A biological signal is transmitted by interactions between signaling molecules in the cell. To date, there have been extensive studies regarding signaling pathways using numerical simulation of kinetic equations that are based on equations of continuity and Fick’s law. To obtain a mathematical formulation of cell signaling, we propose a stability kinetic model of cell biological signaling of a simple two-parameter model based on the kinetics of the diffusion-limiting step. In the present model, the signaling is regulated by the binding of a cofactor, such as ATP. Non-linearity of the kinetics is given by the diffusion fluctuation in the interaction between signaling molecules, which is different from previous works that hypothesized autocatalytic reactions. Numerical simulations showed the presence of a critical concentration of the cofactor beyond which the cell signaling molecule concentration is altered in a chaos-like oscillation with frequency, which is similar to a discontinuous phase transition in physics. Notably, we found that the frequency is given by the logarithm function of the difference of the outside cofactor concentration from the critical concentration. This implies that the outside alteration of the cofactor concentration is transformed into the oscillatory alteration of cell inner signaling. Further, mathematical stability kinetic analysis predicted a discontinuous dynamic phase transition in the critical state at which the cofactor concentration is equivalent to the critical concentration. In conclusion, the present model illustrates a unique feature of cell signaling, and the stability analysis may provide an analytical framework of the cell signaling system and a novel formulation of biological signaling.
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