Stochastic simulation tools and continuum models for describing two-dimensional collective cell spreading with universal growth functions

Wang, Jin; Penington, Catherine J.; McCue, Scott W.; Simpson, Matthew J.

doi:10.1088/1478-3975/13/5/056003

Cited by 34 publications

(73 citation statements)

References 36 publications

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“…While we have primarily focused on model selection across different cell motility mechanisms, others have proposed models including generalisations of the source term in Equation (1) (Browning et al, 2017;Jin et al, 2016a;Tsoularis and Wallace, 2002), the inclusion of growth factors (Jin et al, 2016b;Sherratt and Murray, 1990), or chemotaxis (Bianchi et al, 2016). Such extensions are of interest and should be the subject of future research.…”

Section: Discussion and Outlookmentioning

confidence: 99%

Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology

Warne

Baker

Simpson

2018

Preprint

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Reaction-diffusion models describing the movement, reproduction and death of individuals within a population are key mathematical modelling tools with widespread applications in mathematical biology. A diverse range of such continuum models have been applied in various biological contexts by choosing different flux and source terms in the reaction-diffusion framework. For example, to describe collective spreading of cell populations, the flux term may be chosen to reflect various movement mechanisms, such as random motion (diffusion), adhesion, haptotaxis, chemokinesis and chemotaxis. The choice of flux terms in specific applications, such as wound healing, is usually made heuristically, and rarely is it tested quantitatively against detailed cell density data. More generally, in mathematical biology, the questions of model validation and model selection have not received the same attention as the questions of model development and model analysis. Many studies do not consider model validation or model selection, and those that do often base the selection of the model on residual error criteria after model calibration is performed using nonlinear regression techniques. In this work, we present a model selection case study, in the context of cell invasion, with a very detailed experimental data set. Using Bayesian analysis and information criteria, we demonstrate that model selection and model validation should account for both residual errors and model complexity. These considerations are often overlooked in the mathematical biology literature. The results we present here provide a clear methodology that can be used to guide model selection across a range of applications. Furthermore, the case study we present provides a clear example where neglecting the role of model complexity can give rise to misleading outcomes.

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Section: Discussion and Outlookmentioning

confidence: 99%

Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology

Warne

Baker

Simpson

2018

Preprint

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“…In this work, we take the simplest possible approach use the number of nearest neighbours on a hexagonal lattice to represent the local density. Several generalisations, such as working with next nearest neighbours or working with a weighted average of nearest neighbours, could be incorporated into our modelling framework (Fadai et al, 2019;Jin et al, 2016a). Again, we expect that such extensions would lead to an even richer family of population dynamics models.…”

Section: Discussionmentioning

confidence: 99%

Population dynamics with threshold effects give rise to a diverse family of Allee effects

