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Protein Pattern Formation

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Physics of Biological Membranes

Abstract

Protein pattern formation is essential for the spatial organisation of many intracellular processes like cell division, flagellum positioning, and chemotaxis. A prominent example of intracellular patterns are the oscillatory pole-to-pole oscillations of Min proteins in E. coli whose biological function is to ensure precise cell division. Cell polarisation, a prerequisite for processes such as stem cell differentiation and cell polarity in yeast, is also mediated by a diffusion–reaction process. More generally, these functional modules of cells serve as model systems for self-organisation, one of the core principles of life. Under which conditions spatio-temporal patterns emerge, and how these patterns are regulated by biochemical and geometrical factors are major aspects of current research. Here we review recent theoretical and experimental advances in the field of intracellular pattern formation, focusing on general design principles and fundamental physical mechanisms.

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Notes

  1. 1.

    Of course, such a process would also be limited by the duration of protein synthesis.

  2. 2.

    In general, a given reaction–diffusion equation can generate a plethora of spatio-temporal patterns, as is well known from classical equations like the complex Ginzburg-Landau equation [63] or the Gray-Scott equation [64,65,66,67,68]. Conversely, a given pattern can be produced by a vast variety of mathematical equations. Hence, one must be careful to avoid falling into the trap: “Cum hoc ergo propter hoc” (correlation does not imply causation).

  3. 3.

    It should be noted that the condition on the particle numbers mainly serves to emphasise the sequestration mechanism. In order for MinD to accumulate in polar zones the action of MinE must be disabled, and specifying that there are fewer MinE particles permits them to be spatially confined. Outside of this zone MinD can accumulate on the membrane. Recently, it has been shown that MinE’s conformational switch can transiently attenuate the action of MinE, thereby removing the requirement regarding the relative particle numbers of MinD and MinE [69].

  4. 4.

    This is surprising, because Turing instabilities are generically associated with the existence of a characteristic (or intrinsic) wave length in the literature. This is evidently not the case here.

  5. 5.

    In 1966 Mark Kac published an article entitled “Can one hear the shape of a drum?”[80]. As the dynamics (frequency spectrum) of an elastic membrane whose boundary is clamped is described by the Helmholtz equation ∇2 u + σu = 0 with Dirichlet boundaries, ∇u∣⊥ = 0, this amounts to asking how strongly the eigenvalues σ depend on the shape of the domain boundary. Here we ask a much more intricate question, as the dynamics of pattern-forming systems are nonlinear and we would like to know the nonlinear attractor for a given shape and size of a cell.

  6. 6.

    We note that travelling wave patterns have also been observed in vivo [90], albeit only upon massive over-expression of MinD and MinE, leading to highly elevated intracellular protein densities and pathological phenomenology [91] relative to the wild type. While the exact protein densities in the experiments have not been measured, this observation is consistent with the observation of travelling waves in fully confined compartments, where the protein densities inside microfluidic chambers were also elevated [83]. For further discussion of the effect of protein densities, we refer the reader to Sect. 4.2.

  7. 7.

    Either directly or by complex formation as for MinDE complexes.

  8. 8.

    Assuming a cylindrical geometry for simplicity, the volume-to-surface ratio is ∼r∕2, i.e. well below 1 μm for typical cell radii r.

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Acknowledgements

We thank Fridtjof Brauns, Yaron Caspi, Cees Dekker, Jonas Denk, and Fabai Wu for helpful discussions. This research was supported by the German Excellence Initiative via the program “NanoSystems Initiative Munich” (NIM), and the Deutsche Forschungsgemeinschaft (DFG) via project A09 and B02 within the Collaborative Research Center (SFB 1032) “Nanoagents for spatio-temporal control of molecular and cellular reactions”. SK was supported by a DFG fellowship through QBM.

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Authors and Affiliations

  1. Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, München, Germany

    Erwin Frey & Jacob Halatek

  2. Department of Cellular and Molecular Biophysics, Max-Planck-Institute of Biochemistry, Martinsried, Germany

    Simon Kretschmer & Petra Schwille

  3. Graduate School of Quantitative Biosciences, Ludwig-Maximilians-Universität München, München, Germany

    Simon Kretschmer

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  1. Erwin Frey
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Correspondence to Erwin Frey .

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  1. PhysicoChimie Curie (UMR CNRS 168), Institut Curie, Paris, France

    Patricia Bassereau

  2. PhysicoChimie Curie (UMR CNRS 168), Institut Curie, Paris, France

    Pierre Sens

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Frey, E., Halatek, J., Kretschmer, S., Schwille, P. (2018). Protein Pattern Formation. In: Bassereau, P., Sens, P. (eds) Physics of Biological Membranes. Springer, Cham. https://doi.org/10.1007/978-3-030-00630-3_10

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