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Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~N is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers {\it fitness proportional} selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. ([EPW09]) we propose a parameter scaling which fits well to the ``near-critical'' regime that was in the focus of [EPW09] (and in which the mutation-selection ratio diverges logarithmically as N→∞). Using a Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click rate of the ``classical ratchet'' by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection, which (other than that of the classical ratchet) follows an autonomous dynamics up to the time of its extinction. In [GSW23] it was discovered that the tournament ratchet has a hierarchy of dual processes which can be constructed on top of an Ancestral Selection graph with a Poisson decoration. For a regime in which the mutation/selection-ratio remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics of the click rates as well as that of the type frequency profile between clicks. We will describe how these ideas can be extended to the near-critical regime in which the mutation-selection ratio of the tournament ratchet converges to 1 as N→∞.
Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Under certain conditions, that we call the ``partial order of adaptation'', we can couple these measures. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. Interestingly, the construction also provides a connection to the theory of optimal transport. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.
Therapy evasion – and subsequent disease progression – is a major challenge in current oncology. An important role in this context seems to be played by various forms of cancer cell dormancy. For example, therapy-induced dormancy, over short timescales, can create serious obstacles to aggressive treatment approaches such as chemotherapy, and long-term dormancy may lead to relapses and metastases even many years after an initially successful treatment. The underlying dormancy-related mechanisms are complex and highly diverse, so that the analysis even of basic patterns of the population-level consequences of dormancy requires abstraction and idealization, as well as the identification of the relevant specific scenarios.
In this paper, we focus on a situation in which individual cancer cells may switch into and out of a dormant state both spontaneously as well as in response to treatment, and over relatively short time-spans. We introduce a mathematical ‘toy model’, based on stochastic agent-based interactions, for the dynamics of cancer cell populations involving individual short-term dormancy, and allow for a range of (multi-drug) therapy protocols. Our analysis shows that in our idealized model, even a small initial population of dormant cells can lead to therapy failure under classical (and in the absence of dormancy successful) single-drug treatments. We further investigate the effectiveness of several multidrug regimes (manipulating dormant cancer cells in specific ways) and provide some basic rules for the design of (multi-)drug treatment protocols depending on the types and parameters of dormancy mechanisms present in the population.
Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings ι:Pr↪Pn and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.
We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in about 20 minutes. Relying on our result, we show that the Gröbner structure of TGr0(3,8) refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian Dr(3,8), except for 23 orbits of special cones, for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional cones which belong to positive tropicalizations of algebraic varieties. We compute the positive Grassmannian TGr+(3,8) and compare it to the cluster complex of the classical Grassmannian Gr(3,8).
The Calderón problem with finitely many unknowns is equivalent to convex semidefinite optimization
(2023)
We consider the inverse boundary value problem of determining a coefficient function in an elliptic partial differential equation from knowledge of the associated Neumann-Dirichlet-operator. The unknown coefficient function is assumed to be piecewise constant with respect to a given pixel partition, and upper and lower bounds are assumed to be known a-priori.
We will show that this Calderón problem with finitely many unknowns can be equivalently formulated as a minimization problem for a linear cost functional with a convex non-linear semidefinite constraint. We also prove error estimates for noisy data, and extend the result to the practically relevant case of finitely many measurements, where the coefficient is to be reconstructed from a finite-dimensional Galerkin projection of the Neumann-Dirichlet-operator.
Our result is based on previous works on Loewner monotonicity and convexity of the Neumann-Dirichlet-operator, and the technique of localized potentials. It connects the emerging fields of inverse coefficient problems and semidefinite optimization.