A fundamental trade-off among noise suppression, response sensitivity and speed in biological feedback networks.

Biological regulatory networks rely on feedback control to suppress intrinsic noise while remaining sensitive and responsive to external signals, yet whether these objectives can be achieved simultaneously remains unclear. Here, we show that biological feedback networks face an unavoidable constraint: intrinsic fluctuations cannot be arbitrarily suppressed without sacrificing response sensitivity or slowing response speed via feedback control. Using a general framework for stochastic feedback dynamics, we derive a fundamental trade-off that limits how these three performance objectives can be jointly optimized. Theoretical results and numerical simulations demonstrate that this constraint persists across high-dimensional systems. We further show that nonequilibrium, non-gradient dynamics, which are prevalent in biological regulation, can partially relax but never eliminate this limitation, reducing the minimal cost of noise suppression by at most a factor of two. We validate our theory using a biologically motivated activator-inhibitor feedback motif. Together, our results elucidate a fundamental limitation of feedback control in enhancing the information transmission capacity of biological regulatory networks.