  Living cells deploy many resources to sense their environments, including
receptors, downstream signaling molecules, time and fuel. However, it is not
known which resources fundamentally limit the precision of sensing, like weak
links in a chain, and which can compensate each other, leading to trade-offs
between them. We show by modeling that in equilibrium systems the precision is
limited by the number of receptors; the downstream network can never increase
precision. This limit arises from a trade-off between the removal of extrinsic
noise in the receptor and intrinsic noise in the downstream network.
Non-equilibrium systems can lift this trade-off by storing the receptor state
over time in chemical modification states of downstream molecules. As we
quantify for a push-pull network, this requires i) time and receptors; ii)
downstream molecules; iii) energy (fuel turnover) to drive modification. These
three resource classes cannot compensate each other, and it is the limiting
class which sets the fundamental sensing limit. Within each class, trade-offs
are possible. Energy allows a power-speed trade-off, while time can be traded
against receptors.
