1,721,015 research outputs found
Software for "A Stochastic Population Model for the Impact of Cancer Cell Dormancy on Therapy Success"
No full text<p>This notebook contains the software used to create all the simulations in the preprint "A Stochastic Population Model for the Impact of Cancer Cell Dormancy on Therapy Success" (2023). This publication is part of MATH+ project EF4-7 "The impact of dormancy on the evolutionary, ecological and pathogenic properties of microbial populations".</p>
Fitness Valleys, Metastability and Changing Environments
Full text linkMathematical modelling of biological processes has become an area of high scientific interest. This field has expanded substantially over the past century and developed a wide variety of mathematical approaches and biological applications. The complexity of biological systems requires models taking random effects into account.
This thesis investigates stochastic individual-based models of adaptive dynamics for asexually reproducing populations with mutation, focussing on the interplay between population dynamics, mutation rates, and environmental changes. Evolution is driven by linear birth rates, density-dependent logistic death rates, and mutations occurring along a finite trait graph. The model partially incorporates time-varying parameters, such as periodic changes in the environment or drug concentrations, which impact the evolutionary process. We investigate the behaviour of mutants and their invasion dynamics under small mutation rates and a simultaneously diverging population size, where environmental changes occur on a moderately diverging time scale.
The results of the first part (Appendix A) provide a detailed analysis of transitions between evolutionary stable conditions (ESC) in a constant environment. Here multiple mutations need to be accumulated to cross fitness valleys. The system exhibits metastable behaviour across multiple time scales which are linked to the widths of these fitness valleys. Introducing a meta-graph framework of ESCs, we describe the multi-scale jump chain resulting from concatenated jumps and prove the convergence of the population process to a Markov jump process that visits only ESCs of sufficiently high stability.
We then turn to the study of periodically changing environments. In the second part (Appendix B), we examine the growth of emergent mutants and their invasion of the resident population with a focus on mesoscopic scaling limits and the effective growth rates of mutants. The dynamics are influenced by an averaging effect of invasion fitness across different phases of the environment.
Additionally, we explore the crossing of fitness valleys in a changing environment in the third part (Appendix B), distinguishing two cases: Under the assumption of a strict fitness valley, we can show that the crossing rates are computed as an average taking into account the ability to survive. A particularly interesting scenario is the pit stop phenomenon, where intermediate mutants within a fitness valley experience phases of positive fitness, allowing them to grow to large sizes before going extinct. This accelerates the traversal of the valley and introduces a novel time scale in the evolutionary process
Stochastic and deterministic models for the evolution of heterogeneous populations: Multiscale approximation and applications to melanoma T-cell therapy
Full text linkStochastic modelling at the interface of mathematics and life sciences has gained great attention over the last decades. The study of many complex biological systems requires models taking random effects into account. Both sides benefit from this interdisciplinary collaboration. A structured mathematical analysis can provides a new perspective and helps to gain insights into biological problems. Vice versa, biological research inspires new mathematical questions leading to an interesting theory on their own. In this thesis, we demonstrate how mathematical modelling supports biomedical research in various ways: First, important mechanisms are identified that determine the outcome of experiments. Second, likely causes for the observed phenomena are investigated, which helps to interpret experimental data. Third, the clinical applicability of experimental scenarios is validated. Forth, predictions are made that reach beyond the experiment. Conversely, we study mathematical questions arising from biology. We approximate stochastic and deterministic models for adaptive dynamics under various parameter regimes to investigate the long-term behaviour of a population. Some of these results are again beneficial for applications in biomedicine since they have potential to improve the algorithms for simulations of the studied systems.
