Selecting a common direction II. Peak-like solutions representing total alignment of cell clusters

Alexander Mogilner, Leah Edelstein-Keshet, G. Bard Ermentrout

Research output: Contribution to journalArticle

Abstract

The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the objects is essentially deterministic, and the individuals or objects tend to gather in clusters at various orientations. (Numerical solutions show that the angular distribution develops sharp peaks at various discrete orientations.) To understand the behaviour of the deterministic models with analytic tools, we represent the distribution as a number of δ-like peaks. This approximation of a true solution by a set of (infinitely sharp) peaks will be referred to as the peak ansatz. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to a singular perturbation problem which we investigate. We briefly discuss other applications of similar techniques.

Original languageEnglish (US)
Pages (from-to)811-842
Number of pages32
JournalJournal of Mathematical Biology
Volume34
Issue number8
StatePublished - 1996

Fingerprint

Alignment
Angular distribution
Cell
cells
Singular Perturbation Problems
Deterministic Model
Limiting
Numerical Solution
Tend
Angle
Motion
Object
Direction compound
Dependent
Approximation
methodology

Keywords

  • Orientation selection
  • Parallel cells
  • Peak ansatz
  • Total alignment

ASJC Scopus subject areas

  • Agricultural and Biological Sciences (miscellaneous)
  • Mathematics (miscellaneous)

Cite this

Selecting a common direction II. Peak-like solutions representing total alignment of cell clusters. / Mogilner, Alexander; Edelstein-Keshet, Leah; Ermentrout, G. Bard.

In: Journal of Mathematical Biology, Vol. 34, No. 8, 1996, p. 811-842.

Research output: Contribution to journalArticle

@article{5f04e73c81cd44e2898457e61fc1d625,
title = "Selecting a common direction II. Peak-like solutions representing total alignment of cell clusters",
abstract = "The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the objects is essentially deterministic, and the individuals or objects tend to gather in clusters at various orientations. (Numerical solutions show that the angular distribution develops sharp peaks at various discrete orientations.) To understand the behaviour of the deterministic models with analytic tools, we represent the distribution as a number of δ-like peaks. This approximation of a true solution by a set of (infinitely sharp) peaks will be referred to as the peak ansatz. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to a singular perturbation problem which we investigate. We briefly discuss other applications of similar techniques.",
keywords = "Orientation selection, Parallel cells, Peak ansatz, Total alignment",
author = "Alexander Mogilner and Leah Edelstein-Keshet and Ermentrout, {G. Bard}",
year = "1996",
language = "English (US)",
volume = "34",
pages = "811--842",
journal = "Journal of Mathematical Biology",
issn = "0303-6812",
publisher = "Springer Verlag",
number = "8",

}

TY - JOUR

T1 - Selecting a common direction II. Peak-like solutions representing total alignment of cell clusters

AU - Mogilner, Alexander

AU - Edelstein-Keshet, Leah

AU - Ermentrout, G. Bard

PY - 1996

Y1 - 1996

N2 - The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the objects is essentially deterministic, and the individuals or objects tend to gather in clusters at various orientations. (Numerical solutions show that the angular distribution develops sharp peaks at various discrete orientations.) To understand the behaviour of the deterministic models with analytic tools, we represent the distribution as a number of δ-like peaks. This approximation of a true solution by a set of (infinitely sharp) peaks will be referred to as the peak ansatz. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to a singular perturbation problem which we investigate. We briefly discuss other applications of similar techniques.

AB - The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the objects is essentially deterministic, and the individuals or objects tend to gather in clusters at various orientations. (Numerical solutions show that the angular distribution develops sharp peaks at various discrete orientations.) To understand the behaviour of the deterministic models with analytic tools, we represent the distribution as a number of δ-like peaks. This approximation of a true solution by a set of (infinitely sharp) peaks will be referred to as the peak ansatz. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to a singular perturbation problem which we investigate. We briefly discuss other applications of similar techniques.

KW - Orientation selection

KW - Parallel cells

KW - Peak ansatz

KW - Total alignment

UR - http://www.scopus.com/inward/record.url?scp=0029681920&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029681920&partnerID=8YFLogxK

M3 - Article

C2 - 8858852

AN - SCOPUS:0029681920

VL - 34

SP - 811

EP - 842

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 8

ER -