On non-equilibrium stochastic dynamics for interacting particle systems in continuum

Kondratiev, Yuri; Kutoviy, Oleksandr; Minlos, R. A.

doi:10.1016/j.jfa.2007.12.006

Cited by 44 publications

(59 citation statements)

References 23 publications

(39 reference statements)

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“…IPS models require that the space is discretized, which is a disadvantage with regard to both the biological situation and some technicalities in the model specification and analysis. However, currently the mathematical theory for analogous modeling problems in continuous space is not developed enough to set up and analyze a cell sorting model within this framework [Kondratiev, Kutoviy and Minlos, 2008]. It remains open -and is actually a subject of current mathematical research -to what extent the spatial discretization affects the model predictions.…”

Section: Biological Foundation Of the Modeling Assumptionsmentioning

confidence: 99%

The cellular basis of cell sorting kinetics

Voß-Böhme

Deutsch

2010

Journal of Theoretical Biology

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Cell sorting is a dynamical cooperative phenomenon that is fundamental for tissue morphogenesis and tissue homeostasis. According to Steinberg's differential adhesion hypothesis, the structure of sorted cell aggregates is determined by physical characteristics of the respective tissues, the tissue surface tensions. Steinberg postulated that tissue surface tensions result from quantitative differences in intercellular adhesion. Several experiments in cell cultures as well as in developing organisms support this hypothesis.The question of how tissue surface tension might result from differential adhesion was addressed in some theoretical models. These models describe the cellular interdependence structure once the temporal evolution has stabilized. In general, these models are capable of reproducing sorted patterns. However the model dynamics at the cellular scale are defined implicitly and are not welljustified. The precise mechanism describing how differential adhesion generates the observed sorting kinetics at the tissue level is still unclear.It is necessary to formulate the concepts of cell level kinetics explicitly. Only then it is possible to understand the temporal development at the cellular and tissue scales. Here we argue that individual cell mobility is reduced the more the cells stick to their neighbors. We translate this assumption into a precise mathematical model which belongs to the class of stochastic interacting particle systems. Analyzing this model, we are able to predict the emergent sorting behavior at the population level. We describe qualitatively the geometry of cell segregation depending on the intercellular adhesion parameters. Furthermore, we derive a functional relationship between intercellular adhesion and surface tension and highlight the role of cell mobility in the process of sorting. We show that the interaction between the cells and the boundary of a confining vessel has a major impact on the sorting geometry.Keywords: differential adhesion hypothesis, tissue surface tension, phase segregation, stochastic lattice gas, interacting particle system 2000 MSC: 92C15, 92C17, 92C37 * Corresponding author Email addresses: anja.voss-boehme@tu-dresden.de (A. Voß-Böhme), andreas.deutsch@tu-dresden.de (A. Deutsch) Preprint submitted to Elsevier 4th December 2009A c c e p t e d m a n u s c r i p t

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Section: Biological Foundation Of the Modeling Assumptionsmentioning

confidence: 99%

The cellular basis of cell sorting kinetics

Voß-Böhme

Deutsch

2010

Journal of Theoretical Biology

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“…Within probability theory, a mathematically rigorous toolbox for the analysis of spatiotemporal point processes has recently started to emerge (Kondratiev et al 2006b(Kondratiev et al , 2008a(Kondratiev et al , b, 2010Kondratiev and Skorokhod 2006;Finkelshtein et al 2012), but these tools have attracted little attention in the ecological literature, for two reasons. First, the mathematical literature is written in a highly technical notation, making it difficult for many theoretical ecologists even to assess the relevance of the results.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, in earlier work in theoretical ecology (e.g., Bolker and Pacala 1997;Law et al 2003;Ovaskainen and Cornell 2006b;North et al 2011a), spatial moment equations were derived separately for each of the orders. In contrast, the formalism used by the probabilists (e.g., Kondratiev et al 2008a;Finkelshtein et al 2009) allows one to derive exact equations for spatial moments simultaneously of all orders.…”

Section: Introductionmentioning

confidence: 99%

A general mathematical framework for the analysis of spatiotemporal point processes

et al. 2013

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heuristic rather than mathematically justified assumptions, so that, e.g., the choice of the moment closure can be considered more of an art than a science. In this paper, we build on recent developments in specific branch of probability theory, Markov evolutions in the space of locally finite configurations, to develop a mathematically rigorous and practical framework that we expect to be widely applicable for theoretical ecology. In particular, we show how spatial moment equations of all orders can be systematically derived from the underlying individual-based assumptions. Further, as a new mathematical development, we go beyond mean-field theory by discussing how spatial moment equations can be perturbatively expanded around the mean-field model. While we have suggested such a perturbation expansion in our previous research, the present paper gives a rigorous mathematical justification. In addition to bringing mathematical rigor, the application of the mathematically well-established framework of Markov evolutions allows one to derive perturbation expansions in a transparent and systematic manner, which we hope will facilitate the application of the methods in theoretical ecology.

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“…States of infinite systems are mostly such that μ( 0 ) = 0, which makes direct solving (1.3) with an arbitrary initial state μ 0 rather unaccessible for the method existing at this time, cf. [20]. In this work we continue following the statistical approach [5,10,12,13,20] in which the evolution of states is described by means of the corresponding correlation functions.…”

Section: Posingmentioning

confidence: 99%

“…[20]. In this work we continue following the statistical approach [5,10,12,13,20] in which the evolution of states is described by means of the corresponding correlation functions. To briefly explain its essence let us consider the set of all compactly supported continuous functions θ : R d → (−1, 0].…”

Section: Posingmentioning

confidence: 99%

The Evolution of States in a Spatial Population Model

Kondratiev

Kozitsky

2016

J Dyn Diff Equat

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The evolution of states in a spatial population model is studied. The model describes an infinite system of point entities in R d which reproduce themselves at distant points (disperse) and die with rate that includes a competition term. The system's states are probability measures on the space of configurations, and their evolution is obtained from a hierarchical chain of differential equations for the corresponding correlation functions derived from the Fokker-Planck equation for the states. Under natural conditions imposed on the model parameters it is proved that the correlation functions evolve in a scale of Banach spaces in such a way that at each moment of time the correlation function corresponds to a unique sub-Poissonian state. Some further properties of the evolution of states constructed in this way are described.

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On non-equilibrium stochastic dynamics for interacting particle systems in continuum

Abstract: We propose a general scheme for construction of Markov stochastic dynamics on configuration spaces in continuum. An application to the Glauber-type dynamics with competitions is considered.

Cited by 44 publications

References 23 publications

The cellular basis of cell sorting kinetics

The cellular basis of cell sorting kinetics

A general mathematical framework for the analysis of spatiotemporal point processes

The Evolution of States in a Spatial Population Model

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