Biophysical inference mechanisms face a trade-off between external and internal noise resistance

Cells must extract relevant information from time-varying external signals while ignoring a sea of other irrelevant fluctuations in those signals. We find that biophysical mechanisms that are most effective at projecting out irrelevant external fluctuations are the most vulnerable to the internal noise originated from the mechanisms themselves. We show this trade-off relationship in circadian clocks, in gene regulation, and in biochemical receptors. We trace this trade-off to a fundamental tension in the geometry of dynamical systems. To be robust against external fluctuation, the geometry needs to be flat in the projected dimension of the signal and curved along the projected dimensions of external noise. However, this flat dimension makes the estimator vulnerable to internal fluctuations, which affect all dimensions. We conclude with experimental observations of S. Elongatus, and P. Marinus which have evolved distinct clock geometries in correspondence with their protein copy number as predicted by our theory.