A FUNCTIONAL DIFFERENTIAL EQUATION MODEL FOR BIOLOGICAL CELL SORTING DUE TO DIFFERENTIAL ADHESION

2012 ◽  
Vol 23 (01) ◽  
pp. 93-126 ◽  
Author(s):  
GREG LEMON ◽  
JOHN R. KING
Keyword(s):  
Mathematical Model ◽  
Cell Sorting ◽  
Biological Tissue ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Cell Types ◽  
Equation Model ◽  
Differential Adhesion ◽  
Novel Approach ◽  
Functional Differential

This paper presents a mathematical model to describe the sorting of two different types of cells, arising from differential adhesion mechanisms within biological tissue. The model is based on a continuum approach that takes into account individual cell behavior including aspects of the cell-migration process, dynamics of the adhesions between contacting cells, and finite cell size. Numerical solutions and bifurcation analyses for the case of a collection of two different cell types show a variety of behaviors observed in experiments, including spatially uniform mixing of cells and the formation of two distinct, containing both types of cells or just one. The mathematical model, which is in the form of a set of functional differential equations, represents a novel approach to continuum modeling of cell sorting and migration within biological tissue.

scholarly journals Model-Based Prediction of an Effective Adhesion Parameter Guiding Multi-Type Cell Segregation

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1378
Author(s):  
Philipp Rossbach ◽  
Hans-Joachim Böhme ◽  
Steffen Lange ◽  
Anja Voss-Böhme
Keyword(s):  
Cell Sorting ◽  
Cell Types ◽  
Series Data ◽  
Analytical Prediction ◽  
Differential Migration ◽  
Migration Model ◽  
Differential Adhesion ◽  
Cell Segregation ◽  
Differential Adhesion Hypothesis ◽  
The Impact

The process of cell-sorting is essential for development and maintenance of tissues. Mathematical modeling can provide the means to analyze the consequences of different hypotheses about the underlying mechanisms. With the Differential Adhesion Hypothesis, Steinberg proposed that cell-sorting is determined by quantitative differences in cell-type-specific intercellular adhesion strengths. An implementation of the Differential Adhesion Hypothesis is the Differential Migration Model by Voss-Böhme and Deutsch. There, an effective adhesion parameter was derived analytically for systems with two cell types, which predicts the asymptotic sorting pattern. However, the existence and form of such a parameter for more than two cell types is unclear. Here, we generalize analytically the concept of an effective adhesion parameter to three and more cell types and demonstrate its existence numerically for three cell types based on in silico time-series data that is produced by a cellular-automaton implementation of the Differential Migration Model. Additionally, we classify the segregation behavior using statistical learning methods and show that the estimated effective adhesion parameter for three cell types matches our analytical prediction. Finally, we demonstrate that the effective adhesion parameter can resolve a recent dispute about the impact of interfacial adhesion, cortical tension and heterotypic repulsion on cell segregation.

scholarly journals Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Josef Rebenda ◽  
Zuzana Pátíková
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Recurrence Relation ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Neutral System ◽  
Initial Value ◽  
Differential Transformation ◽  
Functional Differential

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

scholarly journals Numerical Solutions of Stochastic Functional Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 141-161 ◽  
Author(s):  
Xuerong Mao
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Convergence Theory ◽  
Stochastic Functional Differential Equations ◽  
True Solution ◽  
Linear Growth Condition ◽  
Mean Square Convergence ◽  
Functional Differential ◽  
Stochastic Functional

AbstractIn this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the linear growth condition. These two conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here were obtained under quite general conditions.

scholarly journals Approximations of Numerical Method for Neutral Stochastic Functional Differential Equations with Markovian Switching

2012 ◽  
Vol 2012 ◽  
pp. 1-32
Author(s):  
Hua Yang ◽  
Feng Jiang
Keyword(s):  
Differential Equations ◽  
Stochastic Systems ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Approximate Solutions ◽  
Markovian Switching ◽  
Stochastic Functional Differential Equations ◽  
Em Method ◽  
Functional Differential ◽  
Stochastic Functional

Stochastic systems with Markovian switching have been used in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. In this paper, we study the Euler-Maruyama (EM) method for neutral stochastic functional differential equations with Markovian switching. The main aim is to show that the numerical solutions will converge to the true solutions. Moreover, we obtain the convergence order of the approximate solutions.

scholarly journals Self-* properties through gossiping

2008 ◽  
Vol 366 (1881) ◽  
pp. 3747-3757 ◽  
Author(s):  
Ozalp Babaoglu ◽  
Márk Jelasity
Keyword(s):  
Cell Sorting ◽  
Mobile Agents ◽  
Large Scale ◽  
Overlay Networks ◽  
Biological Mechanism ◽  
Human Intervention ◽  
Differential Adhesion ◽  
Novel Approach ◽  
Large Numbers ◽  
Ubiquitous Systems

As computer systems have become more complex, numerous competing approaches have been proposed for these systems to self-configure, self-manage, self-repair, etc. such that human intervention in their operation can be minimized. In ubiquitous systems, this has always been a central issue as well. In this paper, we overview techniques to implement self-* properties in large-scale, decentralized networks through bio-inspired techniques in general, and gossip-based algorithms in particular. We believe that gossip-based algorithms could be an important inspiration for solving problems in ubiquitous computing as well. As an example, we outline a novel approach to arrange large numbers of mobile agents (e.g. vehicles, rescue teams carrying mobile devices) into different formations in a totally decentralized manner. The approach is inspired by the biological mechanism of cell sorting via differential adhesion, as well as by our earlier work in self-organizing peer-to-peer overlay networks.

Numerical Method of Hybrid Stochastic Functional Differential Equations with the Local Lipschitz Coefficients

2011 ◽  
Vol 267 ◽  
pp. 422-426
Author(s):  
Hua Yang ◽  
Feng Jiang ◽  
Jun Hao Hu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Lipschitz Condition ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Stochastic Functional

Recently, hybrid stochastic differential equations have received a great deal of attention. It is surprising that there are not any numerical schemes established for the hybrid stochastic functional differential equations. In this paper, the Euler—Maruyama method is developed, and the main aim is to show that the numerical solutions will converge to the true solutions under the local Lipschitz condition. The result obtained generalizes the earlier results.

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