting our predicted need for supplementation.

### Summary

HD-ClvR offers an efficient, self-limiting, and controllable gene drive strategy. We show that in the spatial model, complete population suppression is achieved approximately 5 years later than in the randomly mating population model. We then explored how the placement of supplemented animals could impact population suppression. Our results show that spatial dynamics of supplementation placement are not prohibitive to the spread of the gene drive, but that in fact, with an optimised strategy, spread at a rate equal to randomly mating population can be achieved. In our models, we have shown that grey squirrels have a spatial life history which facilitates the spread of a gene drive. Therefore, gene drives could be a valuable tool in the conservation toolbox.

## Methods

We describe our methods and materials in two sections. The first section details the randomly mating population model, and the second the spatial model. For the modelling, we adopted the work of Prowse et al.16 and implemented new features. This model is an individual-based, stochastic, discrete-time model of a randomly mating population. Per individual, the model keeps track of several characteristics such as age, sex, parents, and the state of genetic loci involved in the gene drive. For each offspring, we model the homing and subsequent inheritance of the gene drive. By running this stochastic model several times, we obtain an impression of the possible outcomes. Several life history parameters of an organism are needed to run this model. The parameters we used to model a grey squirrel population can be seen in Table 1.

### Randomly mating model

For the randomly mating model, we added three additional features to the model of Prowse et al.16: cleave-and-rescue, daisyfield, and driving-Y. Cleave-and-rescue and daisyfield were not tested by Prowse et al.16, who only compared homing-based gene drives. We also modelled a driving-Y-cleave-and-rescue gene drive, but the homing-cleave-and-rescue was deemed more promising because the identification of a highly-specific spermatogenesis promoter remains a challenge. In addition to these three new features, we extended the supplementation functionality, beacuse daisyfield-based population suppression requires flexible supplementation.

1. 1.

Cleave-and-rescue. In the model, we keep track of each gRNA-targeted site in cleave-and-rescue target genes and their functionality in each individual. The homing gene drive construct contains the recoded rescue copy of this target gene. All wildtype organisms start with two viable target genes, while gene drive organisms start with one viable target gene and one rescue. In general, after germline Cas-gRNA activity, viable target genes are cleaved and the rescue gene homes along with the gene drive. However, as with any sites targeted by a Cas-gRNA, it is possible that resistance alleles form after non-homolgous end joining and on occasion restore functionality of the target gene. Therefore, we implemented cleave-and-rescue gRNA multiplexing in the model. The probability that cleave-and-rescue target genes go from i to j functional cutting sites ($$P_{ij}$$) is:

\begin{aligned} P_{ij}={\genfrac(){0.0pt}0{i}{i-j}}(P_c(1-P_f))^{i-j}(1-P_c)^j{P_b}^{i-j-1}\,\text {,} \end{aligned}
(1)

where $$P_c$$ is the probability of cutting at a gRNA-targeted site, $$P_f$$ is the probability of functional restoration in case of cutting, and $$P_b$$ is the probability that a block of DNA in between two cutting sites is not removed. This formula consists of four factors: first, we multiply by all permutations of cutting sites, because their order is irrelevant. Second, we multiply by the probability that $$i-j$$ cutting sites are all cut and repaired functionally. Third, we multiply by the probability that j sites remain uncut. Fourth, we multiply by the probability that no blocks of DNA in between cut sites were removed. We use $$P_c = 0.95$$ and $$P_f = 0.667$$ following Prowse et al.16. We estimated $$P_b$$ from our unpublished data of 18 mouse embryonic stem cell lines, each cut simultaneously with Cas9 at two sites spaced 36 bp apart. In 3 out of 18 cases, the block of DNA in between the cut sites was not removed and therefore, we use a $$P_b$$ of 0.2. All left-over probability ($$1-P_{ij}$$) is the probability that a target gene is rendered non-functional. An organism needs to have exactly two copies of the target gene (recoded rescue or original) to be viable. We assumed that there is no embryonic Cas-gRNA activity. After random inheritance of parental alleles, we remove non-viable offspring.

2. 2.

Daisyfield. We implemented daisyfield by tracking the number of daisyfield elements in the genome of each individual. Wildtype organisms start without any daisy elements and the number of daisy elements for gene drive organisms is a parameter in the model. Each daisy element contains both the homing and the cleave-and-rescue gRNAs and in case of gRNA multiplexing, it contains one of each different gRNA. Therefore, during germline Cas-gRNA expression, if no daisy elements are present, both homing and cleave-and-rescue can not occur. We assumed that daisy elements remain complete through every meiosis, so there is no crossing over in the middle of them. Also, we assumed that there is no linkage between daisy elements, that is, they are spaced far apart or located on different chromosomes. During inheritance, each daisy element from the parents has a 0.5 probability of being inherited to the offspring.

