When part of a biological system cannot be investigated directly by experimentation, we face the problem of structure identification: how can we construct a model for an unknown part of a mostly known system using measurements gathered from its input and output? This problem is especially difficult to solve when the measurements available are noisy and sparse, i.e. widely and unevenly spaced in time, as is common when measuring biological quantities at the cellular level. Here we present a procedure to identify a static nonlinearity embedded between two dynamical systems using noisy, sparse measurements. To reduce the level of error caused by measurement noise, we introduce the concept of weighted-sum predictability. If we make the input and output subsystems weighted-sum predictable and normalize the measurements to their weighted sum, we achieve better noise reduction than through normalizing to a loading control. We then interpolate the normalized measurements to obtain continuous input and output signals, with which we solve directly for the input-output characteristics of the unknown static nonlinearity. We demonstrate the effectiveness of this structure identification procedure by applying it to identify a model for ergosterol sensing by the proteins Sre1 and Scp1 in fission yeast. Simulations with this model produced outputs consistent with experimental observations. The techniques introduced here will provide researchers with a new tool by which biological systems can be identified and characterized.
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