Fadai

Simpson

2020

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The Allee effect describes populations that deviate from logistic growth models and arises in applications including ecology and cell biology. A common justification for incorporating Allee effects into population models is that the population in question has altered growth mechanisms at some critical density, often referred to as a threshold effect. Despite the ubiquitous nature of threshold effects arising in various biological applications, the explicit link between local threshold effects and global Allee effects has not been considered. In this work, we examine a continuum population model that incorporates threshold effects in the local growth mechanisms. We show that this model gives rise to a diverse family of Allee effects and we provide a comprehensive analysis of which choices of local growth mechanisms give rise to specific Allee effects. Calibrating this model to a recent set of experimental data describing the growth of a population of cancer cells provides an interpretation of the threshold population density and growth mechanisms associated with the population.Here, C(t) ≥ 0 is the density at time t, dC(t)/dt is the growth rate, r > 0 is the low-density growth rate, C(0) is the initial density, and K > 0 is the carrying capacity density. Equation (1) has two equilibria: C * = 0 and C * = K, where an equilibrium is any value C * such that dC(t)/dt = 0 when C(t) ≡ C * . Since densities near C(t) ≡ K will approach K, while densities near C(t) ≡ 0 diverge away from zero, we say that C * = K is a stable equilibrium, while C * = 0 is an unstable equilibrium. This means that the logistic growth model implicitly assumes that all densities, no matter how small, eventually thrive, since lim t→∞ C(t) = K forMathematical models that include an Allee effect relax the assumption that all population densities will thrive and survive, which is inherent in (1) (Edelstein-Keshet, 2005;Murray, 2003;Stephens et al., 1999;Taylor and Hastings, 2005). Consequently, populations described using Allee effect models exhibit more complicated and nuanced dynamics, including reduced growth at low densities (Gerlee, 2013;Johnson et al., 2006;Neufeld et al., 2017) and extinction below a critical density threshold (Allee and Bowen, 1932;Courchamp et al., 1999;Taylor and Hastings, 2005). The phrase Allee effect can have many different interpretations in different parts of the literature. For instance, the Weak Allee effect is used to describe density growth rates that deviate from logistic growth, but do not include additional equilibria (Edelstein-Keshet, 2005;Murray, 2003;Stephens et al., 1999;Taylor and Hastings, 2005). A common mathematical description of the Weak Allee effect iswhere the factor 1 + C(t)/A represents the deviation from the classical logistic growth model. Despite the similarity between (1) and (2), it is not possible to write down an explicit solution for (2) in terms of C(t), like we can for (1). Despite this, we are still able to examine the equilibria of (2) to understand the salient features of the Weak Al...

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“…We use a discrete random walk model to simulate the passaging process and we refer to individual random walkers in the model as cells. All simulations are performed on a hexagonal lattice, with the lattice spacing ∆ taken to be equal to the average cell diameter [19]. The model includes crowding effects by ensuring that there is, at most, one cell per lattice site [36].…”

Section: Model Frameworkmentioning

confidence: 99%

“…such that 1 ≤ i ≤ I and 1 ≤ j ≤ J [19]. In any single realisation of the model, the occupancy of site (i, j) is denoted C i,j , with C i,j = 1 if the site is occupied, and C i,j = 0 if vacant.…”

Section: Model Frameworkmentioning

confidence: 99%

“…In this work, we describe a mathematical model that can be used to study the passaging process in 2D in vitro cell culture [19,43]. A key feature of our model is that we allow individual cells within the population to take on a range of characteristics, such as variable proliferation rates, and therefore it is natural to focus on using a discrete model for this purpose [8,19]. In particular, we are interested in examining whether the apparently contradictory effects of passaging reported in the literature can be recapitulated using a fairly straightforward discrete model.…”

Section: Introductionmentioning

confidence: 99%

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A computational modelling framework to quantify the effects of passaging cell lines

Wang

Penington

McCue

et al. 2017

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In vitro cell culture is routinely used to grow and supply a sufficiently large number of cells for various types of cell biology experiments. Previous experimental studies report that cell characteristics evolve as the passage number increases, and various cell lines can behave differently at high passage numbers. To provide insight into the putative mechanisms that might give rise to these differences, we perform in silico experiments using a random walk model to mimic the in vitro cell culture process. Our results show that it is possible for the average proliferation rate to either increase or decrease as the passaging process takes place, and this is due to a competition between the initial heterogeneity and the degree to which passaging damages the cells. We also simulate a suite of scratch assays with cells from near-homogeneous and heterogeneous cell lines, at both high and low passage numbers. Although it is common in the literature to report experimental results without disclosing the passage number, our results show that we obtain significantly different closure rates when performing in silico scratch assays using cells with different passage numbers. Therefore, we suggest that the passage number should always be reported to ensure that the experiment is as reproducible as possible. Furthermore, our modelling also suggests some avenues for further experimental examination that could be used to validate or refine our simulation results.

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Stochastic simulation tools and continuum models for describing two-dimensional collective cell spreading with universal growth functions

Cited by 34 publications

References 36 publications

Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology

Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology

Population dynamics with threshold effects give rise to a diverse family of Allee effects

A computational modelling framework to quantify the effects of passaging cell lines

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