Stability and response of polygenic traits to stabilizing selection and mutation

When polygenic traits are under stabilizing selection, many different combinations of alleles allow close adaptation to the optimum. If alleles have equal effects, all combinations that result in the same deviation from the optimum are equivalent. Furthermore, the genetic variance that is maintained by mutation-selection balance is $2 \mu/S$ per locus, where $\mu$ is the mutation rate and $S$ the strength of stabilizing selection. In reality, alleles vary in their effects, making the fitness landscape asymmetric, and complicating analysis of the equilibria. We show that that the resulting genetic variance depends on the fraction of alleles near fixation, which contribute by $2 \mu/S$, and on the total mutational effects of alleles that are at intermediate frequency. The interplay between stabilizing selection and mutation leads to a sharp transition: alleles with effects smaller than a threshold value of $2\sqrt{\mu / S}$ remain polymorphic, whereas those with larger effects are fixed. The genetic load in equilibrium is less than for traits of equal effects, and the fitness equilibria are more similar. We find that if the optimum is displaced, alleles with effects close to the threshold value sweep first, and their rate of increase is bounded by $\sqrt{\mu S}$. Long term response leads in general to well-adapted traits, unlike the case of equal effects that often end up at a sub-optimal fitness peak. However, the particular peaks to which the populations converge are extremely sensitive to the initial states, and to the speed of the shift of the optimum trait value.Comment: Accepted in Genetic

Wolbachia versus dengue: Evolutionary forecasts.

A novel form of biological control is being applied to the dengue virus. The agent is the maternally transmitted bacterium Wolbachia, naturally absent from the main dengue vector, the mosquito Aedes aegypti. Three Wolbachia-based control strategies have been proposed. One is suppression of mosquito populations by large-scale releases of males incompatible with native females; this intervention requires ongoing releases. The other interventions transform wild mosquito populations with Wolbachia that spread via the frequency-dependent fitness advantage of Wolbachia-infected females; those interventions potentially require just a single, local release for area-wide disease control. One of these latter strategies uses Wolbachia that shortens mosquito life, indirectly preventing viral maturation/transmission. The other strategy uses Wolbachia that block viral transmission. All interventions can be undermined by viral, bacterial or mosquito evolution; viral virulence in humans may also evolve. We examine existing theory, experiments and comparative evidence to motivate predictions about evolutionary outcomes. (i) The life-shortening strategy seems the most likely to be thwarted by evolution. (ii) Mosquito suppression has a reasonable chance of working locally, at least in the short term, but long-term success over large areas is challenging. (iii) Dengue blocking faces strong selection for viral resistance but may well persist indefinitely at some level. Virulence evolution is not mathematically predictable, but comparative data provide no precedent for Wolbachia increasing dengue virulence. On balance, our analysis suggests that the considerable possible benefits of these technologies outweigh the known negatives, but the actual risk is largely unknown

Facilitating Wolbachia introductions into mosquito populations through insecticide-resistance selection.

Wolbachia infections are being introduced into mosquito vectors of human diseases following the discovery that they can block transmission of disease agents. This requires mosquitoes infected with the disease-blocking Wolbachia to successfully invade populations lacking the infection. While this process is facilitated by features of Wolbachia, particularly their ability to cause cytoplasmic incompatibility, blocking Wolbachia may produce deleterious effects, such as reduced host viability or fecundity, that inhibit successful local introductions and subsequent spatial spread. Here, we outline an approach to facilitate the introduction and spread of Wolbachia infections by coupling Wolbachia introduction to resistance to specific classes of insecticides. The approach takes advantage of very high maternal transmission fidelity of Wolbachia infections in mosquitoes, complete incompatibility between infected males and uninfected females, the widespread occurrence of insecticide resistance, and the widespread use of chemical control in disease-endemic countries. This approach is easily integrated into many existing control strategies, provides population suppression during release and might be used to introduce Wolbachia infections even with high and seasonally dependent deleterious effects, such as the wMelPop infection introduced into Aedes aegypti for dengue control. However, possible benefits will need to be weighed against concerns associated with the introduction of resistance alleles

A one locus, biased mutation model and its equivalence to an unbiased model

Experimental data suggests that for some continuously varying characters under stabilising selection, mutation may cause a mean change in the value of the character. A one locus, mathematical model of a continuously varying biological character with this property of biased mutation is investigated. Via a mathematical transformation, the equilibrium equation describing a large population of individuals is reduced to the equilibrium equation describing a mutationally unbiased problem. Knowledge of an unbiased problem is thus su¢ cient to determine all equilibrium properties of the corresponding biased problem. In the biased mutation problem, the dependence of the mean equilibrium value of the character, as a function of the mutational bias, is non monotonic and remains small, for all levels of mutational bias. The analysis presented in this work sheds new light on Turelli's House of Cards approximation

Commentary: Fisher's infinitesimal model: A story for the ages.

