On the spontaneous emergence of cell polarity

Abstract

Diverse cell polarity networks require positive feedback for locally amplifying distributions of signalling molecules at the plasma membrane. Additional mechanisms, such as directed transport or coupled inhibitors, 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 genetics, 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.

Figure 1

A conceptual model of a positive feedback circuit is characterized by five biologically interpretable parameters. Colored arrows indicate four main sources of molecular transport (see Main Text; purple: spontaneous membrane association, or “input” to positive feedback circuit, with rate k_on; black: spontaneous membrane disassociation, with rate k_off; red: positive feedback, with rate k_fb; blue: lateral diffusion, with rate D). Not shown is N, the total number of signalling molecules.

Figure 2

Numerical simulations reveal an inverse dependence between polarization frequency and large numbers of signalling molecules. a. Positive feedback alone leads to homogeneous steady-state solutions for large numbers of signalling molecules (Methods). Inset: Evolution of total membrane fraction h(t); theoretical equilibrium value h_eq = 0.1. b. Decreasing numbers of signalling molecules leads to increasing levels of spatial segregation. Shown are kymographs from simulations (vertical-axis: membrane position; horizontal-axis: time). Particle density is rescaled to units of equilibrium fraction (color bar). c. Small k_on/k_fb can lead to the establishment of a dominant clan (Text Box 1). Clan color assignment in kymograph is random, and is reset for each molecule after membrane disassociation. Model parameters are as in Methods.

Figure 3

Dependence of polarization on model parameters. a. Illustration of phase-plane portrait. Theory reveals robust parameter regimes for cluster formation (Supplemental Materials): 1-polarization; 2-no polarization; 3-transition zone; 4-too few signalling molecules for polarization (Figure S2). b. Frequency of observing Cdc42 polarization in yeast for increasing numbers of molecules (N). Shown are polarization frequencies estimated from numerical simulations (Top panel) and experimental observation of yeast cells expressing GFP-Cdc42 (Middle panel) (Methods and Figure S3). Green error bars represent standard error from four independent replicates. Bottom panel: Examples of polarized and unpolarized yeast cells. Arrows point to polarized regions; scale bar 1.9 μm.