The thesis is divided into two more theoretical parts and one more applied part. In the theoretical parts we study individual-based Markov processes and their deterministic counterpart as models for the evolution of a heterogeneous population. We consider the limit of large populations and rare mutations. The resulting limit processes show different behaviour and are highly dependent and the scaling of mutation rates and the choice of time scales. The short-term dynamics are governed by Lotka-Volterra interactions of large subpopulations and the invasion of arising mutations can only be witnessed on a divergent time scale. In Chapter 2, we analyse the deterministic system that arises from the stochastic model in a law of large numbers. We study the limit of rare mutations. This corresponds to a scenario of relatively high mutation rates, compared to other limit regimes. It leads to multiple microscopic mutant populations that compete to invade the resident population at the same time. To determine which of the mutant traits succeeds, one has to carefully keep track of the growth of all subpopulations. The general discrete graph that we consider as a trait space induces complex dynamics of mutations between traits. To handle these, we have to introduce a new approach of inductive approximation of the population sizes of different traits, taking into account the influence of different traits at an increasing distance. Moreover, we investigate a couple of interesting special cases that relate to the scenario of adaptive walks and propose a cut-off model that mimics the simultaneous limit of large populations and rare mutations in the stochastic model. In Chapter 3, we combine the mentioned inductive procedure and couplings to branching processes to consider this simultaneous limit. To do so, we have to extend some existing limit results for branching processes to the multidimensional case. We derive a complete characterisation of the limiting jump process in the scenario of power law mutation rates, thus extending previous results for linear trait spaces and specific parameters to general finite graphs and arbitrary fitness landscapes. In the second part of the chapter we present a collection of specific examples that represent interesting and partially counter-intuitive behaviour arising under this scaling. Chapter 4 is dedicated to an application of individual-based models in the field of oncology. We investigate the role of phenotypic and genotypic heterogeneity of melanoma cells in the development of resistance to immunotherapy. Here, we substantially extend the existing model of a previous collaboration to include effects of immunosuppression, aspects of the spatial structure of the tumour, and the possibility of spontaneous mutations. While the previous model was designed to investigate phenotypic switches, we focus on the study of genetic variants. Through simulations we analyse the effect of subclonal fitness variability on the enrichment of resistant cell types
Software for "The impact of dormancy on evolutionary branching"
No full text<p>This file contains the source code for simulations used in the paper "The impact of dormancy on evolutionary branching" (2024). This publication is part of MATH+ project EF4-7 "The impact of dormancy on the evolutionary, ecological and pathogenic properties of microbial populations".</p>
The Impact of Dormancy on the Ecological, Evolutionary and Pathogenic Properties of Microbial Populations
Full text linkDie vorliegende Arbeit behandelt das biologische Phänomen der Dormanz mit Hilfe mathematischer Modellierung. Dormanz beschreibt dabei einen reversiblen Zustand von Individuen, in dem die metabolische Aktivität reduziert wird und die Resistenz gegen Natureinflüsse erhöht ist.
Der erste Teil der Arbeit widmet sich den ökologischen Eigenschaften. Hier wird zunächst ein Moranmodell vorgestellt, welches verschiedene Modellierungsarten von Dormanz aus der Populationsgenetik vereint und unter verschiedenen Skalierungen den schwachen seed-bank Koaleszenten und den starken seed-bank Koaleszenten als anzestralen Prozess innehat. Dadurch werden die Parameter der Koaleszenten vergleichbar. Als Anwendung betrachten wir die sogenannte species abundance distribution, welche mithilfe von Koaleszenten beschrieben werden kann.
Der zweite Teil beschäftigt sich mit den Auswirkungen von Dormanz auf evolutionäre Eigenschaften und beginnt mit einer Einführung in die Theorie von adaptive dynamics. Dort werden auch verschiedene Möglichkeiten der Modellierung von Dormanz in individuenbasierten Modellen besprochen. Danach befassen wir uns mit der Erweiterung eines Modells für sympatrische Speziation um den Aspekt der Dormanz. Die canonical equation of adaptive dynamics wird - motiviert durch ein Modell mit Dormanz - für schnellere Mutationsraten aus dem sogenannten power-law Mutationsregime für einen Grenzfall hergeleitet.
