On a model for the evolution of morphogens in a growing tissue II: $$\varvec{\theta = \log (2)}$$ θ = log ( 2 ) case

Coclite, Giuseppe Maria; Coclite, Mario Michele

doi:10.1007/s00033-017-0838-0

Cited by 10 publications

(4 citation statements)

References 18 publications

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“…We consider the analysis of these cases in [7] and [8]. We present numerical experiments showing that the cases of θ = log(2) and θ < log(2) yield qualitatively similar results to that of θ > log(2).…”

Section: θ < Logmentioning

confidence: 95%

On a Model for the Evolution of Morphogens in a Growing Tissue

Coclite¹,

Coclite²,

Mishra³

2016

SIAM J. Math. Anal.

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We analyze a recently proposed model [I. Averbukh et al., Development, 141 (2014), pp. 2150-2156 for the regulation of growth and patterning in developing tissues by diffusing morphogens. We show that solutions of the underlying coupled systems of nonlinear PDEs exist, are unique, and are stable in a suitable sense. The key tool in the analysis is the transformation of the underlying system to a porous medium equation. Numerical experiments illustrating the model are also presented.

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Section: θ < Logmentioning

confidence: 95%

On a Model for the Evolution of Morphogens in a Growing Tissue

Coclite¹,

Coclite²,

Mishra³

2016

SIAM J. Math. Anal.

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“…Key tool for the analysis of the well-posedness (see [3,4,7]) and of the asymptotic behavior as t → ∞ (see [5,6]) is the definition of a suitable family of "characteristic" curves which start at the points of [0, L 0 ] and "cover" {(t, x) | t ≥ 0, 0 ≤ x ≤ L θ (t)}. Let us briefly recall them because they are also useful in this paper.…”

Section: (): V-volmentioning

confidence: 99%

On a singular limit as $$\theta \rightarrow 0$$ for a model for the evolution of morphogens in a growing tissue

Coclite

2023

Z. Angew. Math. Phys.

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“…The biofilm thickness L(t) represents the free boundary of the mathematical problem. Its evolution is governed by the following ordinary differential equation [7,16,[19][20][21][22][23],…”

Section: Introductionmentioning

confidence: 99%

Free boundary problem for the role of planktonic cells in biofilm formation and development

D’Acunto

Frunzo

Luongo

et al. 2021

Z. Angew. Math. Phys.

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The dynamics of biofilm lifecycle are deeply influenced by the surrounding environment and the interactions between sessile and planktonic phenotypes. Bacterial biofilms typically develop in three distinct stages: attachment of cells to a surface, growth of cells into colonies, and detachment of cells from the colony into the surrounding medium. The attachment of planktonic cells from the surrounding environment plays a prominent role in the initial phase of biofilm lifecycle as it initiates the colony formation. During the maturation stage, biofilms harbor numerous microenvironments which lead to metabolic heterogeneity. Such microniches provide conditions suitable for the growth of new species, which are present in the bulk liquid as planktonic cells and can penetrate the porous biofilm matrix. We present a 1D continuum model on the interaction of sessile and planktonic phenotypes in biofilm lifestyle. Such a model is able to reproduce the key role of planktonic cells in the formation and development of biofilms by considering the initial attachment and colonization phenomena. The model is formulated as a hyperbolic–elliptic free boundary value problem with vanishing initial value which considers the concentrations of planktonic and sessile cells as state variables. Hyperbolic equations reproduce the transport and growth of sessile species, while elliptic equations model the diffusion and conversion of planktonic cells and dissolved substrates. The attachment is modeled as a continuous, deterministic process which depends on the concentrations of the attaching species. The growth of new species is modeled through a reaction term in the hyperbolic equations which depends on the concentration of planktonic species within the biofilm. Existence and uniqueness of solutions are discussed and proved for the attachment regime. Finally, some numerical examples show that the proposed model correctly reproduces the growth of new species within the biofilm and overcomes the ecological restrictions characterizing the Wanner–Gujer-type models.

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On a model for the evolution of morphogens in a growing tissue II: $$\varvec{\theta = \log (2)}$$ θ = log ( 2 ) case

Cited by 10 publications

References 18 publications

On a Model for the Evolution of Morphogens in a Growing Tissue

On a Model for the Evolution of Morphogens in a Growing Tissue

On a singular limit as $$\theta \rightarrow 0$$ for a model for the evolution of morphogens in a growing tissue

Free boundary problem for the role of planktonic cells in biofilm formation and development

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