Exact solutions of linear reaction-diffusion processes on a uniformly growing domain: Criteria for successful colonization

Simpson, Matthew J.

doi:10.1101/011122

Cited by 7 publications

(18 citation statements)

References 32 publications

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“…PLOS 5/36 on 0 < x < L(t), which, at this point, can be solved by using the methods outlined in 82 our previous work for single uncoupled reaction-diffusion equations on growing 83 domains [20,21]. We note that the solution of Equation (4) can be unbounded when 84 k i < 0.…”

Section: Introductionmentioning

confidence: 96%

“…Alternatively, linear growth, L(t) = L(0) + βt, corresponds to σ(t) = β/(L(0) + βt) 99 and v(x, t) = xβ/(L(0) + βt) [20,21].…”

mentioning

confidence: 99%

“…The second reaction term, −σ(t)a i , is proportional to ∂v/∂x, and since ∂v/∂x > 0 this 108 is a sink term that represents a dilution effect caused by the domain growth. To 109 simplify Equation (9) we re-scale the time variable, T (t) = t 0 D/L 2 (s) ds [20], so that 110 the coefficient of the diffusive term is constant. This gives…”

mentioning

confidence: 99%

“…where we choose the Fourier coefficients, Ψ i,n , so that a i (ξ, conditions because previous studies have also used similar boundary conditions [6,20]. 127 We note, however, that greater care is required when applying nonhomogeneous with this is to increase the number of generations by including partial differential 146 equation models for C 5 (x, t), C 6 (x, t), and so on.…”

mentioning

confidence: 99%

“…where L(t) = L(0)e αt and T (t) = D(1 − e −2αt )/(2αL 2 (0)) [20,21]. To ensure that 160 a i (x, 0) matches the appropriate initial condition, we require…”

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confidence: 99%

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Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Simpson

Sharp

Morrow

et al. 2015

Preprint

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Embryonic development involves diffusion and proliferation of cells, as well as diffusionand reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction-diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction-diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction-diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially-confined population diffuses sufficiently slowly that it is PLOS 1/36 unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially-confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate.However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit. PLOS 2/36 1 Several processes during embryonic development are associated with the migration 2 and proliferation of cells within growing tissues. A canonical example of such a process 3 is the development of the enteric nervous system (ENS) [1-5]. This involves a 4 population of precursor cells that is initially confined towards the oral end of the 5 developing gut tissue. Cells within the population undergo individual migration and 6 proliferation events, leading to a population-level front of cells that moves toward the 7 anal end of the gut [6]. The spatial distribution of the population of cells is also 8 affected by the growth of the underlying gut tissue [7, 8]. Normal development of the 9 ENS requires that the moving front reaches the anal end of the developing tissue. 10 Conversely, abnormal ENS development is thought to be associated with situations 11where the front of cells fails to reach the anal end of the tissue [6,7]. 12Previous mathematical models of ENS development involve reaction-diffusion 13 equations on a growing domain [6,9]. These partial differential equation models have 14 been solved numerically, and the numerical solutions used to i...

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Section: Introductionmentioning

confidence: 96%

“…Alternatively, linear growth, L(t) = L(0) + βt, corresponds to σ(t) = β/(L(0) + βt) 99 and v(x, t) = xβ/(L(0) + βt) [20,21].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…where L(t) = L(0)e αt and T (t) = D(1 − e −2αt )/(2αL 2 (0)) [20,21]. To ensure that 160 a i (x, 0) matches the appropriate initial condition, we require…”

mentioning

confidence: 99%

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Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Simpson

Sharp

Morrow

et al. 2015

Preprint

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Modelling collective cell migration: neural crest as a model paradigm

et al. 2019

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A huge variety of mathematical models have been used to investigate collective cell migration. The aim of this brief review is twofold: to present a number of modelling approaches that incorporate the key factors affecting cell migration, including cell-cell and cell-tissue interactions, as well as domain growth, and to showcase their application to model the migration of neural crest cells. We discuss the complementary strengths of microscale and macroscale models, and identify why it can be important to understand how these modelling approaches are related. We consider neural crest cell migration as a model paradigm to illustrate how the application of different mathematical modelling techniques, combined with experimental results, can provide new biological insights. We conclude by highlighting a number of future challenges for the mathematical modelling of neural crest cell migration.

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Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

Allwright¹

2021

Proceedings of the Edinburgh Mathematical Society

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A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.

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Exact solutions of linear reaction-diffusion processes on a uniformly growing domain: Criteria for successful colonization

Abstract: 8Many processes during embryonic development involve transport and reaction of molecules,

Cited by 7 publications

References 32 publications

Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Modelling collective cell migration: neural crest as a model paradigm

Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

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