Bridging the gap between individual-based and continuum models of growing cell populations

Chaplain, M. A. J.; Lorenzi, Tommaso; Macfarlane, Fiona

doi:10.1007/s00285-019-01391-y

Cited by 29 publications

(38 citation statements)

References 66 publications

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“…Our analyses extend this framework to include apparent kinetics emergent from spatial and temporal variation in enzyme and substrate concentrations, which should be broadly applicable for scaling up microbial processes to the ecosystem scale. This emergence of a coarse-scale pattern from localscale kinetics is analogous to results from a cell population study by Chaplain et al (2018), who found that an individual-based model can generate complex spatial patterns of population growth observed in a corresponding continuum model.…”

Section: Eca Captures Kinetics Emergent From Spatial and Temporal Varsupporting

confidence: 76%

Emergent properties of organic matter decomposition by soil enzymes

Wang

Allison

2019

Soil Biology and Biochemistry

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Plant residues and soil organic matter are predominantly decomposed by exoenzymes. Many soil carbon models now represent enzymatic decomposition, but the mathematical formulation of this process has been debated over the last 15 years. Some models apply the traditional "forward" Michaelis-Menten equation to represent enzyme kinetics, whereas others apply the "reverse" Michaelis-Menten equation, which assumes that kinetic rates saturate at high enzyme concentrations. Recently the equilibrium chemistry approximation (ECA) has been proposed as an alternative to both Michaelis-Menten formulations. However, because of methodological limitations, in-situ enzyme kinetics-especially in the context of soil system heterogeneity-have been difficult to verify experimentally. Therefore, the overarching goal of our study was to evaluate different enzyme kinetic formulations using model-based evidence at microbial to ecosystem scales. We used a spatially explicit individual-and trait-based microbial model, DEMENT, to circumvent methodological challenges. Although DEMENT assumes forward Michaelis-Menten kinetics at local scales, at the grid scale we found saturating relationships between degradation rate and both substrate concentrations and enzyme concentrations that fit the forward and reverse Michaelis-Menten equations, respectively, at specific successional stages during decomposition. Although forward and reverse Michaelis-Menten equations emerged under some conditions, only the ECA adequately represented decay rates emerging from the spatial-temporal variation in substrate and enzyme concentrations throughout the decomposition process. Our results support a more widespread adoption of the ECA equation in soil biogeochemical modelling at ecosystem scales.

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Section: Eca Captures Kinetics Emergent From Spatial and Temporal Varsupporting

confidence: 76%

Emergent properties of organic matter decomposition by soil enzymes

Wang

Allison

2019

Soil Biology and Biochemistry

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“…Having rewritten the problem in this form allows us to construct travelling-wave solutions using an approach that builds on the method of proof recently presented in [12,31]. For the sake of brevity, in this section we drop the tildes from all the quantities in problem (4.6).…”

Section: Travelling-wave Solutions Of the Free-boundary Problemmentioning

confidence: 99%

“…These models are formulated in terms of nonlinear partial differential equations for cell densities (or cell volume fractions) and, as such, are amenable to both numerical and analytical approaches that enable insight to be gained into the biological system under study. From a mathematical perspective, over the past few years particular attention has been paid to the existence of travelling-wave solutions with composite shapes [5,12,31,43,48] and to the convergence to free-boundary problems in the asymptotic limit whereby cells are represented as an incompressible fluid [7,30,33,41,42].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, building on a recently presented method for a related system of nonlinear partial differential equations [12,31], we also construct travelling-wave solutions for the free-boundary problem. In this respect, the novelty of our work lies in the fact that we consider a free-boundary problem subject to biomechanical transmission conditions which are different from those considered in [12,31]. This requires a different approach when studying the properties of the solution at the interface between the two cell populations and introduces a significant difference in the qualitative properties of the travelling wave.…”

Section: Introductionmentioning

confidence: 99%

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From individual-based mechanical models of multicellular systems to free-boundary problems

Lorenzi

Murray

Ptashnyk³

2020

Interfaces Free Bound.

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In this paper we present an individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics. An offlattice modelling approach is considered whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described via generic nonlinear force laws; and (iii) cell proliferation is contact inhibited. We formally show that the continuum counterpart of this discrete model is given by a free-boundary problem for the cell densities. The results of the derivation demonstrate how the parameters of continuum mechanical models of multicellular systems can be related to biophysical cell properties. We prove an existence result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations are performed in the case where the cellular interaction forces are described by the celebrated Johnson-Kendall-Roberts model of elastic contact, which has been previously used to model cell-cell interactions. The results obtained indicate excellent agreement between the simulation results for the individual-based model, the numerical solutions of the corresponding free-boundary problem and the travelling-wave analysis.

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“…Spatial structure in biological populations includes both clustering and segregation [32][33][34][35][36][37]. Stochastic individual-based models (IBM) offer a straightforward means of exploring population dynamics without invoking a meanfield approximation [38,39]. However, IBM approaches are computationally prohibitive for large populations and provide limited mathematical insight into the population dynamics, for example how particular biological mechanisms affect the carrying capacity or the Allee threshold [40].…”

Section: Introductionmentioning

confidence: 99%

Population dynamics with spatial structure and an Allee effect

Plank

Surendran

Simpson

2020

Preprint

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Allee effects describe populations in which long-term survival is only possible if the population density is above some threshold level. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to population extinction, whereas initial densities above the threshold eventually asymptote to some positive carrying capacity density. Mean field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as short-range competition and dispersal. The influence of such non mean-field effects has not been studied in the presence of an Allee effect. To address this we develop an individual-based model (IBM) that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework accurately recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure that mean-field models neglect. For example, we show that there are cases where the mean-field model predicts extinction but the population actually survives and vice versa. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

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Bridging the gap between individual-based and continuum models of growing cell populations

Cited by 29 publications

References 66 publications

Emergent properties of organic matter decomposition by soil enzymes

Emergent properties of organic matter decomposition by soil enzymes

From individual-based mechanical models of multicellular systems to free-boundary problems

Population dynamics with spatial structure and an Allee effect

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