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Signal transduction, the information processing mechanism in biological cells, is carried out by a network of biochemical reactions. The dynamics of driven biochemical reactions can be studied in terms of nonequilibrium statistical physics. Such systems may also be studied in terms of Shannon's information theory. We combine these two perspectives in this study of the basic units (modules) of cellular signaling: the phosphorylation dephosphorylation cycle (PdPC) and the guanosine triphosphatase (GTPase). We show that the channel capacity is zero if and only if the free energy expenditure of biochemical system is zero. In fact, a positive correlation between the channel capacity and free energy expenditure is observed. In terms of the information theory, a linear signaling cascade consisting of multiple steps of PdPC can function as a distributed “multistage code.” With increasing number of steps in the cascade, the system trades channel capacity with the code complexity. Our analysis shows that while a static code can be molecular structure based, a biochemical communication channel has to have energy expenditure.