Understanding recessive disease risk in multi-ethnic populations with different degrees of consanguinity.

Population medical genetics aims at translating clinically relevant findings from recent studies of large cohorts into healthcare for individuals. Genetic counseling concerning reproductive risks and options is still mainly based on family history, and consanguinity is viewed to increase the risk for recessive diseases regardless of the demographics. However, in an increasingly multi-ethnic society with diverse approaches to partner selection, healthcare professionals should also sharpen their intuition for the influence of different mating schemes in non-equilibrium dynamics. We, therefore, revisited the so-called out-of-Africa model and studied in forward simulations with discrete and not overlapping generations the effect of inbreeding on the average number of recessive lethals in the genome. We were able to reproduce in both frameworks the drop in the incidence of recessive disorders, which is a transient phenomenon during and after the growth phase of a population, and therefore showed their equivalence. With the simulation frameworks, we also provide the means to study and visualize the effect of different kin sizes and mating schemes on these parameters for educational purposes.

The speed of invasion in an advancing population.

We derive rigorous estimates on the speed of invasion of an advantageous trait in a spatially advancing population in the context of a system of one-dimensional F-KPP equations. The model was introduced and studied heuristically and numerically in a paper by Venegas-Ortiz et al. (Genetics 196:497-507, 2014). In that paper, it was noted that the speed of invasion by the mutant trait is faster when the resident population is expanding in space compared to the speed when the resident population is already present everywhere. We use the Feynman-Kac representation to provide rigorous estimates that confirm these predictions.

From adaptive dynamics to adaptive walks.

We consider an asexually reproducing population on a finite type space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this population can be modelled as a measure-valued Markov process. Multiple variations of this system have been studied in the simultaneous limit of large populations and rare mutations, where the regime is chosen such that mutations are separated. We consider the deterministic system, resulting from the large population limit, and then let the mutation probability tend to zero. This corresponds to a much higher frequency of mutations, where multiple microscopic types are present at the same time. The limiting process resembles an adaptive walk or flight and jumps between different equilibria of coexisting types. The graph structure on the type space, determined by the possibilities to mutate, plays an important role in defining this jump process. In a variation of the above model, where the radius in which mutants can be spread is limited, we study the possibility of crossing valleys in the fitness landscape and derive different kinds of limiting walks.

The recovery of a recessive allele in a Mendelian diploid model.

We study the large population limit of a stochastic individual-based model which describes the time evolution of a diploid hermaphroditic population reproducing according to Mendelian rules. Neukirch and Bovier (J Math Biol 75:145-198, 2017) proved that sexual reproduction allows unfit alleles to survive in individuals with mixed genotype much longer than they would in populations reproducing asexually. In the present paper we prove that this indeed opens the possibility that individuals with a pure genotype can reinvade in the population after the appearance of further mutations. We thus expose a rigorous description of a mechanism by which a recessive allele can re-emerge in a population. This can be seen as a statement of genetic robustness exhibited by diploid populations performing sexual reproduction.

Survival of a recessive allele in a Mendelian diploid model.

In this paper we analyse the genetic evolution of a diploid hermaphroditic population, which is modelled by a three-type nonlinear birth-and-death process with competition and Mendelian reproduction. In a recent paper, Collet et al. (J Math Biol 67(3):569-607, 2013) have shown that, on the mutation time-scale, the process converges to the Trait-Substitution Sequence of adaptive dynamics, stepping from one homozygotic state to another with higher fitness. We prove that, under the assumption that a dominant allele is also the fittest one, the recessive allele survives for a time of order at least [Formula: see text], where K is the size of the population and [Formula: see text].

A stochastic model for immunotherapy of cancer.

