AIMS Biophysics, 2016, 3(1): 119-145. doi: 10.3934/biophy.2016.1.119.

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Patterning of the MinD cell division protein in cells of arbitrary shape can be predicted using a heuristic dispersion relation

1 School of Physics, University of New South Wales, Sydney NSW 2052, Australia
2 The ithree institute, University of Technology Sydney NSW 2007, Australia
3 School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia

Many important cellular processes require the accurate positioning of subcellular structures. Underpinning many of these are protein systems that spontaneously generate spatiotemporal patterns. In some cases, these systems can be described by non-linear reaction-diffusion equations, however, a full description of such equations is rarely available. A well-studied patterning system is the Min protein system that underpins the positioning of the FtsZ contractile ring during cell division in Escherichia coli. Using a coordinate-free linear stability analysis, the reaction terms can be separated from the geometry of a cell. The reaction terms produce a dispersion relation that can be used to predict patterning on any cell shape and size. Applying linear stability analysis to an accurate mathematical model of the Min system shows that while it correctly predicts the onset of patterning, the dispersion relation fails to predict oscillations and quantitative mode transitions. However, we show that data from full solutions of the Min model can be used to generate a heuristic dispersion relation. We show that this heuristic dispersion relation can be used to approximate the Min protein patterning obtained by full simulations of the non-linear reaction-diffusion equations. Moreover, it also predicts Min patterning obtained from experiments where the shapes of E. coli cells have been deformed into rectangles or arbitrary shapes. Using this procedure, it should be possible to generate heuristic dispersion relations from protein patterning data or simulations for any patterning process and subsequently use these to predict patterning for arbitrary cell shapes.
  Article Metrics

Keywords reaction-diffusion; pattern formation; cell division; Min system; Turing patterns

Citation: James C. Walsh, Christopher N. Angstmann, Anna V. McGann, Bruce I. Henry, Iain G. Duggin, Paul M. G. Curmi. Patterning of the MinD cell division protein in cells of arbitrary shape can be predicted using a heuristic dispersion relation. AIMS Biophysics, 2016, 3(1): 119-145. doi: 10.3934/biophy.2016.1.119


