One Receptor-Two Ligands, Stochastic Kinetic Analysis

Santillan, Chemical Kinetics, Stochastic Processes, and Irreversible Thermodynamics, Lecture Notes on Mathematical Modelling in the life Sciences, 

If we consider a single receptor molecule that, due to the binding and xmbinding of ligands X and Y, switches between its four available states (0,0), (X, 0), (0,7), and (X, 7), the master-equation family governing its dynamics is  

A comparison between Eqs.(7.9)-(7.12) and Eqs.(6.18)-(6.21) reveals that cooperativity changes the relative probability of state (X, F). Positive cooperativity kc 1) increases the stationary probability of state X, F). This agrees with the fact that the propensity for the detachment of a ligand (either X or F) decreases in proportion to the value of kc Conversely, the relative probability of state (X, F) decreases in case of negative cooperativity kc )  

Consider now the case in which ny independent receptor molecules interact with a constant number nx of X ligands, and a constant number ny of F ligands. Following the procedure leading to Eq. (6.22) it is straightforward to derive the following master equation for such a system  

This can be tested by differentiating Eq. (7.14), substituting Eqs.(7.9)-(7.12), and proving that the obtained result is equal to the one obtained after substimting Eq. (7.14) into the right-hand side of Eq. (7.13). 

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