A Classical WR Model with $$q$$ q Particle Types

Mazel, A.; Suhov, Yuri; Stuhl, Izabella

doi:10.1007/s10955-015-1219-8

Cited by 21 publications

(26 citation statements)

References 22 publications

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“…The melanoma cells are supposed to have a lower level of contact inhibition than the keratinocyte cells. A mathematical mechanism of a stochastic nature has been considered, based on a Widom-Rowlinson-type model and its modifications [24][25][26] . The model has its own subtleties as stressed in the supplementary information.…”

Section: Discussionmentioning

confidence: 99%

“…A mathematical model explaining the above experimental results is related to the Widom-Rowlinson model [24,25]. Since the data refer to a monolayered co-culture arrangement, we can focus on a twodimensional system composed of two cell types interacting as 'colored' billiard balls of different sizes, where each cell type has two exclusion areas surrounding it (one for balls of the same color, the other for balls of the different color).…”

Section: A Stochastic Model Of Cell Proliferation For a Cell-to-cell mentioning

confidence: 99%

“…The melanoma cells are supposed to have a lower level of contact inhibition than the keratinocyte cells. A mathematical mechanism of a stochastic nature has been considered, based on a Widom-Rowlinson-type model [27] and its modification [25]. This is presented as a Markov chain on admissible cell configurations with specified exclusion diameters for the keratinocyte (healthy) and melanoma (tumor) cells; in the simulations we use the same rates for supposed transitions for cells of both types: division, death, and migration.…”

Section: R a F Tmentioning

confidence: 99%

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Stochastic model of contact inhibition and the proliferation of melanomain situ

Morais

Stuhl

Sabino

et al. 2017

Preprint

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Contact inhibition is a central feature orchestrating cell proliferation in culture experiments; its loss is associated with malignant transformation and tumorigenesis. We performed a co-culture experiment with human metastatic melanoma cell line (SK-MEL-147) and immortalized keratinocyte cells (HaCaT). After 8 days a spatial pattern was detected, characterized by the formation of clusters of melanoma cells surrounded by keratinocytes constraining their proliferation. In addition, we observed that the proportion of melanoma cells within the total population has increased. To explain our results we propose a spatial stochastic model (following a philosophy of the Widom-Rowlinson model from Statistical Physics and Molecular Chemistry) which considers cell proliferation, death, migration, and cell-to-cell interaction through contact inhibition. Our numerical simulations demonstrate that loss of contact inhibition is a sufficient mechanism, appropriate for an explanation of the increase in the proportion of tumor cells and generation of spatial patterns established in the conducted experiments.

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“…The melanoma cells are supposed to have a lower level of contact inhibition than the keratinocyte cells. A mathematical mechanism of a stochastic nature has been considered, based on a Widom-Rowlinson-type model and its modifications [24][25][26] . The model has its own subtleties as stressed in the supplementary information.…”

Section: Discussionmentioning

confidence: 99%

Section: A Stochastic Model Of Cell Proliferation For a Cell-to-cell mentioning

confidence: 99%

“…The melanoma cells are supposed to have a lower level of contact inhibition than the keratinocyte cells. A mathematical mechanism of a stochastic nature has been considered, based on a Widom-Rowlinson-type model [27] and its modification [25]. This is presented as a Markov chain on admissible cell configurations with specified exclusion diameters for the keratinocyte (healthy) and melanoma (tumor) cells; in the simulations we use the same rates for supposed transitions for cells of both types: division, death, and migration.…”

Section: R a F Tmentioning

confidence: 99%

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Stochastic model of contact inhibition and the proliferation of melanomain situ

Morais

Stuhl

Sabino

et al. 2017

Preprint

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“…To obtain a quantitative description of our model we consider the a variation of the Widom-Rowlinson model [45][46][47]. A cartoon of our model is presented in Fig.…”

Section: Cell Level Modelsmentioning

confidence: 99%

Some lessons and perspectives for applications of stochastic models in biological and cancer research

Sabino

Vasconcelos

Sittoni

et al. 2018

Preprint

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Randomness is an unavoidable feature of inner cellular environment and its effects propagate to higher levels of living matter organization such as cells, tissues, and organisms. Approaching those systems experimentally to understand their dynamics is a complex task because of the plethora of compounds interacting in a web that combines intra and inter level elements such that a coordinate behavior come up. Such a characteristic points to the necessity of establishing principles that help on the description, categorization, classification, and the prediction of the behavior of biological systems. The theoretical machinery already available, or the ones to be discovered motivated by biological problems, can play an important role on that quest. Here we exemplify the applicability of theoretical tools by discussing some biological problems that we have approached mathematically: fluctuations in gene expression and cell proliferation in the context of loss of contact inhibition. We discuss the methods that we have employed aiming to provide the reader with a phenomenological, biologically motivated, perspective of the use of theoretical methods. Furthermore, we discuss some of our conclusions after employing our approach and some research perspectives that they motivate.

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“…As far as we know, this model and another model with a particular Kac type potential treated in [17] are the only models, in the continuum setting without spin, for which a non-uniqueness result is proved. Note also that the Area-interaction have been abundantly studied by researchers from different communities in statistical physics, probability theory or spatial statistics [1,5,13,19].…”

Section: Introductionmentioning

confidence: 99%

Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

Dereudre

Houdebert

2018

J Stat Phys

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The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in R d with the formal Hamiltonian defined as the volume of ∪ x∈ω B 1 (x), where ω is a locally finite configuration of points and B 1 (x) denotes the unit closed ball centred at x. The model is also tuned by two other parameters: the activity z > 0 related to the intensity of the process and the inverse temperature β ≥ 0 related to the strength of the interaction. In the present paper we investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than 2r > 0, we show that for any β ≥ 0, there exists 0 < z a c (β, r) < +∞ such that an exponential decay of connectivity at distance n occurs in the subcritical phase (i.e. z < z a c (β, r)) and a linear lower bound of the connection at infinity holds in the supercritical case (i.e. z > z a c (β, r)). These results are in the spirit of recent works using the theory of randomised tree algorithms [7,9,8]. Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters z, β. Old results [22,24] claim that a non-uniqueness regime occurs for z = β large enough and it is conjectured that the uniqueness should hold outside such an half line (z = β ≥ β c > 0). We solve partially this conjecture in any dimension by showing that for β large enough the non-uniqueness holds if and only if z = β. We show also that this critical value z = β corresponds to the percolation threshold z a c (β, r) = β for β large enough, providing a straight connection between these two notions of phase transition.

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A Classical WR Model with $$q$$ q Particle Types

Abstract: A version of the Widom-Rowlinson model is considered, where particles of q types coexist, subject to pairwise hard-core exclusions. For q ≤ 4, in the case of large equal fugacities, we give a complete description of the pure phase picture based on the theory of dominant ground states.

Cited by 21 publications

References 22 publications

Stochastic model of contact inhibition and the proliferation of melanomain situ

Stochastic model of contact inhibition and the proliferation of melanomain situ

Some lessons and perspectives for applications of stochastic models in biological and cancer research

Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

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