Coalescence and sampling distributions for Feller diffusions.

Consider the diffusion process defined by the forward equation ut(t,x)=12{xu(t,x)}xx-α{xu(t,x)}x for t,x≥0 and -∞<α<∞, with an initial condition u(0,x)=δ(x-x0). This equation was introduced and solved by Feller to model the growth of a population of independently reproducing individuals. We explore important coalescent processes related to Feller's solution. For any α and x0>0 we calculate the distribution of the random variable An(s;t), defined as the finite number of ancestors at a time s in the past of a sample of size n taken from the infinite population of a Feller diffusion at a time t since its initiation. In a subcritical diffusion we find the distribution of population and sample coalescent trees from time t back, conditional on non-extinction as t→∞. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.

An estimator for the recombination rate from a continuously observed diffusion of haplotype frequencies.

Recombination is a fundamental evolutionary force, but it is difficult to quantify because the effect of a recombination event on patterns of variation in a sample of genetic data can be hard to discern. Estimators for the recombination rate, which are usually based on the idea of integrating over the unobserved possible evolutionary histories of a sample, can therefore be noisy. Here we consider a related question: how would an estimator behave if the evolutionary history actually was observed? This would offer an upper bound on the performance of estimators used in practice. In this paper we derive an expression for the maximum likelihood estimator for the recombination rate based on a continuously observed, multi-locus, Wright-Fisher diffusion of haplotype frequencies, complementing existing work for an estimator of selection. We show that, contrary to selection, the estimator has unusual properties because the observed information matrix can explode in finite time whereupon the recombination parameter is learned without error. We also show that the recombination estimator is robust to the presence of selection in the sense that incorporating selection into the model leaves the estimator unchanged. We study the properties of the estimator by simulation and show that its distribution can be quite sensitive to the underlying mutation rates.

Soluble ACE2 Is Filtered into the Urine.

Background: ACE2 is a key enzyme in the renin-angiotensin system (RAS) capable of balancing the RAS by metabolizing angiotensin II (AngII). First described in cardiac tissue, abundance of ACE2 is highest in the kidney, and it is also expressed in several extrarenal tissues. Previously, we reported an association between enhanced susceptibility to hypertension and elevated renal AngII levels in global ACE2-knockout mice. Methods: To examine the effect of ACE2 expressed in the kidney, relative to extrarenal expression, on the development of hypertension, we used a kidney crosstransplantation strategy with ACE2-KO and WT mice. In this model, both native kidneys are removed and renal function is provided entirely by the transplanted kidney, such that four experimental groups with restricted ACE2 expression are generated: WT→WT (WT), KO→WT (KidneyKO), WT→KO (SystemicKO), and KO→KO (TotalKO). Additionally, we used nanoscale mass spectrometry-based proteomics to identify ACE2 fragments in early glomerular filtrate of mice. Results: Although significant differences in BP were not detected, a major finding of our study is that shed or soluble ACE2 (sACE2) was present in urine of KidneyKO mice that lack renal ACE2 expression. Detection of sACE2 in the urine of KidneyKO mice during AngII-mediated hypertension suggests that sACE2 originating from extrarenal tissues can reach the kidney and be excreted in urine. To confirm glomerular filtration of ACE2, we used micropuncture and nanoscale proteomics to detect peptides derived from ACE2 in the Bowman's space. Conclusions: Our findings suggest that both systemic and renal tissues may contribute to sACE2 in urine, identifying the kidney as a major site for ACE2 actions. Moreover, filtration of sACE2 into the lumen of the nephron may contribute to the pathophysiology of kidney diseases characterized by disruption of the glomerular filtration barrier.

The transition distribution of a sample from a Wright-Fisher diffusion with general small mutation rates.

The transition distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree of the sample up to the most recent common ancestor with additional mutations occurring on the lineage from the most recent common ancestor to the time origin if complete coalescence occurs before the origin. The sampling distribution leads to an approximation for the transition density in the diffusion with small mutation rates. This new solution has interest because the transition density in a Wright-Fisher diffusion with general mutation rates is not known.

The stationary distribution of a sample from the Wright-Fisher diffusion model with general small mutation rates.

