Geometric cues stabilise long-axis polarisation of PAR protein patterns in C. elegans
Raphaela Geßele, Jacob Halatek, Laeschkir Würthner, Erwin Frey
Many different processes during development and growth, such as cell division, proliferation and motility, depend on spatial cues for proper functioning. Intracellular signaling proteins establish the required spatial orientation by forming polarization patterns along specific axes of the cell body.
In the nematode worm C. elegans, polarization of the zygote is defined by domains of membrane-bound PAR(titioning defective) proteins. Despite free diffusion of the PAR proteins throughout the zygote, the polarized distribution is stably established and maintained along the long axis and directs the first, asymmetric cell division. The mechanistic basis for the formation of the two opposing protein domains on the membrane is the mutual antagonism between two sub-groups of PARs, aPARs (anterior) and pPARs (posterior), which mutually phosphorylate each. Theoretical models for the PAR system have verified that mutual antagonism stabilizes cell polarity. However, we find that mutual antagonism does not stabilize the long axis as polarity axis in cellular geometry. We therefore ask: “Why is polarization along the anterior-posterior axis preferred over polarization along the dorsal-ventral axis?”, and more generally: “What are the biomolecular processes that control axis selection?”
To answer this question, we formulate a reaction-diffusion model in realistic, ellipsoidal cell geometry, based on known biomolecular reactions and fully accounting for the coupling between membrane and cytosolic dynamics. A step by step investigation of the model in two- and three-dimensional ellipsoidal geometry enables us to identify distinct mechanisms for polarity axis selection. We find that the local ratio of membrane surface to cytosolic volume is the main geometric cue that initiates pattern formation, and the decisive parameter that determines initial axis selection is the ratio between the diffusive length of the phosphorylated protein state to cell length. By altering protein numbers, we furthermore find that a reservoir of active proteins counteracts this phosphorylated state. However, performing a full three-dimensional analysis of the pattern formation process in realistic cell geometry we have identified another crucial principle for the selection of the final polarisation axis: the length of the aPAR-pPAR interface on the membrane is always minimised. This process is mediated by flux minimisation and ensures robust long-axis polarization.
We believe that the above principles are generic and should apply broadly to intracellular pattern-forming systems. Moreover, our analysis suggests that selection of a characteristic wavelength and selection of a polarity axis are distinct phenomena. They are, in general, mediated by different underlying mechanisms. We expect the following findings to be generic for mass-conserved intracellular protein systems: (i) cells sense the cellular geometry by means of the local membrane to bulk ratio, (ii) an activation-deactivation cycle impacts the axis of a pattern, (iii) cytosolic protein reservoirs alter the sensitivity to cell geometry, and last but not least (iv) the specific topology of patterns in higher dimensions and their respective protein fluxes strongly impact which patterns stabilize.