3. 3.

Driving-Y. Although the driving-Y is not a part of our final gene drive strategy, we implemented it in the model. The driving-Y gene drive is modelled on the Y-chromosome and skews the sex ratio of offspring towards males. The efficiency of this skew is a parameter in the model and is defined as the probability that offspring of a gene drive animal is male.

4. 4.

Supplementation. We made two changes to the supplementation already implemented by Prowse et al.16. Instead of yearly suplementation of the same amount as the initial gene drive release, we added two parameters to vary supplementation amount and interval. Supplementation amount can be any percentage of the total population size, and supplementation interval can be any decimal number of years as long as they coincide with generations.

### Spatial modelling

For the spatial modelling, we added basic spatial functionality on top of the other additions to the randomly mating model of Prowse et al.16. We model a square, two-dimensional space and assume uniformly distributed resources such as food. The spatial functionality is comprised of four steps: spatial setup, distance-dependent mate allocation, offspring placement, and movement. The spatial setup is only done once at the start of the model and initiates everything necessary for spatial functionality. Mate allocation, offspring placement, and movement occur each generation, and their purpose is to reflect spatial life histories. Distance-dependent mate allocation ensures that squirrels who are close together are more likely to mate than squirrels further apart. Offspring placement demonstrates the location of birth and maternal care of individuals. Movement reflects the migration of individuals whenever overpopulation occurs in an area. With several parameters shown in Table 1, this spatial functionality can be adapted to reflect the spatial life history of many species. Additionally, we have added spatial placement strategies for supplementation.

1. 1.

Spatial setup. The first step in spatial modelling is to determine the size of the area in which the simulations take place. As we use a square two-dimensional space, we need to know the length of the side of this area A. We calculate A using the carrying capacity of the population K, the radius of the home range of the organism r, and the density at carrying capacity D:

\begin{aligned} A = \frac{\sqrt{K\pi r^2}}{\sqrt{D}}\,\text {.} \end{aligned}
(2)

Essentially, this formula transforms a circular home range radius into an area, multiplies it by the number of individuals, transforms it into the length of a square area, and makes it smaller according to the density at carrying capacity. Using this formula, the area is exactly large enough to hold K number of individuals at D density. In this two-dimensional area, we track the x and y coordinates of individuals. Each individual starts at a random location within the area. Where gene drive individuals are placed depends on the placement strategy.

2. 2.

Distance-dependent mate allocation. During the reproduction step of the model, instead of random mate allocation, we use distance-dependent mate allocation. We do this in three steps. First, we calculate the Euclidian distance between all females and males. Second, we use a Gaussian radial basis function to calculate the probability of a male approaching the female to mate ($$P_a$$), depending on the distance s between them:

\begin{aligned} P_a = e^{-(\epsilon s)^2}\,\text {,} \end{aligned}
(3)

where the value $$\epsilon$$ determines the shape of the radial basis function and is calculated from the mating range parameter. In the case of the grey squirrel, the maximum observed mating range was 600 out of 30 observations55. Therefore, we assumed that the probability of a mating range of 600 was 1/30 and from this, we calculate $$\epsilon$$. Third, from the males that do approach the female, we choose a random one as the father of the offspring. In the case that no males approach the female, she does not reproduce.

3. 3.

Offspring placement. We place offspring at the location of the female at the moment of reproduction.

4. 4.

Movement. In grey squirrels, migration is the driving force behind a stable population size58. Therefore, we implemented density-dependent migration and not density-dependent mortality. In the model, we make a distinction between the movement of migrants and residents. Firstly, we determine which individuals migrate and which remain as residents. This distinction is density dependent, that is, the density at the location of an individual determines the probability that they migrate ($$P_m$$):

\begin{aligned} P_m = {\left\{ \begin{array}{ll} 0 &{} d \le D \\ 1-\frac{D}{d} &{} d > D \end{array}\right. }\,\text {,} \end{aligned}
(4)

where the local density d and the density at carrying capacity D are measures of the number of individuals that are in the home range of an individual. Therefore, when the local density is below maximum density, individuals will not migrate. When the local density is higher than the maximum density, the probability of migration is equal to the proportion of individuals that need to migrate to leave the local density at the maximum density. Next, for both the resident and the migrant movement, we choose a direction and a distance to determine a new location. We choose a random direction and a distance from two seperate gamma distributions for residents and emmigrants with shape and scale parameters: $$distance\sim \Gamma (k,\theta )\equiv {\text {Gamma}} (5,r/5 )$$ for residents and $$distance\sim \Gamma (k,\theta )\equiv {\text {Gamma}} (5,3r/5 )$$ for migrants, r being the home range. We use a broader distribution for migrants than for residents as migrants tend to travel greater distances55. The residents move to a random location in a single step. If the new location is out of the boundaries of the spatial space, we pick a new direction and distance. In contrast, migrants move in multiple steps within a certain migrational range to a place where there is space available, that is, where the local density d is lower than the density at carrying capacity D. The migrant searches for a new location in a lazy manner, which means that an animal will first try nearby locations, and incrementally migrate further if necessary. In each step, we pick a random direction and add a new distance from the gamma distribution to the previous distance. If the maximum migration distance is surpassed, the distance is set to zero and the process starts again. To ease the computational burden of this algorithm, we limit the number of steps to 50 and then, we keep the last location regardless of density.