  • 1. Turing AM (1952) The Chemical Basis of Morphogenesis. Philos T R Soc B 237: 37–72.    
  • 2. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetika 12: 30–39.    
  • 3. Murray JD (1990) Developmental Biology. B Math Biol 52: 119–152.    
  • 4. Rudner DZ, Losick R (2010) Protein subcellular localization in bacteria. CSH Perspect Biol 2: a000307.
  • 5. Ames P, Parkinson JS (2006) Conformational suppression of inter-receptor signaling defects. Proc Natl Acad Sci U S A 103: 9292–9297.    
  • 6. Alley MR, Maddock JR, Shapiro L (1992) Polar localization of a bacterial chemoreceptor. Genes Dev 6: 825–836.    
  • 7. Errington J (2003) Regulation of endospore formation in Bacillus subtilis. Nat Rev Microbiol 1: 117–126.    
  • 8. Stragier P, Losick R (1996) Molecular genetics of sporulation in Bacillus subtilis. Annu Rev Genet 30: 297–241.    
  • 9. Driks A, Losick R (1991) Compartmentalized expression of a gene under the control of sporulation transcription factor sigma E in Bacillus subtilis. Proc Natl Acad Sci U S A 88: 9934–9938.    
  • 10. Margolin W (2005) FtsZ and the division of prokaryotic cells and organelles. Nat Rev Mol Cell Biol 6: 862–871.    
  • 11. Bi EF, Lutkenhaus J (1991) FtsZ ring structure associated with division in Escherichia coli. Nature 354: 161–164.    
  • 12. de Boer PA (2010) Advances in understanding E coli cell fission. Curr Opin Microbiol 13: 730–737.    
  • 13. Harry E, Monahan L, Thompson L (2006) Bacterial cell division: the mechanism and its precision. Int Rev Cytol 253: 27–94.    
  • 14. Loose M, Fischer-Friedrich E, Ries J, et al. (2008) Spatial regulators for bacterial cell division self-organize into surface waves in vitro. Science 320: 789–792.    
  • 15. Zaikin AN, Zhabotinsky AM (1970) Concentration wave propagation in two-dimensional liquid-phase self-oscillating system. Nature 225: 535–537.
  • 16. Meacci G, Kruse K (2005) Min-oscillations in Escherichia coli induced by interactions of membrane-bound proteins. Phys Biol 2: 89–97.    
  • 17. Huang KC, Meir Y, Wingreen NS (2003) Dynamic structures in Escherichia coli: spontaneous formation of MinE rings and MinD polar zones. Proc Natl Acad Sci U S A 100: 12724–12728.    
  • 18. Kruse K (2002) A dynamic model for determining the middle of Escherichia coli. Biophys J 82: 618–627.    
  • 19. Meinhardt H, de Boer PA (2001) Pattern formation in Escherichia coli: a model for the pole-to-pole oscillations of Min proteins and the localization of the division site. Proc Natl Acad Sci U S A 98: 14202–14207.    
  • 20. Walsh JC, Angstmann CN, Duggin IG, et al. (2015) Molecular Interactions of the Min Protein System Reproduce Spatiotemporal Patterning in Growing and Dividing Escherichia coli Cells. PLoS One 10: e0128148.    
  • 21. Moseley JB, Nurse P (2010) Cell division intersects with cell geometry. Cell 142: 184–188.    
  • 22. Murray JD (2002) Mathematical biology, 3rd ed, New York: Springer.
  • 23. Wu F, van Schie BG, Keymer JE, et al. (2015) Symmetry and scale orient Min protein patterns in shaped bacterial sculptures. Nat Nanotechnol 10: 719–726.    
  • 24. Mannik J, Wu F, Hol FJ, et al. (2012) Robustness and accuracy of cell division in Escherichia coli in diverse cell shapes. Proc Natl Acad Sci U S A 109: 6957–6962.    
  • 25. Wu W, Park KT, Holyoak T, et al. (2011) Determination of the structure of the MinD-ATP complex reveals the orientation of MinD on the membrane and the relative location of the binding sites for MinE and MinC. Mol Microbiol 79: 1515–1528.    
  • 26. Mileykovskaya E, Fishov I, Fu X, et al. (2003) Effects of phospholipid composition on MinD-membrane interactions in vitro and in vivo. J Biol Chem 278: 22193–22198.    
  • 27. Taghbalout A, Ma L, Rothfield L (2006) Role of MinD-membrane association in Min protein interactions. J Bacteriol 188: 2993–3001.    
  • 28. Park KT, Wu W, Battaile KP, et al. (2011) The Min oscillator uses MinD-dependent conformational changes in MinE to spatially regulate cytokinesis. Cell 146: 396–407.    
  • 29. Kang GB, Song HE, Kim MK, et al. (2010) Crystal structure of Helicobacter pylori MinE, a cell division topological specificity factor. Mol Microbiol 76: 1222–1231.    
  • 30. Ghasriani H, Goto NK (2011) Regulation of symmetric bacterial cell division by MinE: What is the role of conformational dynamics? Commun Integr Biol 4: 101–103.    
  • 31. King GF, Shih YL, Maciejewski MW, et al. (2000) Structural basis for the topological specificity function of MinE. Nat Struct Biol 7: 1013–1017.    
  • 32. Park KT, Wu W, Lovell S, et al. (2012) Mechanism of the asymmetric activation of the MinD ATPase by MinE. Mol Microbiol 85: 271–281.    
  • 33. Edelstein-Keshet L (2005) Mathematical models in biology. Philadelphia: Society for Industrial and Applied Mathematics.
  • 34. Woolley TE, Baker RE, Gaffney EA, et al. (2011) Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation. Phys Rev E 84: 046216.    


This article has been cited by

  • 1. James C. Walsh, Christopher N. Angstmann, Iain G. Duggin, Paul M. G. Curmi, Ruben Claudio Aguilar, Non-linear Min protein interactions generate harmonics that signal mid-cell division in Escherichia coli, PLOS ONE, 2017, 12, 10, e0185947, 10.1371/journal.pone.0185947
  • 2. Lukas Wettmann, Mike Bonny, Karsten Kruse, Lev Tsimring, Effects of geometry and topography on Min-protein dynamics, PLOS ONE, 2018, 13, 8, e0203050, 10.1371/journal.pone.0203050
  • 3. James C. Walsh, Christopher N. Angstmann, Alexandre W. Bisson‐Filho, Ethan C. Garner, Iain G. Duggin, Paul M. G. Curmi, Division plane placement in pleomorphic archaea is dynamically coupled to cell shape, Molecular Microbiology, 2019, 10.1111/mmi.14316

Reader Comments

your name: *   your email: *  

Copyright Info: 2016, Paul M. G. Curmi, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved