This chapter describes the development of a math trail for high school students. In 2016, we developed this trail through Leiden (The Netherlands) during a student project for the Science... Show moreThis chapter describes the development of a math trail for high school students. In 2016, we developed this trail through Leiden (The Netherlands) during a student project for the Science Communication and Society specialization, a track for master students at the Faculty of Science at Leiden University. Our aim was to provide a guided trail through the city that links everyday sights to mathematical concepts within the curriculum of high school students between 13 and 15 years old. The entire project was carried out in 3 weeks. We did background research, consisting of literature reviews, target audience surveys with school children, and focus groups with teachers. Based on the conclusions from this background research, we developed questions that suited both the goal to make the math trail a fun experience that makes math less abstract and the goal to include questions from across the curriculum. In this chapter, we would like to share our insights from the background research and our experiences in developing a math trail. We moreover aim to provide those who are interested in designing a math trail in their city with a practical step-by-step plan and checklist. Show less
All tissue development and replenishment relies upon the breaking of symmetries leading to the morphological and operational differentiation of progenitor cells into more specialized cells. One of... Show moreAll tissue development and replenishment relies upon the breaking of symmetries leading to the morphological and operational differentiation of progenitor cells into more specialized cells. One of the main engines driving this process is the Notch signal transduction pathway, a ubiquitous signalling system found in the vast majority of metazoan cell types characterized to date. Broadly speaking, Notch receptor activity is governed by a balance between two processes: 1) intercellular Notch transactivation triggered via interactions between receptors and ligands expressed in neighbouring cells; 2) intracellular cis inhibition caused by ligands binding to receptors within the same cell. Additionally, recent reports have also unveiled evidence of cis activation. Whilst context-dependent Notch receptor clustering has been hypothesized, to date, Notch signalling has been assumed to involve an interplay between receptor and ligand monomers. In this study, we demonstrate biochemically, through a mutational analysis of DLL4, both in vitro and in tissue culture cells, that Notch ligands can efficiently self-associate. We found that the membrane proximal EGF-like repeat of DLL4 was necessary and sufficient to promote oligomerization/dimerization. Mechanistically, our experimental evidence supports the view that DLL4 ligand dimerization is specifically required for cis-inhibition of Notch receptor activity. To further substantiate these findings, we have adapted and extended existing ordinary differential equation-based models of Notch signalling to take account of the ligand dimerization-dependent cis-inhibition reported here. Our new model faithfully recapitulates our experimental data and improves predictions based upon published data. Collectively, our work favours a model in which net output following Notch receptor/ligand binding results from ligand monomer-driven Notch receptor transactivation (and cis activation) counterposed by ligand dimer-mediated cis-inhibition.Author summary The growth and maintenance of tissues is a fundamental characteristic of metazoan life, controlled by a highly conserved core of cell signal transduction networks. One such pathway, the Notch signalling system, plays a unique role in these phenomena by orchestrating the generation of the phenotypic and genetic asymmetries which underlie tissue development and remodeling. At the molecular level, it achieves this via two specific types of receptor/ligand interaction: intercellular binding of receptors and ligands expressed in neighbouring cells, which triggers receptor activation (transactivation); intracellular receptor/ligand binding within the same cell which blocks receptor activation (cis inhibition). Together, these counterposed mechanisms determine the strength, the direction and the specificity of Notch signalling output. Whilst, the basic mechanisms of receptor transactivation have been delineated in some detail, the precise nature of cis inhibition has remained enigmatic. Through a combination of experimental approaches and computational modelling, in this study, we present a new model of Notch signalling in which ligand monomers promote Notch receptor transactivation, whereas cis inhibition is induced optimally via ligand dimers. This is the first model to include a concrete molecular distinction, in terms of ligand configuration, between the main branches of Notch signalling. Our model faithfully recapitulates both our presented experimental results as well as the recently published work of others, and provides a novel perspective for understanding Notch-regulated biological processes such as embryo development and angiogenesis.Competing Interest StatementThe authors have declared no competing interest. Show less