5. 5.

Supplementation. The placement of individuals for supplementation is important. Therefore, we have implemented five placement strategies that can be used, although further exploration of this aspect is interesting. The six placement strategies are: middle of the area, mean of population location, mode of population location, random location at each supplementation, divided into 10 groups and placed at random locations at each supplementation, and divided into 10 groups and placed as a moving front in 10 steps.

## Code availability

Code is available from the Highlanderlab gitlab: https://git.ecdf.ed.ac.uk/HighlanderLab_public/nfaber_squirrel_gd.

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## Acknowledgements

We thank Craig Shuttleworth for all his squirrel-related expertise and advice. C.B.A.W. acknowledges support from BBSRC ISP through BB/P013732/1 and BB/P013759/1. G.G. acknowledges support from the BBSRC to The Roslin Institute (BBS/E/D/30002275) and The University of Edinburgh’s Data-Driven Innovation Chancellor’s fellowship.

## Author information

Authors

### Contributions

C.B.A.W. and G.R.M. conceived the grey squirrel project. N.R.F. and G.R.M. conceived the novel gene drive. N.R.F. conducted the modelling with assistance from R.C.G., I.P., and G.G.. N.R.F wrote the manuscript and all authors reviewed it.

### Corresponding author

Correspondence to Nicky R. Faber.

## Ethics declarations

### Competing interest

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Faber, N.R., McFarlane, G.R., Gaynor, R.C. et al. Novel combination of CRISPR-based gene drives eliminates resistance and localises spread. Sci Rep 11, 3719 (2021). https://doi.org/10.1038/s41598-021-83239-4

• Accepted:

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• DOI: https://doi.org/10.1038/s41598-021-83239-4


text:
Covering spaces, R^n, homomorphism

• May 17th 2011, 11:40 PM
aharonidan
Covering spaces, R^n, homomorphism
Hello all,
Let $A$ be a subset of $R^n$ . let $h : (A, a_0) \rightarrow (Y, y_0)$ . show that if $h$ is extendable to a continuous map of $R^n$ into $Y$, then $h_*$ is the zero homomorphism. (the trivial homomorphism that maps everything to the identity element)

thanks
• May 21st 2011, 07:24 PM
ojones
Hint: Let $[f]\in \pi_1(A,a_0)$. Show that $h\circ f$ is nullhomotopic.

format: plain


text:
Review question

# What can we say if a normal to an ellipse passes through this point? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5237

## Suggestion

Prove that the equation of the normal to the ellipse $x^2/a^2+y^2/b^2=1$ at the point $P\,(a\cos\theta,b\sin\theta)$ is $\frac{ax}{\cos \theta} - \frac{by}{\sin \theta} = a^2-b^2.$

If $x = f(t), y = g(t)$, what is $\dfrac{dy}{dx}$ in terms of $t$?

Or could we use implicit differentiation?

If there is a value of $\theta$ between $0$ and $\pi/2$ such that the normal at $P$ passes through one end of the minor axis, show that the eccentricity of the ellipse must be greater than $1/\sqrt{2}$.

You might use this applet to explore the problem.


text:
[tex-live] Serious (?) problem with Babel and language attributes

Ulrike Fischer news3 at nililand.de
Mon Jun 2 10:08:15 CEST 2014

Am Mon, 2 Jun 2014 09:54:59 +0900 schrieb Norbert Preining:

> I don't know *when* this mess started, but it seems that
> the other language attributes are also broken.

Well I can't read russian so I can't say if the actual words are
okay. But if I compare some of the differences of "normal" and
"ancient" russian I get the results I would expect from the coding,
here for \bibname:

\documentclass[12pt]{article}
\usepackage[T2A]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[russian]{babel}
%\languageattribute{russian}{ancient}
\begin{document}
current value:

\bibname

modern code:  %in \captionsrussian

\cyr\CYRL\cyri\cyrt\cyre\cyrr\cyra\cyrt\cyru\cyrr\cyra

ancient code: % in \captionsrussian at ancient

\cyr\CYRB\cyri\cyrb\cyrl\cyrii\cyro\cyrg\cyrr\cyra\cyrf\cyrii\cyrya
\end{document}

--
Ulrike Fischer
http://www.troubleshooting-tex.de/


text:
MathSciNet bibliographic data MR290088 (44 #7273) 46.35 Kanter, Marek A representation theorem for \$L\sp{p}\$$L\sp{p}$ spaces. Proc. Amer. Math. Soc. 31 1972 472–474. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
### Home > AC > Chapter 6 > Lesson 6.1.1 > Problem6-7

6-7.

