From Random Walks to Fully Anisotropic Diffusion Models for Cell and Animal Movement

Painter, Kevin J.; Hillen, Thomas

doi:10.1007/978-3-319-96842-1_5

Cited by 11 publications

(10 citation statements)

References 49 publications

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“…We expect such an approach to reduce the computational complexity of the subcellular model, thus allowing us to run larger scale simulations. We also plan to scale up the model in order to obtain a continuum PDE limit description of it [81,[94][95][96]. This will further reduce the computational complexity of the simulations.…”

Section: Discussionmentioning

confidence: 99%

A multiscale model of complex endothelial cell dynamics in early angiogenesis

et al. 2021

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We introduce a hybrid two-dimensional multiscale model of angiogenesis, the process by which endothelial cells (ECs) migrate from a pre-existing vascular bed in response to local environmental cues and cell-cell interactions, to create a new vascular network. Recent experimental studies have highlighted a central role of cell rearrangements in the formation of angiogenic networks. Our model accounts for this phenomenon via the heterogeneous response of ECs to their microenvironment. These cell rearrangements, in turn, dynamically remodel the local environment. The model reproduces characteristic features of angiogenic sprouting that include branching, chemotactic sensitivity, the brush border effect, and cell mixing. These properties, rather than being hardwired into the model, emerge naturally from the gene expression patterns of individual cells. After calibrating and validating our model against experimental data, we use it to predict how the structure of the vascular network changes as the baseline gene expression levels of the VEGF-Delta-Notch pathway, and the composition of the extracellular environment, vary. In order to investigate the impact of cell rearrangements on the vascular network structure, we introduce the mixing measure, a scalar metric that quantifies cell mixing as the vascular network grows. We calculate the mixing measure for the simulated vascular networks generated by ECs of different lineages (wild type cells and mutant cells with impaired expression of a specific receptor). Our results show that the time evolution of the mixing measure is directly correlated to the generic features of the vascular branching pattern, thus, supporting the hypothesis that cell rearrangements play an essential role in sprouting angiogenesis. Furthermore, we predict that lower cell rearrangement leads to an imbalance between branching and sprout elongation. Since the computation of this statistic requires only individual cell trajectories, it can be computed for networks generated in biological experiments, making it a potential biomarker for pathological angiogenesis.

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Section: Discussionmentioning

confidence: 99%

A multiscale model of complex endothelial cell dynamics in early angiogenesis

et al. 2021

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“…As described above, our model only accounts for diffusion in two directions. Further extension of our model could therefore incorporate anisotropic diffusion (Othmer et al, 1988;Painter & Hillen, 2018). Another possibility could be the introduction of stochastic process components, in order to convert the partial differential equations into stochastic differential equations (Brown et al, 2000).…”

Section: Discussionmentioning

confidence: 99%

Modelling the spatiotemporal spread of beneficial alleles using ancient genomes

Muktupavela

Petr

Ségurel

et al. 2021

Preprint

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Ancient genome sequencing technologies now provide the opportunity to study natural selection in unprecedented detail. Rather than making inferences from indirect footprints left by selection in present-day genomes, we can directly observe whether a given allele was present or absent in a particular region of the world at almost any period of human history within the last 10,000 years. Methods for studying selection using ancient genomes often rely on partitioning individuals into discrete time periods or regions of the world. However, a complete understanding of natural selection requires more nuanced statistical methods which can explicitly model allele frequency changes in a continuum across space and time. Here we introduce a method for inferring the spread of a beneficial allele across a landscape using two-dimensional partial differential equations. Unlike previous approaches, our framework can handle time-stamped ancient samples, as well as genotype likelihoods and pseudohaploid sequences from low-coverage genomes. We apply the method to a panel of published ancient West Eurasian genomes, to produce dynamic maps showcasing the inferred spread of candidate beneficial alleles over time and space. We also provide estimates for the strength of selection and diffusion rate for each of these alleles. Finally, we highlight possible avenues of improvement for accurately tracing the spread of beneficial alleles in more complex scenarios.

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“…They are therefore often rewarded by new insights when the mechanism underlying a spatial effect can be worked into a process model [1]. Redistribution kernels are a popular means to this end, with applications as diverse as predator-prey interactions [2]; range expansion and invasion biology [3]; grouping/swarming behaviour [4]; chemical communication [5]; and cellular transport [6].…”

Section: Introductionmentioning

confidence: 99%

A unifying theory for two-dimensional spatial redistribution kernels with applications in population spread modelling

Koch

Lewis

Lele

2020

J. R. Soc. Interface.

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When building models to explain the dispersal patterns of organisms, ecologists often use an isotropic redistribution kernel to represent the distribution of movement distances based on phenomenological observations or biological considerations of the underlying physical movement mechanism. The Gaussian, two-dimensional (2D) Laplace and Bessel kernels are common choices for 2D space. All three are special (or limiting) cases of a kernel family, the Whittle–Matérn–Yasuda (WMY), first derived by Yasuda from an assumption of 2D Fickian diffusion with gamma-distributed settling times. We provide a novel derivation of this kernel family, using the simpler assumption of constant settling hazard, by means of a non-Fickian 2D diffusion equation representing movements through heterogeneous 2D media having a fractal structure. Our derivation reveals connections among a number of established redistribution kernels, unifying them under a single, flexible modelling framework. We demonstrate improvements in predictive performance in an established model for the spread of the mountain pine beetle upon replacing the Gaussian kernel by the Whittle–Matérn–Yasuda, and report similar results for a novel approximation, the product-Whittle–Matérn–Yasuda, that substantially speeds computations in applications to large datasets.

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From Random Walks to Fully Anisotropic Diffusion Models for Cell and Animal Movement

Cited by 11 publications

References 49 publications

A multiscale model of complex endothelial cell dynamics in early angiogenesis

A multiscale model of complex endothelial cell dynamics in early angiogenesis

Modelling the spatiotemporal spread of beneficial alleles using ancient genomes

A unifying theory for two-dimensional spatial redistribution kernels with applications in population spread modelling

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