Die Arbeit schließt mit dem dritten Teil, in welchem ein individuenbasiertes Modell für die Entwicklung von Krebs unter dem Einfluss von Chemotherapie und unter Berücksichtigung von Dormanz vorgestellt wird. In Simulationsstudien wird untersucht, inwiefern Dormanz zu Misserfolg einer Therapie beiträgt. Ein weiteres Ziel ist die Analyse von Kombinationsbehandlung mit einem Medikament welches mit dormanten Zellen interagieren kann insbesondere unter Betrachtung verschiedener Therapieansätze zur Behandlung von dormanten Krebszellen.The present thesis uses mathematical modelling to investigate the consequences of dormancy. Dormancy describes a reversible and protected state of reduced metabolic activity which enhances an individual's resilience to hazardous conditions. In this sense, dormancy acts as a protection mechanism against habitats with unfavourable environments. The thesis considers the impact of dormancy on ecological, evolutionary and in its broadest sense pathogenic properties of microbial populations.
The first part is concerned with studying the impact of dormancy on ecology. For this, a Moran model is presented which unifies different models of dormancy from population genetics and exhibits the weak seed-bank coalescent and the strong seed-bank coalescent as the scaling limit of the ancestral process. As an application we consider the species abundance distribution which can be described using coalescent theory.
In the second part we consider the influence of dormancy on evolutionary properties. The modelling framework for this is the theory of adaptive dynamics. We then show that competition-induced dormancy may favour sympatric speciation. A key aspect in the derivation of this result is the canonical equation of adaptive dynamics. We extend this equation - motivated by a model including dormancy - to power-law mutations in a limiting case.
We conclude the thesis with the third part where we provide an individual-based model for the treatment of cancer with chemotherapy under consideration of dormant cancer cells. Using simulation studies, we investigate how dormancy may contribute to treatment failure. Another goal of this chapter is to analyse combination treatment with a drug which directly targets dormant cancer cells and to formulate general observations regarding various strategies to counter cancer cell dormancy
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
On/off super-Brown'sche Bewegung: eine super-Brown'sche Bewegung mit Dormanz
No full textWe introduce and construct on/off super-Brownian motion (on/off SBM) as a measure-valued scaling limit of critical on/off branching Brownian motions. The distinguishing feature of this process is that its infinitesimal particles can switch individually into and out of a state of dormancy, in which they neither move nor reproduce. Related dormancy traits have received interest in mathematical population biology recently, introducing memory and delay (in the form of a seed bank) into the corresponding processes. It turns out that the properties of on/off SBM differ significantly from those of classical super-Brownian motion. In particular, the process does not die out in finite time with probability one despite criticality of reproduction. However, the size of the active subpopulation does hit zero in finite time with positive probability, a result which can be shown using methods from polynomial diffusion theory.
In dimension one we further show that the measure-valued states of on/off super-Brownian motion have a density that can be characterized as a solution to a two-component stochastic partial differential equation (SPDE). Additionally, we provide a delay SPDE representation for the densities that explains how the on/off mechanism leads to a memory effect in the system. This representation will be useful in the analysis of support and range properties of the process. In particular this is leading to a representation of the range of the process in terms of its active component.
Finally, in the last chapter, we will couple an on/off super-Brownian motion with a classical super-Brownian motion using a Brownian snake. The resulting coupling of the ranges allows to transfer range properties of classical super-Brownian motion to on/off super-Brownian motion. The core argument for the coupling is a suitable approximation of local time. The dormancy shuffles, in some sense, the local time field. The representation of on/off super-Brownian motion seems to be the first representation of a nonlocal superprocess in terms of a Brownian snake.Wir konstruieren und führen die on/off super-Brown’sche Bewegung (on/off SBM) als einen maßwertigen Grenzwert einer kritischen verzweigenden on/off Brown’schen Bewegung ein. Das Besondere an diesem Prozess ist, dass seine infinitesimal kleinen Partikel individuell in einen Zustand der Dormanz fallen können, in welchem sie sich weder bewegen noch fortpflanzen. Verwandte Dormanzphänomene haben in letzter Zeit verstärkt Aufmerksamkeit im Teilgebiet der mathematischen Populationsbiologie erhalten, wo sie Gedächtnis- und Verzögerungseffekte erzeugen. Wie sich herausstellt unterscheiden sich viele Eigenschaften der on/off super-Brown’schen Bewegung grundlegend von denen der klassischen super-Brown’schen Bewegung. Insbesondere stirbt der Prozess mit Wahrscheinlichkeit eins nicht nach endlicher Zeit aus, trotz seines kritischen Reproduktionsmechanismus. Allerdings trifft seine aktive Komponente nach endlicher Zeit die Null mit echt positiver Wahrscheinlichkeit, ein Resultat, welches mit Hilfe von Methoden der polynomiellen Diffusionstheorie gezeigt werden kann.