Solve the problem below using a Guess and Check table. Note: Be sure to put your work in a safe place, because you will need it for the next lesson.

The perimeter of a triangle is $31$ cm. Sides #$1$ and #$2$ have equal length, while Side #$3$ is one centimeter shorter than twice the length of Side #$1$. How long is each side?

What must you guess and what must you check?

Guess side 1 and check that the perimeter $=31$ What is the relationship between side 1 and the other sides?

Side $1=$ Side $2$

Side $3=2$(Side $1$) $-1$

Use this information to create your Guess and Check chart.

Make sure the sides sum to $31$.


text:
Next: Matrix4 Up: Display objects Previous: DisplayDevice   Contents

Subsections

## Displayable

 Files: Displayable.h, Displayable.C Derived from: \bf ColorUser, \bf Pickable Global instance (if any): none Used in optional component: Part of main VMD code

### Description

The Displayable base class is the parent class for all items which need to be drawn to a DisplayDevice. For example, the set of axes which appear in the corner of the VMD display screen are maintained as a Displayable, and each Molecule is one as well. The Scene (section 24.7) object maintains a list of all the Displayable's that are to be drawn to the screen. Each Displayable consists of the following components:
• A transformation matrix which describes how the graphical object represented by the Displayable is transformed (via translations, rotations, and scaling).
• A display list which contains a set of drawing tokens, in a device-independent format. The display list is used by a DisplayDevice to actually draw the object.
• A parent Displayable, which acts as an owner'; transformations applied to the parent are also applied to its children. If an object does not have a parent, it is a top-level Displayable.
• A set of child Displayable objects.
The Displayable class is the fundamental building block for all items which want to draw something to the screen. Consequently, it is a rather complicated beast.

### Constructors

• Displayable::Displayable(TransMethod, char *, int, Scene *, int)
TransMethod is an enumeration value indicating how the objects transformation matrix should affect the matrix of the DisplayDevice (either it should multiply the matrix (MULT), replace it (LOAD), or it should be ignored altogether (NONE)). The following arguments are the name of the object, the dimension (2 or 3), the Scene with which this object should register, and the initial size of the display list storage (in kilobytes).
• Displayable::Displayable(TransMethod, char *, int, Displayable *, int)
This is the same as the first constructor, but instead of specifying a Scene to register with, a parent Displayable is given. The new Displayable will become a child of the given parent, instead of registering with a Scene.

### Enumerations, lists or character name arrays

TransMethod is a public enumeration which is used to indicate the method of use of the objects transformation matrix. It is one of the following:
• MULT: multiply the DisplayDevice's transformation matrix by this object's matrix when the object is drawn.
• LOAD: replace the DisplayDevice's transformation matrix by this object's matrix when the object is drawn.
• NONE: do not use the object's transformation matrix when rendering.

### Internal data structures

• Displayable *parent - each Displayable has a parent, which may be null. The parent is in charge of collecting children and displaying them as a single entity with separate display lists. Operations which are applied to the parent are also applied ot the children, such as rotations, commands to turn the item on or off, etc. Note that the reverse is NOT true: operations applied to children are not applied to their parent. Thus, if you rotate a parent, all the children will rotate in the same way, but it is possible to rotate each child separately.

• ResizeArray<Displayable *> children - the list of children for this object. This is quite often empty. But other Displayable's have many children, for example MoleculeList has as its children all the Molecule objects.

• void *cmdListBeg - the start of the display list for this object

• void *cmdListPos - the location in the display list where the next drawing command should be placed

• Scene *origScene - each Displayable must register itself with a Scene object, and this item stores a pointer to that Scene

• ResizeArray<Scene *> sceneList - if the Displayable is registered with more than one Scene, they are stored here

• Matrix4 tm - the current transformation matrix. This is a public variable so it may be accessed quickly. The transformation matrix is actually just four different matrices multiplied together; these matrices are centm, rotm, globm, and scalem. The order of multiplication is as follows:
tm = globm * rotm * scalem * centm
When a vector is multiplied by tm, the operations these matrices represent are:
• centm - a centering translation. When tm multiplies a vector, this centering translation is applied first.
• scalem - scales the vector. This is done after the centering translation, so the 'units' for the centering are in the same units as used for the world' coordinates.
• rotm - rotates the vector. This follows the scaling operation (although the two may be done in either order).
• globm - a 'global' translation, which is the last operation done to the vector. Since this is done after the scaling, the units are in post-scaled world coordinates.
Displayable contains many routines which add to or set the values of these matrices.

