| Abstract | Finite nonlinear difference equations can be used to study responses of a multi-step biochemical reaction chain to a stimulus at any step of the chain. This reaction chain can be thought of as signaling pathway activation at the receptor level by a stimulus, such as drugs, and the subsequent changes in the protein levels at each step in the reaction chain as responses. In this study, we developed two mathematical models for two hypothetical multi-step reaction chains involving loops to study transient and long term behaviors of these systems as we go down the chain. The first model is for a reaction chain with a negative feedforward loop, and the second one is for a negative feedback loop. Although both of the models have the same steady state equations and values, we saw that negative feedforward and negative feedback loops can produce significantly different behaviors. The former can bring the system into oscillations with various periods if the loop isstrong enough as the length of the reaction chain increases, whereas the latter is not capable of producing oscillations and more complicated dynamics. We also observed that the negative feedforward loop can produce chaotic behavior. Even though our models are simplified assuming that rate constants at each reaction step after receptor activation are equal, our model for the negative feedforward loop can still produce rich dynamics which have to be achieved if one or more of our assumptions are relaxed. |
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