Cell Modeling Based on Bubbles with Weighted Membranes

Abstract

Mathematicization for cell modeling provides an effective tool to verify the biological theory, and the existing research mainly focuses on the description of cell structures. This article then addresses the pattern question of cell division or morphogenesis by means of bubble model with weighted membranes. In this study, we show that cell shapes including intersection angles at junction points depend on weights on membranes. For convenience, adhesion and contractile force are considered together as a factor in construction of patterning model. This model is also used to compare with experimental data. And the consistency between our model and experiments is also obtained consequently. This system of differential equations with their boundary conditions theorizes the existing experimental models, and improves the rationality of these models.

Get full access to this article

View all access options for this article.

References

Ahlstrom

E.H.

, and Tiffany

L.H.

1934. The algal genus tetrastrum. Am. J. Bot. 21, 499–507.

Besson

, and Dumais

2011. Universal rule for the symmetric division of plant cells. Proc. Natl. Acad. Sci. U. S. A. 108, 6294–6299.

Chan

H.Y.

, Shivakumar

P.C.

, Clément

, et al. 2017. Patterned cortical tension mediated by N-cadherin controls cell geometric order in the Drosophila eye. eLife. 6; DOI: 10.7554/eLife.22796.

Cox

, Harrison

, Hutchings

, et al. 1994. The shortest enclosure of three connected areas in

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \rm R}^2}$$ \end{document}

. Real Anal. Exch. 20, 313–335.

Dupuy

, Mackenzie

, and Haseloff

2010. Coordination of plant cell division and expansion in a simple morphogenetic system. Proc. Natl. Acad. Sci. U. S. A. 107, 2711–2716.

Foisy

, Alfaro

, Brock

, et al. 1993. The standard double soap bubble in

uniquely minimizes perimeter. Pac. J. Math. 159, 47–59.

Fukushima

, Fujita

, Yamaguchi

, et al. 2015. Oriented cell division shapes carnivorous pitcher leaves of Sarracenia purpurea. Nat. Commun. 6, 6450.

Gawlik

S.R.

, and Millington

W.F.

1969. Pattern formation and the fine structure of the developing cell wall in colonies of Pediastrum boryanum. Am. J. Bot. 56, 1084–1093.

Gemp

I.M.

, Carthew

R.W.

, and Hilgenfeldt

2011. Cadherin-dependent cell morphology in an epithelium: Constructing a quantitative dynamical model. PLoS Comput. Biol. 7, e1002115.

10.

Goldberg

1969. Elementary problems: E2185. Am. Math. Mon. 76, 825.

11.

Haig

2015. Coleochaete and the origin of sporophytes. Am. J. Bot. 102, 417–422.

12.

Hayashi

, and Carthew

R.W.

2004. Surface mechanics mediate pattern formation in the developing retina. Nature, 431, 647–652.

13.

Hilgenfeldt

, Erisken

, and Carthew

R.W.

2008. Physical modeling of cell geometric order in an epithelial tissue. Proc. Natl. Acad. Sci. U. S. A. 105, 907–911.

14.

Käfer

J.K.

, Hayashi

, Mareé

A.F.M.

, et al. 2007. Cell adhesion and cortex contractility determine cell patterning in the Drosophila retina. Proc. Natl. Acad. Sci. U. S. A. 104, 18549–18554.

15.

Karol

K.G.

, McCourt

R.M.

, Cimino

M.T.

, et al. 2001. The Closest living relatives of land plants. Science, 294:2351–2353.

16.

Klamkin

M.S.

1992. Book reviews. SIAM Rev. 34, 335–338.

17.

Krienitz

, and Wachsmuth

1991. Zur variabilität der zellwandornamentierung in der gattung Tetrastrum Chodat (Chlorophyceae, Chlorellales) und einige taxonomische schlußfolgerungen. Arch. Protistenkunde. 139, 39–51.

18.

Maître

J.-L.

, Berthoumieux

, Krens

, et al. 2012. Adhesion functions in cell sorting by mechanically coupling the cortices of adhering cells. Science, 338, 253–256.

19.

Millington

W.F.

, Chubb

G.T.

, and Seed

T.M.

1981. Cell shape in the alga Pediastrum (Hydrodictyaceae; Chlorophyta). Protoplasma, 105, 169–176.

20.

Morgan

2009. Geometric Measure Theory: A Beginner's Guide, 4th ed. Academic Press, London.

21.

Rasmussen

C.G.

, Wright

A.J.

, and Müller

2013. The role of the cytoskeleton and associated proteins in determination of the plant cell division plane. Plant J. 75, 258–269.

22.

Ruhfel

B.R.

, Gitzendanner

M.A.

, Soltis

P.S.

, et al. 2014. From algae to angiosperms-inferring the phylogeny of green plants (Viridiplantae) from 360 plastid genomes. BMC Evol. Biol. 14, 23.

23.

Shi

B.L.

, and Wang

Y.D.

2012. Evolution modeling on growth image of cells. Comput. Simul. 29, 307–311 (in Chinese).

24.

Sinnott

E.M.

, and Bloch

1940. Cytoplasmic behavior during division of vacuolate plant cells. Proc. Natl. Acad. Sci. U. S. A. 26, 223–227.

25.

Smith

L.G.

2001. Plant cell division: Building walls in the right places. Nat. Rev. Mol. Cell Biol. 2, 33–39.

26.

Timme

R.E.

, and Delwiche

C.F.

2010. Uncovering the evolutionary origin of plant molecular processes: Comparison of Coleochaete (Coleochaetales) and Spirogyra (Zygnematales) transcriptomes. BMC Plant Biol. 10, 96.

27.

Wang

Y.D.

, and Cong

J.Y.

2016. An improved model of non-uniform Coleochaete cell division. J. Comput. Biol. 23, 693–709.

28.

Wang

Y.D.

, Dou

M.Y.

, and Zhou

Z.G.

2015. The fencing problem and Coleochaete cell division. J. Math. Biol. 70, 893–912.

29.

Wang

Y.D.

, and Zeng

2018. Analysis on mechanism for edge formation for Coleochaete cell in non-uniform division. J. Math. Pract. Theory (in Chinese), accepted.

30.

Wesley

O.C.

1928. Asexual reproduction in Coleochaete. Bot. Gaz. 86, 1–31.

31.

Wichiramala

2004. Proof of the planar triple bubble conjecture. J. Reine Angew. Math. 567, 1–49.

32.

Wiener

1914. The shortest line dividing an area in a given ratio. Proc. Camb. Philos. Soc. 18, 56–58.

33.

Yang

J.G.

1986. Coleochaete and the origin of terrestrial plants. Chin. Bull. Bot., 4, 114–116) (in Chinese).

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.81 MB