Phenotypic lag and population extinction in the moving-optimum model: insights from a small-jumps limit.

Continuous environmental change-such as slowly rising temperatures-may create permanent maladaptation of natural populations: Even if a population adapts evolutionarily, its mean phenotype will usually lag behind the phenotype favored in the current environment, and if the resulting phenotypic lag becomes too large, the population risks extinction. We analyze this scenario using a moving-optimum model, in which one or more quantitative traits are under stabilizing selection towards an optimal value that increases at a constant rate. We have recently shown that, in the limit of infinitely small mutations and high mutation rate, the evolution of the phenotypic lag converges to an Ornstein-Uhlenbeck process around a long-term equilibrium value. Both the mean and the variance of this equilibrium lag have simple analytical formulas. Here, we study the properties of this limit and compare it to simulations of an evolving population with finite mutational effects. We find that the "small-jumps limit" provides a reasonable approximation, provided the mean lag is so large that the optimum cannot be reached by a single mutation. This is the case for fast environmental change and/or weak selection. Our analysis also provides insights into population extinction: Even if the mean lag is small enough to allow a positive growth rate, stochastic fluctuations of the lag will eventually cause extinction. We show that the time until this event follows an exponential distribution, whose mean depends strongly on a composite parameter that relates the speed of environmental change to the adaptive potential of the population.

Functional Law of Large Numbers and PDEs for Epidemic Models with Infection-Age Dependent Infectivity.

We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time since infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic Volterra-type integral equations, which has a further characterization using PDEs under some regularity conditions. The solutions are characterized with boundary conditions that are given by a system of Volterra equations. We also characterize the equilibrium points for the PDEs in the SIS model with infection-age dependent infectivity. To establish the FLLNs, we employ a useful criterion for weak convergence for the two-parameter processes together with useful representations for the relevant processes via Poisson random measures.

Fluctuations Around a Homogenised Semilinear Random PDE.

We consider a semilinear parabolic partial differential equation in R + × [ 0 , 1 ] d , where d = 1 , 2 or 3, with a highly oscillating random potential and either homogeneous Dirichlet or Neumann boundary condition. If the amplitude of the oscillations has the right size compared to its typical spatiotemporal scale, then the solution of our equation converges to the solution of a deterministic homogenised parabolic PDE, which is a form of law of large numbers. Our main interest is in the associated central limit theorem. Namely, we study the limit of a properly rescaled difference between the initial random solution and its LLN limit. In dimension d = 1 , that rescaled difference converges as one might expect to a centred Ornstein-Uhlenbeck process. However, in dimension d = 2 , the limit is a non-centred Gaussian process, while in dimension d = 3 , before taking the CLT limit, we need to subtract at an intermediate scale the solution of a deterministic parabolic PDE, subject (in the case of Neumann boundary condition) to a non-homogeneous Neumann boundary condition. Our proofs make use of the theory of regularity structures, in particular of the very recently developed methodology allowing to treat parabolic PDEs with boundary conditions within that theory.

Estimating the state of the COVID-19 epidemic in France using a model with memory.

In this paper, we use a deterministic epidemic model with memory to estimate the state of the COVID-19 epidemic in France, from early March until mid-December 2020. Our model is in the SEIR class, which means that when a susceptible individual (S) becomes infected, he/she is first exposed (E), i.e. not yet contagious. Then he/she becomes infectious (I) for a certain length of time, during which he/she may infect susceptible individuals around him/her, and finally becomes removed (R), that is, either immune or dead. The specificity of our model is that it assumes a very general probability distribution for the pair of exposed and infectious periods. The law of large numbers limit of such a model is a model with memory (the future evolution of the model depends not only upon its present state, but also upon its past). We present theoretical results linking the (unobserved) parameters of the model to various quantities which are more easily measured during the early stages of an epidemic. We then apply these results to estimate the state of the COVID-19 epidemic in France, using available information on the infection fatality ratio and on the distribution of the exposed and infectious periods. Using the hospital data published daily by Santé Publique France, we gather some information on the delay between infection and hospital admission, intensive care unit (ICU) admission and hospital deaths, and on the proportion of people who have been infected up to the end of 2020.

Evolution of the ancestral recombination graph along the genome in case of selective sweep.

We consider the genome of a sample of n individuals taken at the end of a selective sweep, which is the fixation of an advantageous allele in the population. When the selective advantage is high, the genealogy at a locus under selective sweep can be approximated by a comb with n teeth. However, because of recombinations during the selective sweep, the hitchhiking effect decreases as the distance from the selected site increases, so that far from this locus, the tree can be approximated by a Kingman coalescent tree, as in the neutral case. We first give the distribution of the tree at a given locus. Then we focus on the evolution of this tree along the genome. Since this tree-valued process is not Markovian, we study the evolution of the Ancestral Recombination Graph along the genome in case of selective sweep.

Soluble HLA-G Expression Inversely Correlates With Fetal Microchimerism Levels in Peripheral Blood From Women With Scleroderma.