We propose an extension of a standard stochastic individual-based model in population dynamics which broadens the range of biological applications. Our primary motivation is modelling of immunotherapy of malignant tumours. In this context the different actors, T-cells, cytokines or cancer cells, are modelled as single particles (individuals) in the stochastic system. The main expansions of the model are distinguishing cancer cells by phenotype and genotype, including environment-dependent phenotypic plasticity that does not affect the genotype, taking into account the effects of therapy and introducing a competition term which lowers the reproduction rate of an individual in addition to the usual term that increases its death rate. We illustrate the new setup by using it to model various phenomena arising in immunotherapy. Our aim is twofold: on the one hand, we show that the interplay of genetic mutations and phenotypic switches on different timescales as well as the occurrence of metastability phenomena raise new mathematical challenges. On the other hand, we argue why understanding purely stochastic events (which cannot be obtained with deterministic models) may help to understand the resistance of tumours to therapeutic approaches and may have non-trivial consequences on tumour treatment protocols. This is supported through numerical simulations.

Stochastic modelling of T-cell activation.

We investigate a specific part of the human immune system, namely the activation of T-cells, using stochastic tools, especially sharp large deviation results. T-cells have to distinguish reliably between foreign and self peptides which are both presented to them by antigen presenting cells. Our work is based on a model studied by Zint et al. (J Math Bio 57(6):841-861, 2008). We are able to dispense with some restrictive distribution assumptions that were used previously, i.e., we establish a higher robustness of the model. A central issue is the analysis of two new perspectives to the scenario (two different quenched systems) in detail. This means that we do not only analyse the total probability of a T-cell activation (the annealed case) but also consider the probability of an activation of one certain clonotype and the probability of a T-cell activation by a certain antigen presentation profile (the quenched cases). Finally, we see analytically that the probability of T-cell activation increases with the number of presented foreign peptides in all three cases.

Optimization of transcription factor binding map accuracy utilizing knockout-mouse models.

Genome-wide assessment of protein-DNA interaction by chromatin immunoprecipitation followed by massive parallel sequencing (ChIP-seq) is a key technology for studying transcription factor (TF) localization and regulation of gene expression. Signal-to-noise-ratio and signal specificity in ChIP-seq studies depend on many variables, including antibody affinity and specificity. Thus far, efforts to improve antibody reagents for ChIP-seq experiments have focused mainly on generating higher quality antibodies. Here we introduce KOIN (knockout implemented normalization) as a novel strategy to increase signal specificity and reduce noise by using TF knockout mice as a critical control for ChIP-seq data experiments. Additionally, KOIN can identify 'hyper ChIPable regions' as another source of false-positive signals. As the use of the KOIN algorithm reduces false-positive results and thereby prevents misinterpretation of ChIP-seq data, it should be considered as the gold standard for future ChIP-seq analyses, particularly when developing ChIP-assays with novel antibody reagents.

Plasticity of tumour and immune cells: a source of heterogeneity and a cause for therapy resistance?

Immunotherapies, signal transduction inhibitors and chemotherapies can successfully achieve remissions in advanced stage cancer patients, but durable responses are rare. Using malignant melanoma as a paradigm, we propose that therapy-induced injury to tumour tissue and the resultant inflammation can activate protective and regenerative responses that represent a shared resistance mechanism to different treatments. Inflammation-driven phenotypic plasticity alters the antigenic landscape of tumour cells, rewires oncogenic signalling networks, protects against cell death and reprogrammes immune cell functions. We propose that the successful combination of cancer treatments to tackle resistance requires an interdisciplinary understanding of these resistance mechanisms, supported by mathematical models.

An asymptotic maximum principle for essentially linear evolution models.

Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional maximum principle in the limit N-->infinity (where N, or N(d) with d> or =1, is proportional to the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible matrices of asymptotic dimension N(d) and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order 1/N. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types.

Aging in the random energy model.

The random energy model (REM) has become a key reference model for glassy systems. In particular, it is expected to provide a prime example of a system whose dynamics shows aging, a universal phenomenon characterizing the dynamics of complex systems. The analysis of its activated dynamics is based on so-called trap models, introduced by Bouchaud, that are also used to mimic the dynamics of more complex disordered systems. In this Letter we report the first results that justify rigorously the trap model predictions in the REM.