The stationary distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to the first order in the rates. The sample probabilities characterize an approximation for the stationary distribution from the Wright-Fisher diffusion. The approach is different from Burden and Tang (Theor Popul Biol 112:22-32, 2016; Theor Popul Biol 113:23-33, 2017) who use a probability flux argument to obtain the same results from a forward diffusion generator equation. The solution has interest because the solution is not known when rates are not small. An analogous solution is found for the configuration of alleles in a general exchangeable binary coalescent tree. In particular an explicit solution is found for a pure birth process tree when individuals reproduce at rate [Formula: see text].

Stationary distribution of a 2-island 2-allele Wright-Fisher diffusion model with slow mutation and migration rates.

The stationary distribution of the diffusion limit of the 2-island, 2-allele Wright-Fisher with small but otherwise arbitrary mutation and migration rates is investigated. Following a method developed by Burden and Tang (2016, 2017) for approximating the forward Kolmogorov equation, the stationary distribution is obtained to leading order as a set of line densities on the edges of the sample space, corresponding to states for which one island is bi-allelic and the other island is non-segregating, and a set of point masses at the corners of the sample space, corresponding to states for which both islands are simultaneously non-segregating. Analytic results for the corner probabilities and line densities are verified independently using the backward generator and for the corner probabilities using the coalescent.

Vascular type 1 angiotensin receptors control blood pressure by augmenting peripheral vascular resistance in female mice.

Angiotensin II (ANG II) is a major mediator of hypertension pathogenesis. In addition, there are well-documented differences in expression of the renin-angiotensin system (RAS) components and ANG II responses between males and females, which may explain sex differences in blood pressure (BP) and hypertension epidemiology. We previously showed that type 1A angiotensin (AT1A) receptors in vascular smooth muscle cells (VSMCs) play a critical role in BP regulation and hypertension pathogenesis, but these studies were carried out in male mice. Therefore, the major goal of the current studies was to examine the impact of VSMC AT1A receptors on BP and hypertension pathogenesis in female mice. We found that elimination of VSMC AT1A receptors in female mice reduced (≈8 mmHg) baseline BP without altering sodium sensitivity. The severity of ANG II-induced hypertension was diminished (≈33% reduction in BP), particularly during the last 2 wk of chronic ANG II infusion, compared with controls, but natriuresis was not altered during the first 5 days of ANG II infusion. Urinary norepinephrine levels were enhanced in female SMKO compared with control mice. There was a virtually complete elimination of ANG II-induced kidney hemodynamic responses with attenuation of acute vasoconstrictor responses in the systemic vasculature. These findings demonstrate that direct vascular actions of AT1A receptors play a prominent role in BP control and hypertension pathogenesis in female mice.

Ancestral inference from haplotypes and mutations.

We consider inference about the history of a sample of DNA sequences, conditional upon the haplotype counts and the number of segregating sites observed at the present time. After deriving some theoretical results in the coalescent setting, we implement rejection sampling and importance sampling schemes to perform the inference. The importance sampling scheme addresses an extension of the Ewens Sampling Formula for a configuration of haplotypes and the number of segregating sites in the sample. The implementations include both constant and variable population size models. The methods are illustrated by two human Y chromosome datasets.

Wright-Fisher diffusion bridges.

The trajectory of the frequency of an allele which begins at x at time 0 and is known to have frequency z at time T can be modelled by the bridge process of the Wright-Fisher diffusion. Bridges when x=z=0 are particularly interesting because they model the trajectory of the frequency of an allele which appears at a time, then is lost by random drift or mutation after a time T. The coalescent genealogy back in time of a population in a neutral Wright-Fisher diffusion process is well understood. In this paper we obtain a new interpretation of the coalescent genealogy of the population in a bridge from a time t∈(0,T). In a bridge with allele frequencies of 0 at times 0 and T the coalescence structure is that the population coalesces in two directions from t to 0 and t to T such that there is just one lineage of the allele under consideration at times 0 and T. The genealogy in Wright-Fisher diffusion bridges with selection is more complex than in the neutral model, but still with the property of the population branching and coalescing in two directions from time t∈(0,T). The density of the frequency of an allele at time t is expressed in a way that shows coalescence in the two directions. A new algorithm for exact simulation of a neutral Wright-Fisher bridge is derived. This follows from knowing the density of the frequency in a bridge and exact simulation from the Wright-Fisher diffusion. The genealogy of the neutral Wright-Fisher bridge is also modelled by branching Pólya urns, extending a representation in a Wright-Fisher diffusion. This is a new very interesting representation that relates Wright-Fisher bridges to classical urn models in a Bayesian setting.

A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning.

Duality plays an important role in population genetics. It can relate results from forwards-in-time models of allele frequency evolution with those of backwards-in-time genealogical models; a well known example is the duality between the Wright-Fisher diffusion for genetic drift and its genealogical counterpart, the coalescent. There have been a number of articles extending this relationship to include other evolutionary processes such as mutation and selection, but little has been explored for models also incorporating crossover recombination. Here, we derive from first principles a new genealogical process which is dual to a Wright-Fisher diffusion model of drift, mutation, and recombination. The process is reminiscent of the ancestral recombination graph, a widely-used multilocus genealogical model, but here ancestral lineages are typed and transition rates are regarded as being conditioned on an observed configuration at the leaves of the genealogy. Our approach is based on expressing a putative duality relationship between two models via their infinitesimal generators, and then seeking an appropriate test function to ensure the validity of the duality equation. This approach is quite general, and we use it to find dualities for several important variants, including both a discrete L-locus model of a gene and a continuous model in which mutation and recombination events are scattered along the gene according to continuous distributions. As an application of our results, we derive a series expansion for the transition function of the diffusion. Finally, we study in further detail the case in which mutation is absent. Then the dual process describes the dispersal of ancestral genetic material across the ancestors of a sample. The stationary distribution of this process is of particular interest; we show how duality relates this distribution to haplotype fixation probabilities. We develop an efficient method for computing such probabilities in multilocus models.

Vascular Type 1A Angiotensin II Receptors Control BP by Regulating Renal Blood Flow and Urinary Sodium Excretion.

Inappropriate activation of the type 1A angiotensin (AT1A) receptor contributes to the pathogenesis of hypertension and its associated complications. To define the role for actions of vascular AT1A receptors in BP regulation and hypertension pathogenesis, we generated mice with cell-specific deletion of AT1A receptors in smooth muscle cells (SMKO mice) using Loxp technology and Cre transgenes with robust expression in both conductance and resistance arteries. We found that elimination of AT1A receptors from vascular smooth muscle cells (VSMCs) caused a modest (approximately 7 mmHg) yet significant reduction in baseline BP and exaggerated sodium sensitivity in mice. Additionally, the severity of angiotensin II (Ang II)-dependent hypertension was dramatically attenuated in SMKO mice, and this protection against hypertension was associated with enhanced urinary excretion of sodium. Despite the lower BP, acute vasoconstrictor responses to Ang II in the systemic vasculature were largely preserved (approximately 80% of control levels) in SMKO mice because of exaggerated activity of the sympathetic nervous system rather than residual actions of AT1B receptors. In contrast, Ang II-dependent responses in the renal circulation were almost completely eliminated in SMKO mice (approximately 5%-10% of control levels). These findings suggest that direct actions of AT1A receptors in VSMCs are essential for regulation of renal blood flow by Ang II and highlight the capacity of Ang II-dependent vascular responses in the kidney to effect natriuresis and BP control.

Analysis and rejection sampling of Wright-Fisher diffusion bridges.

We investigate the properties of a Wright-Fisher diffusion process starting at frequency x at time 0 and conditioned to be at frequency y at time T. Such a process is called a bridge. Bridges arise naturally in the analysis of selection acting on standing variation and in the inference of selection from allele frequency time series. We establish a number of results about the distribution of neutral Wright-Fisher bridges and develop a novel rejection-sampling scheme for bridges under selection that we use to study their behavior.

Thromboxane receptors in smooth muscle promote hypertension, vascular remodeling, and sudden death.

The prostanoid thromboxane A2 has been implicated to contribute to the pathogenesis of many cardiovascular diseases, including hypertension. To study the role of vascular thromboxane-prostanoid (TP) receptors in blood pressure regulation, we generated mice with cell-specific deletion of TP receptors in smooth muscle using Cre/Loxp technology. We crossed the KISM22α-Cre transgenic mouse line expressing Cre recombinase in smooth muscle cells with a mouse line bearing a conditional allele of the Tbxa2r gene (Tp(flox)). In KISM22α-Cre(+)Tp(flox/flox) (TP-SMKO) mice, TP receptors were efficiently deleted from vascular smooth muscle cells. In TP-SMKOs, acute vasoconstrictor responses to the TP agonist U46619 were attenuated to a similar extent in both the peripheral and renal circulations. Yet, acute vascular responses to angiotensin II were unaffected at baseline and after chronic angiotensin II administration. Infusion of high-dose U46619 caused circulatory collapse and death in a majority of control mice but had negligible hemodynamic effects in TP-SMKOs, which were completely protected from U46619-induced sudden death. Baseline blood pressures were normal in TP-SMKOs. However, the absence of TP receptors in vascular smooth muscle cells was associated with significant attenuation of angiotensin II-induced hypertension and diminished vascular remodeling. This was also associated with reduced urinary thromboxane production after chronic angiotensin II. Thus, TP receptors in vascular smooth muscle cells play a major role in mediating the actions of thromboxane A(2) in TP agonist-induced shock, hypertension, and vascular remodeling of the aorta.

Angiotensin II type 1A receptors in vascular smooth muscle cells do not influence aortic remodeling in hypertension.

Vascular injury and remodeling are common pathological sequelae of hypertension. Previous studies have suggested that the renin-angiotensin system acting through the type 1 angiotensin II (AT(1)) receptor promotes vascular pathology in hypertension. To study the role of AT(1) receptors in this process, we generated mice with cell-specific deletion of AT(1) receptors in vascular smooth muscle cells using Cre/Loxp technology. We crossed the SM22α-Cre transgenic mouse line expressing Cre recombinase in smooth muscle cells with a mouse line bearing a conditional allele of the Agtr1a gene (Agtr1a (flox)), encoding the major murine AT(1) receptor isoform (AT(1A)). In SM22α-Cre(+)Agtr1a (flox/flox) (SMKO) mice, AT(1A) receptors were efficiently deleted from vascular smooth muscle cells in larger vessels but not from resistance vessels such as preglomerular arterioles. Thus, vasoconstrictor responses to angiotensin II were preserved in SMKO mice. To induce hypertensive vascular remodeling, mice were continuously infused with angiotensin II for 4 weeks. During infusion of angiotensin II, blood pressures increased significantly and to a similar extent in SMKO and control mice. In control mice, there was evidence of vascular oxidative stress indicated by enhanced nitrated tyrosine residues in segments of aorta; this was significantly attenuated in SMKO mice. Despite these differences in oxidative stress, the extent of aortic medial expansion induced by angiotensin II infusion was virtually identical in both groups. Thus, vascular AT(1A) receptors promote oxidative stress in the aortic wall but are not required for remodeling in angiotensin II-dependent hypertension.

Inference from samples of DNA sequences using a two-locus model.

Performing inference on contemporary samples of DNA sequence data is an important and challenging task. Computationally intensive methods such as importance sampling (IS) are attractive because they make full use of the available data, but in the presence of recombination the large state space of genealogies can be prohibitive. In this article, we make progress by developing an efficient IS proposal distribution for a two-locus model of sequence data. We show that the proposal developed here leads to much greater efficiency, outperforming existing IS methods that could be adapted to this model. Among several possible applications, the algorithm can be used to find maximum likelihood estimates for mutation and crossover rates, and to perform ancestral inference. We illustrate the method on previously reported sequence data covering two loci either side of the well-studied TAP2 recombination hotspot. The two loci are themselves largely non-recombining, so we obtain a gene tree at each locus and are able to infer in detail the effect of the hotspot on their joint ancestry. We summarize this joint ancestry by introducing the gene graph, a summary of the well-known ancestral recombination graph.

Renal actions of RGS2 control blood pressure.