Women with scleroderma (SSc) maintain significantly higher quantities of persisting fetal microchimerism (FMc) from complete or incomplete pregnancies in their peripheral blood compared to healthy women. The non-classical class-I human leukocyte antigen (HLA) molecule HLA-G plays a pivotal role for the implantation and maintenance of pregnancy and has often been investigated in offspring from women with pregnancy complications. However data show that maternal HLA-G polymorphisms as well as maternal soluble HLA-G (sHLA-G) expression could influence pregnancy outcome. Here, we aimed to investigate the underlying role of maternal sHLA-G expression and HLA-G polymorphisms on the persistence of FMc. We measured sHLA-G levels by enzyme linked immunosorbent assay in plasma samples from 88 healthy women and 74 women with SSc. Male Mc was quantified by DYS14 real-time PCR in blood samples from 58 women who had previously given birth to at least one male child. Furthermore, eight HLA-G 5'URR/3'UTR polymorphisms, previously described as influencing HLA-G expression, were performed on DNA samples from 96 healthy women and 106 women with SSc. Peripheral sHLA-G was at lower concentration in plasma from SSc (76.2 ± 48.3 IU/mL) compared to healthy women (117.5 ± 60.1 IU/mL, p < 0.0001), independently of clinical subtypes, autoantibody profiles, disease duration, or treatments. Moreover, sHLA-G levels were inversely correlated to FMc quantities (Spearman correlation, p < 0.01). Finally, women with SSc had lower sHLA-G independently of the eight HLA-G 5'URR/3'UTR polymorphisms, although they were statistically more often homozygous than heterozygous for HLA-G polymorphism genotypes -716 (G/T), -201 (G/A), 14 bp (ins/del), and +3,142 (G/A) than healthy women. In conclusion, women with SSc display less sHLA-G expression independently of the eight HLA-G polymorphisms tested. This decreased production correlates with higher quantities of persisting FMc commonly observed in blood from SSc women. These results shed some lights on the contribution of the maternal HLA-G protein to long-term persistent fetal Mc and initiate new perspectives in this field.

TMEM187-IRAK1 Polymorphisms Associated with Rheumatoid Arthritis Susceptibility in Tunisian and French Female Populations: Influence of Geographic Origin.

Polymorphisms have been identified in the Xq28 locus as risk loci for rheumatoid arthritis (RA). Here, we investigated the association between three polymorphisms in the Xq28 region containing TMEM187 and IRAK1 (rs13397, rs1059703, and rs1059702) in two unstudied populations: Tunisian and French. The rs13397 G and rs1059703 T major alleles were significantly increased in RA patients (n = 408) compared with age-matched controls (n = 471) in both Tunisian and French women. These results were confirmed by a meta-analysis replication study including two independent Greek and Korean cohorts. The rs1059702 C major allele was significantly associated with RA, only with French women. In the French population, the GTC haplotype displayed a protective effect against RA, while the ATC, GCC, and GTT haplotypes conferred significant risk for RA. No association for these haplotypes was found in the Tunisian population. Our results replicated for the first time the association of the three Xq28 polymorphisms with RA risk in Tunisian and French populations and suggested that RA susceptibility is associated with TMEM187-IRAK1 polymorphisms in women. Our data further support the involvement of X chromosome in RA susceptibility and evidence ethnicities differences that might be explained by differences in the frequencies of SE HLA-DRB1 alleles between both populations.

HLA-DRB1 genotypes and the risk of developing anti citrullinated protein antibody (ACPA) positive rheumatoid arthritis.

OBJECTIVE: To provide a table indicating the risk for developing anti citrullinated protein antibody (ACPA) positive rheumatoid arthritis (RA) according to one's HLA-DRB1 genotype. METHODS: We HLA-DRB1 genotyped 857 patients with ACPA positive RA and 2178 controls from South Eastern and Eastern France and calculated Odds Ratios (OR) for developing RA for 106 of 132 possible genotypes accounting for 97% of subjects. RESULTS: HLA-DRB1 genotypic ORs for developing ACPA positive RA range from 28 to 0.19. HLA-DRB1 genotypes with HLA-DRB1*04SE (HLA-DRB1*0404, HLA-DRB1*0405, HLA-DRB1*0408), HLA-DRB1*04∶01, HLA-DRB1*01 are usually associated with high risk for developing RA. The second HLA-DRB1 allele in genotype somewhat modulates shared epitope associated risk. We did not identify any absolutely protective allele. Neither the Reviron, nor the du Montcel models accurately explains our data which are compatible with the shared epitope hypothesis and suggest a dosage effect among shared epitope positive HLA-DRB1 alleles, double dose genotypes carrying higher ORs than single dose genotypes. CONCLUSION: HLA-DRB1 genotypic risk for developing ACPA positive RA is influenced by both HLA-DRB1 alleles in genotype. We provide an HLA-DRB1 genotypic risk table for ACPA positive RA.