Cancer cell migration between different body parts is the driving force behind cancer metastasis, which is the main cause of mortality of patients. Migration of cancer cells often proceeds by... Show moreCancer cell migration between different body parts is the driving force behind cancer metastasis, which is the main cause of mortality of patients. Migration of cancer cells often proceeds by penetration through narrow cavities in locally stiff, yet flexible tissues. In our previous work, we developed a model for cell geometry evolution during invasion, which we extend here to investigate whether leader and follower (cancer) cells that only interact mechanically can benefit from sequential transmigration through narrow micro-channels and cavities. We consider two cases of cells sequentially migrating through a flexible channel: leader and follower cells being closely adjacent or distant. Using Wilcoxon’s signed-rank test on the data collected from Monte Carlo simulations, we conclude that the modelled transmigration speed for the follower cell is significantly larger than for the leader cell when cells are distant, i.e. follower cells transmigrate after the leader has completed the crossing. Furthermore, it appears that there exists an optimum with respect to the width of the channel such that cell moves fastest. On the other hand, in the case of closely adjacent cells, effectively performing collective migration, the leader cell moves 12% faster since the follower cell pushes it. This work shows that mechanical interactions between cells can increase the net transmigration speed of cancer cells, resulting in increased invasiveness. In other words, interaction between cancer cells can accelerate metastatic invasion. Show less
A minimal group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is... Show moreA minimal group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is equivalent to the action not being essentially free, which means that the set of points with non-trivial stabilizer has positive measure. In this paper, we investigate the relation between the property of having essential holonomy and structure of the acting group for minimal equicontinuous actions on Cantor sets. We show that if such a group action is locally quasi-analytic and has essential holonomy, then every commutator subgroup in the group lower central series has elements with positive measure set of points with non-trivial holonomy. In particular, we prove that a minimal equicontinuous Cantor action by a nilpotent group has no essential holonomy. We also show that the property of having essential holonomy is preserved under return equivalence and continuous orbit equivalence of minimal equicontinuous Cantor actions.Finally, we give examples to show that the assumption on the action that it is locally quasi-analytic is necessary. Show less
We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal... Show moreWe define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautological. Also we show that rings of tautological forms are always finite dimensional. Finally we characterize the Kawazumi–Zhang invariant as essentially the only smooth function on the moduli space of curves whose Levi form is a tautological form. Show less
A minimal equicontinuous action by homeomorphisms of a discrete group $\Gamma$ on a Cantor set $X$ is locally quasi-analytic, if each homeomorphism has a unique extension from small open sets... Show moreA minimal equicontinuous action by homeomorphisms of a discrete group $\Gamma$ on a Cantor set $X$ is locally quasi-analytic, if each homeomorphism has a unique extension from small open sets to open sets of uniform diameter on $X$. A minimal action is stable if the action on $X$ of the closure of $\Gamma$ in the group of homeomorphisms of $X$ is locally quasi-analytic. When $\Gamma$ is virtually nilpotent, we say that $\Phi \colon \Gamma \times \mathfrak{X} \to \mathfrak{X}$ is a nilpotent Cantor action. We show that a nilpotent Cantor action with finite prime spectrum must be stable. We also prove there exist uncountably many distinct Cantor actions of the Heisenberg group, necessarily with infinite prime spectrum, which are not stable. Show less
In this work, we propose a novel joint frailty model assuming bivariate discretely- distributed non-parametric frailties, with an unknown finite number of mass points. This ap- proach allows to... Show moreIn this work, we propose a novel joint frailty model assuming bivariate discretely- distributed non-parametric frailties, with an unknown finite number of mass points. This ap- proach allows to detect a latent structure among subjects, clustering them in sub-populations where individuals are characterized by a common frailty value. Our method can be interpreted as an unsupervised classification tool and motivates further investigation into the reasons for similarities within the clustered subjects and dissimilarities across the clusters. This work is motivated by a study of patients with Heart Failure (HF) undergoing ACE inhibitors treatment in the Lombardia region of Italy. Recurrent events of interest are hos- pitalizations due to HF and terminal event is death for any cause. Show less
Smeets, I.; Willems, S.J.W.; Burger, J.P.; Wijnker, W. 2022
Grafieken kunnen verduidelijken, maar ze kunnen ook misleiden. Er zijn verschillende manieren om misleidende grafieken effectief te debunken, blijkt uit ons onderzoek dat zojuist verscheen in het ...Show moreGrafieken kunnen verduidelijken, maar ze kunnen ook misleiden. Er zijn verschillende manieren om misleidende grafieken effectief te debunken, blijkt uit ons onderzoek dat zojuist verscheen in het Journal of Science Communication. Nieuwscheckers geeft zelf het goede voorbeeld met de nieuwe rubriek Grafiekpolitie. Show less
Jong, R. de; Geer, G. van der; Schoof, R.; Top, J. 2022
Op 16 januari 2022 overleed Bas Edixhoven, hoogleraar meetkunde aan de Universiteit Leiden. Naast zijn onderzoek en onderwijs was Bas betrokken bij vele internationale en nationale wiskunde... Show moreOp 16 januari 2022 overleed Bas Edixhoven, hoogleraar meetkunde aan de Universiteit Leiden. Naast zijn onderzoek en onderwijs was Bas betrokken bij vele internationale en nationale wiskunde-organisaties. In Nederland onder andere bij de commissie Onderzoek en de commissie Onderwijs van Platform Wiskunde Nederland, Mastermath, Wisk4all, DIAMANT, Vierkant voor Wiskunde, Compositio Mathematica, Indagationes Mathematicae, en de Bèta-lerarenkamer. Tevens was hij lid van de Koninklijke Nederlandse Akademie van Wetenschappen. Zijn nabije collega’s Robin de Jong, Jaap Top, Gerard van der Geer en René Schoof herdenken hem. Show less
