We introduce a new type of kinetic model and theory for biological networks, useful for a quantitative description of chemical dynamics in living cells. One advantage of our approach is its applicability to networks producing biomolecules with arbitrary lifetime distributions to which the conventional approaches, such as the classical chemical kinetics, chemical master equation, and chemical Langevin equation, are not applicable. Our approach also enables quantitative investigation into biological networks composed of multi-step or multi-channel reactions whose rates may fluctuate due to their coupling to cell environments. We demonstrate these advantages by providing an unprecedented quantitative explanation of non-classical chemical dynamics observed in various biological systems including single enzymes [1], in vivo motor-protein multiplexes [2], and cell systems with various synthetic gene networks [3]. Time-permitting, we will also introduce a new transport equation governing thermal motion of biomolecules in living cells [4].