A pressure-driven valve is disclosed. The valve includes a housing and a flow control portion disposed within the housing. The housing includes a fluid inlet and a fluid outlet. The flow control portion has a first side subject to fluid flow pressure in a fluid flow channel, and a second side subject to an outlet pressure representative of pressure at the fluid outlet. The flow control portion is deformable to increase and decrease flow through the fluid flow channel based on pressure differentials between the fluid flow pressure, a tube pressure, and the outlet pressure. In some instances, the flow control portion comprises a flow control membrane and a radially-fluctuating pressure tube attached to the periphery of the membrane.