We report the first study of a network of connected enzyme-catalyzed reactions, with added chemical and enzymatic processes that incorporate the recently developed biochemical filtering steps into the functioning of this biocatalytic cascade. New theoretical expressions are derived to allow simple, few-parameter modeling of network components concatenated in such cascades, both with and without filtering. The derived expressions are tested against experimental data obtained for the realized network's responses, measured optically, to variations of its input chemicals' concentrations with and without filtering processes. We also describe how the present modeling approach captures and explains several observations and features identified in earlier studies of enzymatic processes when they were considered as potential network components for multistep information/signal processing systems.