1. Green Center for Systems Biology, Department of Pharmacology and Simmons Cancer Center, University of Texas Southwestern Medical Center, Dallas, Texas 75390, USA
2. Mathematics Department, University of Wisconsin, Madison, Wisconsin 53706, USA

Correspondence to: Steven J. Altschuler¹Lani F. Wu¹ Correspondence and requests for materials should be addressed to S.J.A. (Email: steven.altschuler@utsouthwestern.edu) or L.F.W. (Email: lani.wu@utsouthwestern.edu).

Diverse cell polarity networks require positive feedback for locally amplifying distributions of signalling molecules at the plasma membrane¹. Additional mechanisms, such as directed transport² or coupled inhibitors^{3, 4}, have been proposed to be required for reinforcing a unique axis of polarity. Here we analyse a simple model of positive feedback, with strong analogy to the 'stepping stone' model of population genetics⁵, in which a single species of diffusible, membrane-bound signalling molecules can self-recruit from a cytoplasmic pool. We identify an intrinsic stochastic mechanism through which positive feedback alone is sufficient to account for the spontaneous establishment of a single site of polarity. We find that the polarization frequency has an inverse dependence on the number of signalling molecules: the frequency of polarization decreases as the number of molecules becomes large. Experimental observation of polarizing Cdc42 in budding yeast is consistent with this prediction. Our work suggests that positive feedback can work alone or with additional mechanisms to create robust cell polarity.

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