G protein-coupled receptors (GPCRs) have key roles in cardiovascular regulation and are important targets for the treatment of hypertension. GTPase-activating proteins, such as RGS2, modulate downstream signaling by GPCRs. RGS2 displays regulatory selectivity for the Gαq subclass of G proteins, and mice lacking RGS2 develop hypertension through incompletely understood mechanisms. Using total body RGS2-deficient mice, we used a kidney crosstransplantation strategy to examine separately the contributions of RGS2 actions in the kidney from those in extrarenal tissues with regard to BP regulation. Loss of renal RGS2 was sufficient to cause hypertension, whereas the absence of RGS2 from all extrarenal tissues including the peripheral vasculature did not significantly alter BP. Accordingly, these results suggest that RGS2 acts within the kidney to modulate BP and prevent hypertension. These data support a critical role for the renal epithelium and/or vasculature as the final determinants of the intra-arterial pressure in hypertension.

Gene expression profiles linked to AT1 angiotensin receptors in the kidney.

To characterize gene expression networks linked to AT(1) angiotensin receptors in the kidney, we carried out genome-wide transcriptional analysis of RNA from kidneys of wild-type (WT) and AT(1A) receptor-deficient mice (KOs) at baseline and after 2 days of angiotensin II infusion (1,000 ng·kg(-1)·min(-1)). At baseline, 405 genes were differentially expressed (>1.5×) between WT and KO kidneys. Of these, >80% were upregulated in the KO group including genes involved in inflammation, oxidative stress, and cell proliferation. After 2 days of angiotensin II infusion in WT mice, expression of ≈805 genes was altered (18% upregulated, 82% repressed). Genes in metabolism and ion transport pathways were upregulated while there was attenuated expression of genes protective against oxidative stress including glutathione synthetase and mitochondrial superoxide dismutase 2. Angiotensin II infusion had little effect on blood pressure in KOs. Nonetheless, expression of >250 genes was altered in kidneys from KO mice during angiotensin II infusion; 14% were upregulated, while 86% were repressed including genes involved in immune responses, angiogenesis, and glutathione metabolism. Between WT and KO kidneys during angiotensin II infusion, 728 genes were differentially expressed; 10% were increased and 90% were decreased in the WT group. Differentially regulated pathways included those involved in ion transport, immune responses, metabolism, apoptosis, cell proliferation, and oxidative stress. This genome-wide assessment should facilitate identification of critical distal pathways linked to blood pressure regulation.

A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit.

The genealogical structure of neutral populations in which reproductive success is highly-skewed has been the subject of many recent studies. Here we derive a coalescent dual process for a related class of continuous-time Moran models with viability selection. In these models, individuals can give birth to multiple offspring whose survival depends on both the parental genotype and the brood size. This extends the dual process construction for a multi-type Moran model with genic selection described in Etheridge and Griffiths (2009). We show that in the limit of infinite population size the non-neutral Moran models converge to a Markov jump process which we call the lamda-Fleming-Viot process with viability selection and we derive a coalescent dual for this process directly from the generator and as a limit from the Moran models. The dual is a branching-coalescing process similar to the Ancestral Selection Graph which follows the typed ancestry of genes backwards in time with real and virtual lineages. As an application, the transition functions of the non-neutral Moran and lamda-coalescent models are expressed as mixtures of the transition functions of the dual process.

IMPORTANCE SAMPLING AND THE TWO-LOCUS MODEL WITH SUBDIVIDED POPULATION STRUCTURE.

The diffusion-generator approximation technique developed by De Iorio and Griffiths (2004a) is a very useful method of constructing importance sampling proposal distributions. Being based on general mathematical principles, the method can be applied to various models in population genetics. In this paper we extend the technique to the neutral coalescent model with recombination, thus obtaining novel sampling distributions for the two-locus model. We consider the case with subdivided population structure, as well as the classic case with only a single population. In the latter case we also consider the importance sampling proposal distributions suggested by Fearnhead and Donnelly (2001), and show that their two-locus distributions generally differ from ours. In the case of the infinitely-many-alleles model, our approximate sampling distributions are shown to be generally closer to the true distributions than are Fearnhead and Donnelly's.