Mendel (1866) suggested that if many heritable "factors" contribute to a trait, near-continuous variation could result. Fisher (1918) clarified the connection between Mendelian inheritance and continuous trait variation by assuming many loci, each with small effect, and by informally invoking the central limit theorem. Barton et al. (2017) rigorously analyze the approach to a multivariate Gaussian distribution of the genetic effects for descendants of parents who may be related. This commentary distinguishes three nested approximations, referred to as "infinitesimal genetics," "Gaussian descendants" and "Gaussian population," each plausibly called "the infinitesimal model." The first and most basic is Fisher's "infinitesimal" approximation of the underlying genetics - namely, many loci, each making a small contribution to the total variance. As Barton et al. (2017) show, in the limit as the number of loci increases (with enough additivity), the distribution of genotypic values for descendants approaches a multivariate Gaussian, whose variance-covariance structure depends only on the relatedness, not the phenotypes, of the parents (or whether their population experiences selection or other processes such as mutation and migration). Barton et al. (2017) call this rigorously defensible "Gaussian descendants" approximation "the infinitesimal model." However, it is widely assumed that Fisher's genetic assumptions yield another Gaussian approximation, in which the distribution of breeding values in a population follows a Gaussian - even if the population is subject to non-Gaussian selection. This third "Gaussian population" approximation, is also described as the "infinitesimal model." Unlike the "Gaussian descendants" approximation, this third approximation cannot be rigorously justified, except in a weak-selection limit, even for a purely additive model. Nevertheless, it underlies the two most widely used descriptions of selection-induced changes in trait means and genetic variances, the "breeder's equation" and the "Bulmer effect." Future generations may understand why the "infinitesimal model" provides such useful approximations in the face of epistasis, linkage, linkage disequilibrium and strong selection

Polisemia di un gesto: l'emittere hastam dei duces e dei feziali

L’articolo presenta una riflessione intorno a un testo del commentario di Servio all’Eneide (Serv. Aen. 9.52), nel quale si descrivono taluni aspetti della procedura feziale per la dichiarazione di guerra. Interessa, in particolare, il gesto della emissio hastae in territorio nemico, quale atto conclusivo della fase ‘dialettica’ e inizio della fase di scontro armato. Servio cita un passaggio di Varrone (Varr. Logistorici frg. 2 Semi), nel quale viene ricordato l’identico gesto compiuto dal comandante militare al momento di entrare con l’esercito in «agrum hostilem», allo scopo di fissare un luogo per l’accampamento. Nell’ambito della dottrina ipercritica, tale frammento è stato letto come indizio di una scarsa conoscenza e diffusione dello ius fetiale (e delle sue pratiche) al tempo di Varrone, e dunque a conferma della teoria dello ius fetiale quale prodotto di una ricostruzione arcaizzante. Il presente lavoro tende, invece, a porre in evidenza come nel testo serviano siano sovrapposti due atti distinti e autonomi, accomunati, tuttavia, dall’identità materiale del gesto posto in essere. Una attenta lettura del documento di Servio, infatti, consente di cogliere come, pur nell’identità comportamentale, altro sia l’atto del condottiero, altro quello del sacerdote feziale: gesti identici, ma con funzioni e significati diversi

'Res incorporales' e 'beni immateriali': categorie affini, ma non congruenti

Il presente lavoro si propone un confronto tra la categoria romana delle res incorporales e quella moderna dei beni immateriali. Le due categorie non sono congruenti: infatti, la categoria romana riguarda i diritti (elementi del patrimonio) e serve a sistemare tutto il diritto privato entro la tre classi personae, res, actiones; invece, la categoria moderna riguarda gli oggetti del diritto, in particolare i beni immateriali. Tuttavia, tra Ottocento e Novecento, la scienza giuridica ha ragionato sulla categoria romana, ipotizzando che essa potesse venire impiegata per collocare i beni immateriali – in particolare i prodotti dell’attività intellettuale – all’interno dei ‘sistemi’ codicistici che in quegli anni si venivano elaborando. Si sono profilati in dottrina due atteggiamenti in proposito. Da un lato coloro che hanno ‘esteso’ la categoria romana dai ‘diritti’ agli ‘oggetti di diritto’; dall’altra, coloro che hanno visto una soluzione di continuità tra antico e moderno: essi hanno mantenuto l’espressione ‘cose incorporali’, ma ne hanno rinnovato il contenuto. In questo graduale rinnovamento sembra avere giocato un ruolo significativo Windscheid. Tra le due categorie si profila così, sul piano della speculazione scientifica, un nesso che si può definire di affinità, nel senso etimologico del termine (< lat. ‘ad-finis’)

Deploying dengue-suppressing Wolbachia : Robust models predict slow but effective spatial spread in Aedes aegypti.