• char *name - the name of the object

• TransMethod transMethod - the method for use of the transmforamtion matrix. At the beginning of each display list is placed commands to either multiply the current DisplayDevice's trans matrix, or to replace it, or to not modify it at all. This variable indicates how this should be done. In almost all cases it should be MULT

• doCent, doRot, doGlob, doScale - flags indicating whether this object should obey commands to modify the respective transformation components. Routines exist to toggle them on or off

• int *displayObj, *isFixed, *Dim - flags indicating whether to draw the object, whether to keep it fixed (so that it does not respond to any transformation commands), and of what dimension it is (2 or 3). These values are actually stored at the beginning of the display list, and these pointers reference those locations

### Virtual member functions

• virtual void prepare(DisplayDevice *) - this should be supplied by each derived class that needs to do some preparation before drawing. It is called by the Scene right before the scene is to be rendered. By default, it does nothing.

### Method of use

A Displayable must either be registered with a parent (i.e. be a child of some parent), or it must be registered with a Scene (in which case it cannot be the child of any other Displayable). To create a new class derived from Displayable, you need to provide two constructors, one taking a Scene pointer, one a Displayable pointer.

Once created, then in the derived-class-provided prepare routine, the display list of for the Displayable must be set up. If this display list will never change during the program, it can be done just once in the constructor and no prepare is necessary. The display list is created by using DispCmd objects.

The display list for each Displayble must be given to a Scene, regardless of whether it is child or parent or both. It is these display lists which are given to the DisplayDevice to be rendered. The Scene keeps a list of registered Displayable's as well so that it can prepare them for drawing. When a parent is prepared for drawing, it also prepares all its children, so only top-level parents need to be registered with the Scene.

Since this class is derived from Pickable and ColorUser, there are several virtual functions which may need to be supplied in order for the derived Displayable to be picked by the pointers, or to have access to the color database. See the descriptions of these objects for more info.

One note about the display lists: the reason they are done in the current scheme is that the CAVE requires the use of shared memory to hold the information for the rendering processes to draw, while a separate update process keeps track of updating the information in shared memory. Since the shared memory cannot have pointers to non-shared memory locations, the display lists are designed to ONLY hold integer and floating-point data. When drawing in the CAVE, a special CaveScene supplies shared memory blocks for the Displayable's to use for their display lists.


text:
230. Kth Smallest Element in a BST

Given a binary search tree, write a function kthSmallest to find the kth smallest element in it.

Note:
You may assume k is always valid, 1 ? k ? BST's total elements.

What if the BST is modified (insert/delete operations) often and you need to find the kth smallest frequently? How would you optimize the kthSmallest routine?

Credits:
Special thanks to @ts for adding this problem and creating all test cases.

Seen this question in a real interview before?
When did you encounter this question?
Which company?


text:
# Eigenvalues of Matrix Sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum?

What about the special case when they are Hermitian and positive-definite?

I am investigating this with regard to finding the normalized graph cut under general convex constraints.

Any pointers will be very helpful.

Thanks. B.

-

The problem of describing the possible eigenvalues of the sum of two hermitian matrices in terms of the spectra of the summands leads into deep waters. The most complete description was conjectured by Horn, and has now been proved by work of Knutson and Tao (and others?) - for a good discussion, see the Notices AMS article by those two authors

Depending on what you want, there should be simpler results giving estimates on the eigenvalues of the sum. A book like Bhatia's Matrix Analysis might have some helpful material.

-

A simple estimate which is often useful is that, if A and B are Hermitian matrices with eigenvalues a_1 > a_2 > ... > a_n and b_1 > b_2 > ... > b_n and the eigenvalues of the sum are c_1 > c_2 > ... > c_n, then

c_{i+j-1} ≤ a_i + b_j and c_{n-i-j} ≥ a_{n-i} + b_{n-j}.

The above conditions are necessary but not sufficient for A+B=C to have a solution; see the Knutson-Tao article if you want sufficient conditions.

If you do not impose that A and B are Hermitian then there are very few restrictions besides the trace being equal. More specifically, the 3n-tuples (a_1, ..., a_n, b_1, ..., b_n, c_1,...,c_n) which occur as eigenvalues of (A,B,C) with A+B=C are dense in the hyperplane \sum a_i + \sum b_i = \sum c_i.

