Spontaneous emergence of cell polarity intrinsically lies at the localization of signaling molecules on a particular region of cell membrane. Such a process necessarily contains a positive feedback loop to amplify the localized cluster. To describe the polarizing process and explore different feedback functions involved, deterministic and stochastic models with non-local kinetics are discussed in this paper. Stochastic Simulation Algorithm (SSA) is used to numerically simulate the polarizing behavior and analytical analysis by the power spectrum is applied to approximate the parameter regime for the spontaneous emergence of cell polarity. Compared to the results from the deterministic model, we can understand how the stochastic effect extends the parameter regime for achieving cell polarization under different types of feedback, including the forms of quadratic function, linear function, and Hill function. Both deterministic and stochastic methods fail to yield the polarity at a low number of molecules. Our results suggest that the parameter region for cell polarization under the Hill function feedback is smaller than that with the quadratic function feedback.
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