Navigating the flow: individual and continuum models for homing in flowing environments

Painter, Kevin J.; Hillen, Thomas

doi:10.1098/rsif.2015.0647

Cited by 41 publications

(36 citation statements)

References 47 publications

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“…In many studies of animal or cell movement, a primary point of concern is the cue (or cues) used to navigate, necessitating statistical correlation between a preferred movement direction and orientating information in the environment. Typical examples include the orientation of animals in response to the earth's magnetic field [31], or the migration of invasive tumour cells through their local extracellular environment [30]. In ecological applications a quasi twodimensional scenario generally suffices -for example, the habitat of a land-bound population or the typical swimming depth of marine organisms -and in such cases an animal's bearing can be described by a single angular coordinate.…”

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confidence: 99%

Moments of von mises and fisher distributions and applications

Hillen

Painter²,

Swan

et al. 2017

Mathematical Biosciences and Engineering

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The von Mises and Fisher distributions are spherical analogues to the Normal distribution on the unit circle and unit sphere, respectively. The computation of their moments, and in particular the second moment, usually involves solving tedious trigonometric integrals. Here we present a new method to compute the moments of spherical distributions, based on the divergence theorem. This method allows a clear derivation of the second moments and can be easily generalized to higher dimensions. In particular we note that, to our knowledge, the variance-covariance matrix of the three dimensional Fisher distribution has not previously been explicitly computed. While the emphasis of this paper lies in calculating the moments of spherical distributions, their usefulness is motivated by their relationship to population statistics in animal/cell movement models and demonstrated in applications to the modelling of sea turtle navigation, wolf movement and brain tumour growth.

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confidence: 99%

Moments of von mises and fisher distributions and applications

Hillen

Painter²,

Swan

et al. 2017

Mathematical Biosciences and Engineering

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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 [78,[87][88][89]. This will further reduce the computational complexity of the simulations.…”

Section: Discussionmentioning

confidence: 99%

“…Prior to migration, cells develop a polarity which depends on their local environment. Following [78,79], we consider the local cell polarity, φ, (see Fig 4) to be a random quantity sampled from the von Mises distribution. The probability density function (pdf) of the von Mises distribution reads…”

Section: Individual Cell System (1)mentioning

confidence: 99%

A multiscale model of complex endothelial cell dynamics in early angiogenesis

Stepanova

Byrne

Maini

et al. 2020

Preprint

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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. Author summaryAngiogenesis, the process by which new blood vessels are formed by sprouting from the pre-existing vascular bed, plays a key role in both physiological and pathological processes, including tumour growth. The structure of a growing vascular network is determined by the coordinated behaviour of endothelial cells in response to various signalling cues. Recent experimental studies have highlighted the importance of cell rearrangements as a driver for sprout elongation. However, the functional role of this phenomenon remains unclear. We formulate a new multiscale model of angiogenesis which, by accounting explicitly for the complex dynamics of endothelial cells within growing angiogenic sprouts, is able to produce generic features of angiogenic structures (branching, chemotactic sensitivity, cell mixing, etc.) as emergent properties of its dynamics. We validate our model against experimental data and then use it to quantify the pheno...

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“…In conclusion, we would like to remark that experience in a variety of contexts has demonstrated the value of relating individual-based stochastic models to deterministic continuum models [5,7,8,10,31,32,38], which make it possible to complement numerical simulations with rigorous analytical results, to achieve conclusions with broad structural stability under parameter changes. In this regard, a fundamental problem is what method of proof can be employed to derive deterministic mesoscopic models from the type of stochastic individual-based models which are defined in our framework.…”

Section: Discussionmentioning

confidence: 99%

Dynamical Patterns of Coexisting Strategies in a Hybrid Discrete-continuum Spatial Evolutionary Game Model

Burgess

Schofield

Hubbard

et al. 2016

Math. Model. Nat. Phenom.

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Abstract. We present a novel hybrid modelling framework that takes into account two aspects which have been largely neglected in previous models of spatial evolutionary games: random motion and chemotaxis. A stochastic individual-based model is used to describe the player dynamics, whereas the evolution of the chemoattractant is governed by a reaction-diffusion equation. The two models are coupled by deriving individual movement rules via the discretisation of a taxis-diffusion equation which describes the evolution of the local number of players. In this framework, individuals occupying the same position can engage in a two-player game, and are awarded a payoff, in terms of reproductive fitness, according to their strategy. As an example, we let individuals play the Hawk-Dove game. Numerical simulations illustrate how random motion and chemotactic response can bring about self-generated dynamical patterns that create favourable conditions for the coexistence of hawks and doves in situations in which the two strategies cannot coexist otherwise. In this sense, our work offers a new perspective of research on spatial evolutionary games, and provides a general formalism to study the dynamics of spatially-structured populations in biological and social contexts where individual motion is likely to affect natural selection of behavioural traits.

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Navigating the flow: individual and continuum models for homing in flowing environments

Cited by 41 publications

References 47 publications

Moments of von mises and fisher distributions and applications

Moments of von mises and fisher distributions and applications

A multiscale model of complex endothelial cell dynamics in early angiogenesis

Dynamical Patterns of Coexisting Strategies in a Hybrid Discrete-continuum Spatial Evolutionary Game Model

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