-


text:
# An observer is moving towards a stationary source of frequency 320 Hz with a velocity of 10 m/s . The speed of sound in air is 343 m/s . The new frequency heard by the observer is

$(a)\;310 \;Hz\qquad(b)\;336 \;Hz\qquad(c)\;340 \;Hz\qquad(d)\;329 \;Hz$

Answer : (d) $\;329 \;Hz$
Explanation :
New frequency $\;f^{|} =(\large\frac{v_{0}+v}{v})\;f$
$v_{0}\;$ velocity of observer
$v\;$ velocity of sound
By Doppler's effect
therefore , $\;f^{|}=(\large\frac{10+343}{343})\times320$
$=\large\frac{353}{343}\times320$
$=329\;Hz \;.$


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Single image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction

Image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction aims to recover detailed information from low-resolution images and reconstruct them into high-resolution images. Due to the limited amount of data and information retrieved from low-resolution images, it is difficult to restore clear, artifact-free images, while still preserving enough structure of the image such as the texture. This paper presents a new single image super-resolution method which is based on adaptive fractional-order gradient interpolation and reconstruction. The interpolated image gradient via optimal fractional-order gradient is first constructed according to the image similarity and afterwards the minimum energy function is employed to reconstruct the final high-resolution image. Fractional-order gradient based interpolation methods provide an additional degree of freedom which helps optimize the implementation quality due to the fact that an extra free parameter $\alpha$-order is being used. The proposed method is able to produce a rich texture detail while still being able to maintain structural similarity even under large zoom conditions. Experimental results show that the proposed method performs better than current single image super-resolution techniques.
Submitted 18 Mar 2017 to Computer Vision and Pattern Recognition [cs.CV]
Published 21 Mar 2017
Author comments: 24 pages, 13 figures, it is to appear in ISA Transactions
http://arxiv.org/abs/1703.06260
http://arxiv.org/pdf/1703.06260.pdf


text:
# Kronecker product of two block diagonal matrices

Consider two block diagonal matrices $$\mathbf A = \begin{bmatrix} A_{1} & O\\ 0 & A_{2} \end{bmatrix} \mathbf B = \begin{bmatrix} B_{1} & O\\ 0 & B_{2} \end{bmatrix}$$ $$A_1, A_2, B_1, B_2$$ have same dimension(may not necessary). Could we show after some row and column permutations of $$\mathbf A\otimes \mathbf B$$, it will be same as $$diag\{A_1\otimes B_1, A_2\otimes B_1, A_1\otimes B_2, A_2\otimes B_2\}$$?

$$A\otimes B=\begin{bmatrix}A_1\otimes B&O\\O&A_2\otimes B\end{bmatrix}$$and $$A_i\otimes B=\begin{bmatrix}A_i\otimes B_1&O\\O&A_i\otimes B_2\end{bmatrix}$$which are direct consequences of the Kronecker product.


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# The 3rd square number and 9th square number added together

The 3rd square number and 9th square number added together

## This Post Has 4 Comments

1. Expert says:

X=14 hope it !

2. Expert says:

yes the data was not prsented

step-by-step explanation:

3. Expert says:

y= -3x is the required function.

step-by-step explanation:

we have been given a table:

x             y

0             0

1             -3

3             -9

-2             6

we can see that the values of y is -3 times the value of x

hence, the function would be: y= -3x.

put the values of x will give the same y as in the table that means function is satisfying the given table.

4. Expert says:

i believe the answer is letter c) 8

step-by-step explanation:


text:
# Insights The Complexity of Modern Science - Comments

1. Jan 10, 2016

### Staff: Mentor

2. Jan 10, 2016

### phinds

I think the biggest problem is that people find it SO much easier to watch pop science on TV than to do any actual study of science, and you know how those shows get so much wrong. I think they do sometimes inspire young people to study but overall I'm not sure but what they do more harm than good and they certainly give those adults who are not likely to further pursue actual science a very poor view of actual science of the kind you talk about. The producers of the TV shows can't be blamed for this any more than McDonalds can be blamed for serving tasty junk food. People sell what other people buy and there are lots of buyers for junk food and junk science, especially since they LOOK so tasty, what with all the nifty graphics and tomato sauce and all.

The first thing we COULD do (and won't) would be to insist that people who teach science, at any level but particularly below the college level, be required to have at least some idea what they are talking about. Teachers below the high school level in particular have no idea, generally, what science is really all about.

3. Jan 10, 2016

### Buzz Bloom

Hi mfb:

This is a great topic to discuss. I hope that some useful ideas will emerge, but I confess I am very pessimistic. NOVA is an excellent source for informing the public about science, but I am guessing that for each viewer of NOVA there are more than a hundred viewers of FOX NEWS.

Regards,
Buzz

4. Jan 10, 2016

### Isaac0427

So first of all, I find this insight to be very interesting. However, as a crackpot by some definitions, I must disagree with you and phinds on one point. I do study actual science and calculus with that science. I learned on this forum to stay away from popsci, but still, I come up with "theories". No, as a middle schooler I do not believe these theories will go anywhere. What I do is I make a theory based on math and what I have learned so far from other sources, and then I look for what I did wrong. Sometimes I need to ask a professional about what I did wrong. In fact, I learn better from actually thinking about it and then finding a problem in my thinking and/or math. I do agree that some people do need to learn real science before making a statement like "special relativity is wrong" but some people (like me) learn from thinking, challenging the theories and finding out why I'm wrong. Just something to think about. Overall, however, a great insight.

