Dataset Information

Comparative analysis of continuum angiogenesis models.

ABSTRACT: Although discrete approaches are increasingly employed to model biological phenomena, it remains unclear how complex, population-level behaviours in such frameworks arise from the rules used to represent interactions between individuals. Discrete-to-continuum approaches, which are used to derive systems of coarse-grained equations describing the mean-field dynamics of a microscopic model, can provide insight into such emergent behaviour. Coarse-grained models often contain nonlinear terms that depend on the microscopic rules of the discrete framework, however, and such nonlinearities can make a model difficult to mathematically analyse. By contrast, models developed using phenomenological approaches are typically easier to investigate but have a more obscure connection to the underlying microscopic system. To our knowledge, there has been little work done to compare solutions of phenomenological and coarse-grained models. Here we address this problem in the context of angiogenesis (the creation of new blood vessels from existing vasculature). We compare asymptotic solutions of a classical, phenomenological "snail-trail" model for angiogenesis to solutions of a nonlinear system of partial differential equations (PDEs) derived via a systematic coarse-graining procedure (Pillay et al. in Phys Rev E 95(1):012410, 2017. https://doi.org/10.1103/PhysRevE.95.012410 ). For distinguished parameter regimes corresponding to chemotaxis-dominated cell movement and low branching rates, both continuum models reduce at leading order to identical PDEs within the domain interior. Numerical and analytical results confirm that pointwise differences between solutions to the two continuum models are small if these conditions hold, and demonstrate how perturbation methods can be used to determine when a phenomenological model provides a good approximation to a more detailed coarse-grained system for the same biological process.

SUBMITTER: Martinson WD

PROVIDER: S-EPMC7900093 | biostudies-literature | 2021 Feb

REPOSITORIES: biostudies-literature

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Comparative analysis of continuum angiogenesis models.

Martinson W Duncan WD Ninomiya Hirokazu H Byrne Helen M HM Maini Philip K PK

Journal of mathematical biology 20210223 4

Although discrete approaches are increasingly employed to model biological phenomena, it remains unclear how complex, population-level behaviours in such frameworks arise from the rules used to represent interactions between individuals. Discrete-to-continuum approaches, which are used to derive systems of coarse-grained equations describing the mean-field dynamics of a microscopic model, can provide insight into such emergent behaviour. Coarse-grained models often contain nonlinear terms that d ...[more]

PMID: 33619643

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Project description:BackgroundUncertainty in comparative analyses can come from at least two sources: a) phylogenetic uncertainty in the tree topology or branch lengths, and b) uncertainty due to intraspecific variation in trait values, either due to measurement error or natural individual variation. Most phylogenetic comparative methods do not account for such uncertainties. Not accounting for these sources of uncertainty leads to false perceptions of precision (confidence intervals will be too narrow) and inflated significance in hypothesis testing (e.g. p-values will be too small). Although there is some application-specific software for fitting Bayesian models accounting for phylogenetic error, more general and flexible software is desirable.MethodsWe developed models to directly incorporate phylogenetic uncertainty into a range of analyses that biologists commonly perform, using a Bayesian framework and Markov Chain Monte Carlo analyses.ResultsWe demonstrate applications in linear regression, quantification of phylogenetic signal, and measurement error models. Phylogenetic uncertainty was incorporated by applying a prior distribution for the phylogeny, where this distribution consisted of the posterior tree sets from Bayesian phylogenetic tree estimation programs. The models were analysed using simulated data sets, and applied to a real data set on plant traits, from rainforest plant species in Northern Australia. Analyses were performed using the free and open source software OpenBUGS and JAGS.ConclusionsIncorporating phylogenetic uncertainty through an empirical prior distribution of trees leads to more precise estimation of regression model parameters than using a single consensus tree and enables a more realistic estimation of confidence intervals. In addition, models incorporating measurement errors and/or individual variation, in one or both variables, are easily formulated in the Bayesian framework. We show that BUGS is a useful, flexible general purpose tool for phylogenetic comparative analyses, particularly for modelling in the face of phylogenetic uncertainty and accounting for measurement error or individual variation in explanatory variables. Code for all models is provided in the BUGS model description language.

Dataset Information

Comparative analysis of continuum angiogenesis models.

Publications

Comparative analysis of continuum angiogenesis models.

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