In Kapitel II zeigen wir, dass in Dimension eins die maßwertigen Zustände der on/off super-Brown’schen Bewegung eine Dichte besitzen, welche als Lösung einer stochastischen partiellen Differentialgleichung mit zwei Komponenten charakterisiert werden kann. Des Weiteren stellen wir eine Verzögerungsdarstellung der stochastischen partiellen Differentialgleichung für die Dichten bereit, welche erklären kann wie der Dormanzmechanismus zu Gedächtniseffekten in dem System führt. Diese Darstellung ist ebenfalls für die Analyse der Träger- und Spannweiteneigenschaften (engl. support and range properties) nützlich. Insbesondere führt dies zu einer Darstellung der Spannweite des Prozesses durch seine aktive Komponente.
Im letzten Kapitel koppeln wir mit Hilfe einer on/off Schlange eine on/off super-Brown’sche Bewegung mit einer klassischen super-Brown’schen Bewegung. Die gleichzeitig resultierende Kopplung der Spannweiten beider Prozesse erlaubt es unmittelbar die Spannweiteneigenschaften von der klassischen auf die on/off super-Brown’sche Bewegung zu transferieren. Das Kernargument ist dabei eine passende Lokalzeitapproximation. In gewissem Sinne bringt die Dormanz das Lokalzeitfeld durcheinander. Die Darstellung der on/off super-Brown’schen Bewegung scheint die erste Darstellung eines nicht lokalen Superprozesses bezüglich einer Brown’schen Schlange zu sein.DFG, 410208580, GRK 2544: Stochastische Analysis in InteraktionDFG, 390685689, EXC 2046: MATH+: Berlin Mathematics Research Cente
Long term behaviour of spatial population models with heterozygous or asymmetric homozygous selection
Full text linkWe investigate the long term behaviour of two models for the spatial distribution of alleles in a diploid population, one with asymmetric selection in favour of the homozygotes, and one with selection in favour of the heterozygote.
We model the population with asymmetric homozygous selection using a version of the spatial Î-Fleming-Viot process. We identify three regimes. For very small values of the asymmetry, the limiting behaviour of the process is the same as for the case with symmetric selection. This case was studied in Etheridge, Freeman and Penington (2017), and they showed, under certain rescaling and initial conditions, that the hybrid zone, the interface between two homogeneous regions of each homozygote, evolves according to mean curvature flow. However, for larger, but not that much larger, values of asymmetry, we show a new behaviour, that the hybrid zone evolves according to a different type of flow, which we call constant curvature flow. Furthermore, there is a strength of asymmetry for which elements of both types of curvature flow are present in the limit. This suggests that the behaviour found in Etheridge, Freeman and Penington (2017) is more sensitive to perturbations than first thought.
We then go on to investigate the fluctuations of this process about its limit. To do this, we specialise to the one-dimensional case. We show that, when time, space and the strength of the asymmetry are appropriately rescaled, the hybrid zone, which is a single point in one dimension, evolves according to a Brownian motion, with drift proportional to the asymmetry.
Finally, we turn to the model with heterozygous selection. We restrict ourselves to two dimensions, and investigate this process through its dual, the branching annihilating random walk. We show that, up to an arbitrary time, and with arbitrarily high probability, there exists a branching rate such that we may couple a branching annihilating random walk to a ternary
branching Brownian motion. While interesting in its own right, this result lends support to a conjecture of Blath, Etheridge and Meredith (2007), that a branching annihilating random walk in two dimensions has a positive
probability of survival for all time for any positive branching rate.</p
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