Kimograph for a simulation of the full, spatially distributed, chemotaxis system.

<p>In the simulation, before starting to stimulate cells with a uniform concentration of cAMP, the system is left to relax with zero signal until the levels of the relevant factors become stationary. Then, the stimulation is switched on at time , when we also impose a gaussian noise on the uniform concentration background in order to mimick random inhomogeneities. We compare the experimental results reported in Reference <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030977#pone.0030977-Postma1" target="_blank">[19]</a> with the simulations of model (1–8). The kimograph shows the time evolution of simulated PIP3 levels along the major cell perimeter. Time in the simulation is to be compared with time s in the experiment.</p

Model of chemotactic polarization.

<p>With respect to the abstract scheme in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030977#pone-0030977-g001" target="_blank">Fig. 1</a> we have the identification  = PIP3,  = PIP2, REC. The PIP3-rich domain corresponds to the presence of a high concentration of chemoattractant factor.</p

Prototypical model of cell polarization.

<p>A system of receptors transduces an external distribution of chemotactic cues into an internal distribution of activated enzymes , which catalyze the switch of a signaling molecule from an unactivated state to an activated state . A counteracting enzyme transforms the state back into . The network contains a couple of amplifying feedback loops: the signaling molecule activates and acvivates . The signaling molecules , are permanently bound to the cell surface and perform diffusive motions on it, while the , enzymes are free to shuttle between the cytosolic reservoir and the membrane. The result of the polarization process is the formation of separate domains with -rich patches and, respectively, -rich patches.</p

In the equilibrium state the circular patches occupied by the

<p><b> and </b><b> phases have areas, respectively, </b><b> and </b><b>.</b> Here we show the ratio at different values of the stimulation . Curves are plotted from top to bottom with increasing ratio of the initial enzymes quantities . Each curve shows two plateaux that are approximatively independent of the signal . For small the system is dominated by the mutual interaction between and , i.e., by the feedback loop, whilst for large the system is dominated by the interaction with receptors, i.e., by the external signal.</p

Physical analogy: membrane polarization and precipitation from a supersaturated solution.

<p>At initial time, the concentration of some molecule is higher than the critical value , so that a small fluctuation, or an impurity, can easily give rise to the formation of small germs of precipitate. Germs larger than a critical size grow steadily, while germs smaller than are dissolved by diffusion. As the size of the germs grows, the molecule is extracted from the hydrated phase and transferred to the solid phase, moving the concentration closer to the critical value , increasing the value of , and correspondingly slowing down the process of germ growth. Grains that were initially larger than are dissolved, so that larger grains grow at the expense of the smaller grains. Eventually, an equilibrium is reached when and a single large grain of precipitate survives.</p

Growth of the PIP2-rich phase (blue lower patch).

<p>The color scale shows the gradation of PIP2 content: the color is the relative concentration difference between PIP3 and PIP2 at a given site. The system at initial time is in a uniform PIP3-rich phase (red), apart from an initial PIP2-rich seed germ of size larger than the threshold radius. Then, a PIP2-rich patch becomes apparent and its radius saturates to an equilibrium value.</p

We simulated a spatially homogeneous version of the Model.

<p>We mimick the experimental conditions by switching on receptor activation at initial time. The phospholipid concentration field is driven by the slow variation in time of the effective potential , that follows the slow variation of the enzyme ratio . Receptor activation at (blue line) induces a uniform increase of PI3K, PIP3 on the whole plasmamembrane, which corresponds to the appearance of a single potential well centered in the PIP3-rich region. The enzyme ratio decreases, corresponding to PI3K recruitment to the plasmamembrane and PTEN relocation to the cytosol (blue line). When the enzyme ratio crosses the boundary of the bistable region (light blue area) the effective potential develops a secondary potential well centered in the PIP2-rich region.</p

Parameters for simulations of budding yeast.

<p>Exposure to mating pheromone of haploid <i>Saccharomyces cerevisiae</i> cells results in polarized growth towards the mating partner. Proteins involved in signaling, polarization, cell adhesion, and fusion are localized at the tip of the mating cell (shmoo) where fusion will eventually occur. Polarization involves localization of the small GTPase Cdc42 and of its guanine nucleotide exchange factor (GEF), Cdc24. The expression of a constitutively activated form of Cdc42 is sufficient to cause polarization in otherwise nonpolarized cells.</p

Reactions belonging to the signaling pathway of Fig. 1 and corresponding transition rates .

<p>We denote by and the number of free , molecules available in the cytosolic reservoir, by the number of molecules on site that are activated by the external signal via receptors, by the number of molecules on site that are activated via the feedback loop in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030977#pone-0030977-g001" target="_blank">Fig. 1</a>.</p

Parameters used in the simulations of epithelial polarization.

<p>We have simulated Model (1–8) with parameter values compatible with the interactions described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030977#pone-0030977-g010" target="_blank">Fig. 10</a> for the process of epithelial polarization. At initial time the plasmamembrane is in a uniform PIP3-rich state. We then create a circular patch of the PIP2-rich phase of radius and investigate its dynamics to check whether a stable polarization state is attained.</p