5. Jan 10, 2016

### Isaac0427

BTW, for some reason part of the quote is not in the quote part of my post.

6. Jan 10, 2016

### Staff: Mentor

That is perfectly fine, as long as challenging the theories is based on actual knowledge of those theories.

I don't see what would be wrong with the quote.

7. Jan 10, 2016

### Greg Bernhardt

As we've seen in the thread: https://www.physicsforums.com/threa...is-not-weird-unless-presented-as-such.850860/ most concepts can't be genuinely described using word language. Susskind also mentioned this in an interview I watched where the interviewer asked some quantum mechanics question and Susskind basically said he couldn't answer it with words, only in math. Not long ago I debated a man who claimed to understand the concepts of the Big Bang very well and didn't need to know the math. I promptly left the conversation. Fact is that the general public certainty can't handle or have the patience for reading research papers so pop sci news agencies water the research down into cookie cutter pieces with catchy headlines which in the end only vaguely resemble what it really means. This is good for the public's imagination but doesn't do justice for how complex their research is.

8. Jan 10, 2016

### Isaac0427

On the actual insights page it turned out differently.

9. Jan 10, 2016

### phinds

I do not find that to be at all in disagree w/ my statements or beliefs and in fact I think it's a fine way to forge ahead in science for someone your age. It keeps you interested but as long as you stay grounded in the knowledge that your theories are based, at this level of your development, more in ignorance than in knowledge, then you are using your process as a learning tool and that's great.

10. Jan 10, 2016

### Isaac0427

Ok, I just picked up an implication (that may or may not have been there) in both your post and the insight that these theories all come from ignorance and stubbornness to accept mainstream theories, and not from an attempt to learn more about these theories by challenging them (and I am not saying people don't create theories out of ignorance and stubbornness to accept mainstream theories, because they do). Most of what I have learned about relativity has been from challenging it and finding a list of reasons why my challenge was incorrect.

11. Jan 10, 2016

### Isaac0427

And we all know how great Fox News is at explaining/acknowledging proven scientific fact.

12. Jan 10, 2016

### Hornbein

A substantial majority of

People believe what they want to believe.
They disregard information/misinformation that contradicts what they believe.
They absorb information/misinformation that reinforces what they believe.
What they believe is usually self-flattering.
They choose a person or organization in which to place their trust. They believe all information/misinformation that issues from this source.

Corollary:
Sources of flattering information are more likely to earn such trust.

To believe that you easily have come up with a simple insight that a thousands of hard-working geniuses have missed is very self-flattering.
To believe that you are ignorant and incompetent in some area is anti-self-flattering.

Ergo, crackpottery is a basic human tendency.

You can coerce a student into learning, but each student is free disbelieve in or subsequently forget what was learned. Once no longer a student, untrusted sources no longer have any power over their learning or beliefs.

-----

I think the educational system is obsolete, ineffective, inflicts pain on the student, and needs a complete overhaul. It has hardly changed since ancient times. Can you blame people for hating it? There is room for improvement.

13. Jan 10, 2016

### JakeBrodskyPE

When I was a child, getting in to ham radio, I was amazed at what the retail for a radio was versus what the parts cost. What I didn't realize is that the radio had a lot of marketing and engineering expenses that needed to be recouped.

This may sound crass, but we need MORE marketing. Science invokes a sense of wonder in its practitioners that is only rarely ever described well. Carl Sagan did that. We don't realize how good that was, until watching the remake of Cosmos. Even his masterful and charismatic protege Neil deGrasse Tyson is only a pale reflection of the kind of science marketing that Carl Sagan did. Don't get me wrong, Tyson is brilliant; but Sagan's presentation was the very embodiment of artistry.

If you can't be as brilliant as Sagan, you can make up for it with quantity. That's where we need to go. Relying upon public enthusiasm will get you only so far. The biggest successes in business and history in general were accompanied by masterful marketing.

We need more.

14. Jan 10, 2016

### JorisL

I think we need another Feynman. However I still have to watch/read some work by Sagan to fully appreciate his efforts.
However I find it hard to believe anyone can match let alone surpass Feynman's QED.

He not only simplifies the theory but also mentions why it is difficult to do cutting edge science.
The latter is what lacks in most popular accounts but also in education. Which I understand since most teachers haven't gotten close to the "cutting edge" over here.
But that last point is for another discussion.

One could argue this is the way to study science and especially physics.
However it would take ages while the time we have (in high school) is severely limited.

