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Simplified theoretical models of biological processes, such as receptor-ligand binding mediated cellular signaling, can provide crucial mechanistic insight such complex processes. I have developed a simplified minimal model of immune receptor diffusion and clustering in the form of an immunological synapse during the course of antigen recognition by adaptive immune cells. This minimal model is based on the continuous time random walk of receptors with a given waiting time distribution that arises from binding with antigens of varying affinity. In the case of a spherical cell surface, space and time variables get coupled in the waiting time distribution in a non-separable manner leading to a sub-diffusive behavior that depends on the antigen affinity. Results obtained from the analytical model are corroborated by Monte Carlo simulations of the minimal model. In a separate study, I have developed a minimal signaling network that can respond to an external stimulus in a strength-dependent manner. Using Gillespie's stochastic simulation algorithm we show that under the condition of weak stimulus a three-step fast-slow-fast pathway is activated where large cell-to-cell stochastic fluctuation dominates the signaling behavior. Interestingly, results obtained from our minimal model can capture the essential stochastic signaling in a full-scale complex signaling network of apoptotic cell death signaling and thus can serve as a general model of apoptosis signaling.