A novel strategy for controlling the spread of arboviral diseases such as dengue, Zika and chikungunya is to transform mosquito populations with virus-suppressing Wolbachia. In general, Wolbachia transinfected into mosquitoes induce fitness costs through lower viability or fecundity. These maternally inherited bacteria also produce a frequency-dependent advantage for infected females by inducing cytoplasmic incompatibility (CI), which kills the embryos produced by uninfected females mated to infected males. These competing effects, a frequency-dependent advantage and frequency-independent costs, produce bistable Wolbachia frequency dynamics. Above a threshold frequency, denoted pˆ, CI drives fitness-decreasing Wolbachia transinfections through local populations; but below pˆ, infection frequencies tend to decline to zero. If pˆ is not too high, CI also drives spatial spread once infections become established over sufficiently large areas. We illustrate how simple models provide testable predictions concerning the spatial and temporal dynamics of Wolbachia introductions, focusing on rate of spatial spread, the shape of spreading waves, and the conditions for initiating spread from local introductions. First, we consider the robustness of diffusion-based predictions to incorporating two important features of wMel-Aedes aegypti biology that may be inconsistent with the diffusion approximations, namely fast local dynamics induced by complete CI (i.e., all embryos produced from incompatible crosses die) and long-tailed, non-Gaussian dispersal. With complete CI, our numerical analyses show that long-tailed dispersal changes wave-width predictions only slightly; but it can significantly reduce wave speed relative to the diffusion prediction; it also allows smaller local introductions to initiate spatial spread. Second, we use approximations for pˆ and dispersal distances to predict the outcome of 2013 releases of wMel-infected Aedes aegypti in Cairns, Australia, Third, we describe new data from Ae. aegypti populations near Cairns, Australia that demonstrate long-distance dispersal and provide an approximate lower bound on pˆ for wMel in northeastern Australia. Finally, we apply our analyses to produce operational guidelines for efficient transformation of vector populations over large areas. We demonstrate that even very slow spatial spread, on the order of 10-20 m/month (as predicted), can produce area-wide population transformation within a few years following initial releases covering about 20-30% of the target area

Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of Allee effects

Unlike unconditionally advantageous “Fisherian” variants that tend to spread throughout a species range once introduced anywhere, “bistable” variants, such as chromosome translocations, have two alternative stable frequencies, absence and (near) fixation. Analogous to populations with Allee effects, bistable variants tend to increase locally only once they become sufficiently common, and their spread depends on their rate of increase averaged over all frequencies. Several proposed manipulations of insect populations, such as using Wolbachia or “engineered underdominance” to suppress vector-borne diseases, produce bistable rather than Fisherian dynamics. We synthesize and extend theoretical analyses concerning three features of their spatial behavior: rate of spread, conditions to initiate spread from a localized introduction, and wave stopping caused by variation in population densities or dispersal rates. Unlike Fisherian variants, bistable variants tend to spread spatially only for particular parameter combinations and initial conditions. Wave initiation requires introduction over an extended region, while subsequent spatial spread is slower than for Fisherian waves and can easily be halted by local spatial inhomogeneities. We present several new results, including robust sufficient conditions to initiate (and stop) spread, using a one-parameter cubic approximation applicable to several models. The results have both basic and applied implications

Deploying dengue-suppressing Wolbachia: Robust models predict slow but effective spatial spread in Aedes aegypti

A novel strategy for controlling the spread of arboviral diseases such as dengue, Zika and chikungunya is to transform mosquito populations with virus-suppressing Wolbachia. In general, Wolbachia transinfected into mosquitoes induce fitness costs through lower viability or fecundity. These maternally inherited bacteria also produce a frequency-dependent advantage for infected females by inducing cytoplasmic incompatibility (CI), which kills the embryos produced by uninfected females mated to infected males. These competing effects, a frequency-dependent advantage and frequency-independent costs, produce bistable Wolbachia frequency dynamics. Above a threshold frequency, denoted pˆ, CI drives fitness-decreasing Wolbachia transinfections through local populations; but below pˆ, infection frequencies tend to decline to zero. If pˆ is not too high, CI also drives spatial spread once infections become established over sufficiently large areas. We illustrate how simple models provide testable predictions concerning the spatial and temporal dynamics of Wolbachia introductions, focusing on rate of spatial spread, the shape of spreading waves, and the conditions for initiating spread from local introductions. First, we consider the robustness of diffusion-based predictions to incorporating two important features of wMel-Aedes aegypti biology that may be inconsistent with the diffusion approximations, namely fast local dynamics induced by complete CI (i.e., all embryos produced from incompatible crosses die) and long-tailed, non-Gaussian dispersal. With complete CI, our numerical analyses show that long-tailed dispersal changes wave-width predictions only slightly; but it can significantly reduce wave speed relative to the diffusion prediction; it also allows smaller local introductions to initiate spatial spread. Second, we use approximations for pˆ and dispersal distances to predict the outcome of 2013 releases of wMel-infected Aedes aegypti in Cairns, Australia, Third, we describe new data from Ae. aegypti populations near Cairns, Australia that demonstrate long-distance dispersal and provide an approximate lower bound on pˆ for wMel in northeastern Australia. Finally, we apply our analyses to produce operational guidelines for efficient transformation of vector populations over large areas. We demonstrate that even very slow spatial spread, on the order of 10-20 m/month (as predicted), can produce area-wide population transformation within a few years following initial releases covering about 20-30% of the target area