Often people also have the idea that since physics is called exact science the approximate models we have are useless.
Also the interplay theory-experiment in the scientific method is not a one-way street. It's a complicated interplay of ideas and results. (A dance if you like metaphors)

I'd say keep on the good work and remain critical.

15. Jan 10, 2016

### Amrator

Awesome Insight, mfb. I'm going to show it to one of my physics professors if you don't mind.

16. Jan 10, 2016

### Monsterboy

One can't expect science T.V shows to tell all the intricate details ,if that is the case the public will stop watching because everyone in the public aren't scientists and most of the things will start going over their head and will scare the public away from science ,the shows are only aimed to give a brief idea to the lay public (and to motivate them to pursue science) who are completely or a large extent, disconnected from modern science, i think the problem mentioned in this article can be overcome if the science documentaries are screened for errors by the actual scientists who work on the topic (either retired or active scientists) and they should make it clear to the public that they are simplifying very complicated science in order for them to understand and that real knowledge only comes with hard study.

17. Jan 10, 2016

### rbelli1

Unfortunately you are using theory in the layman's sense. What you are coming up with are at best hypotheses. It is good to think up new ideas but calling them theories without rigorous testing does an injustice to science.

BoB

18. Jan 10, 2016

### mrspeedybob

I think Richard Feynman did a great job of accurately explaining concepts to lay people without loosing them in the math. That's a very difficult thing to do, but that does not mean that it can't be done, or that it should not be attempted. It should.

A good recent example was "The Space Doctor's Big Idea"
http://www.newyorker.com/tech/elements/the-space-doctors-big-idea-einstein-general-relativity

19. Jan 10, 2016

### Staff: Mentor

Non-Science is a societal problem. It forwards the interest of the idiot fringe and of FUD mongers.

Idiot fringe:
In the US, there are so-called 'anti-vaxxers'. They are people who are vehemently against vaccinating children. Therein lies the issue with junk science in general. These folks made decisions based on pulp news sources and magazines that impact not only their immediate family, but others in the community.
Measles was essentially unreported by the US CDC for years, there were very, very few cases. Not anymore. Measles can result in death and lifelong medical problems. Example SSPE: https://www.nlm.nih.gov/medlineplus/ency/article/001419.htm

The same thing happens with a variety of issues that are subjected to *FUD attacks, in order to further the economic positions of very powerful companies.

Example - The US tobacco lobby's very effective attacks against anti-tobacco legislation. The stupidity did not abate until Science made discoveries that were so very plain, smack-you-in-the-face, that even the pop-science idiots could not get them wrong. FUD stills lives on with the climate "debate" - as newspapers call it. It is not a debate. You cannot legislate scientific observations and results.

*FUD - fear, uncertainty, and doubt.

20. Jan 10, 2016

### jerromyjon

I think there will always be two camps, those who try to strictly adhere to the rules of science which relies mainly on the math which is enough to communicate. and the laypeople who know the world from their own perspectives, which are numerous, I'm certain. The perspective they have built in their mind to comprehend the reality they are familiar with typically fails and so they grapple for a mechanism to leave their sole understanding intact. Many don't have the neural plasticity to drop lifelong thought processes and start over, let alone the determinism to seek a deeper, more accurate understanding, and the mental ability to assemble intricate models. So pop-sci caters to their inadequacies with "fantastic" phrases like "wave-particle duality" and life goes on.

21. Jan 10, 2016

### epistememe

I agree with most of you assessment. Now how about offering solutions.

22. Jan 10, 2016

### toumaza

the complexity of any science field happens at the level of our mind. same as one could say mathss is easier than any other science subject. also some people remind and understand better when you explain the subject to the minor details to them why others (which i am part of ) prefer to understand it on its own than waiting for lecturers to give them the explanations.
just trying to say complexity is at the level of individuals

23. Jan 10, 2016

### Isaac0427

Yes, it is more of a hypothesis than a theory.

24. Jan 10, 2016

### Jeff Rosenbury

Physics Forum is dedicated to advancing the standard model. This discussion doesn't seem to be part of that.

As I see it, every part of the standard model was once part of fringe science at least to the extent of not being accepted/proven. While some fringe science is clearly wrong (or not even wrong), some tiny fraction of it will someday work its way into the standard model.

A discussion like this opens up the question of what is right or wrong about fringe science. That's a problem for a website that tries to avoid fringe science.

I do think anyone attacking the standard needs to have a good understanding of it before declaring it wrong. This forum helps to provide that understanding. Hopefully the work we do here will one day allow us to replace the standard model with one closer to the truth. (Not that the standard model is off by a lot, but there are still outstanding problems.)

Historically liberal arts have always ruled science. That is because a good story has always received more funding than being right. (NSF budget: $7 billion (2012), Disney:$11 billion)

25. Jan 10, 2016

### Isaac0427

Is it? If it was, we would not have a forum for relativity, biology, chemistry, homework help, etc.