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The Turing completeness of continuous Chemical Reaction Networks (CRNs)states that any computable real function can be computed by a continuous CRN ona finite set of molecular species, possibly restricted to elementary reactions,i.e. with at most two reactants and mass action law kinetics. In this paper, weintroduce a more stringent notion of robust online analog computation, andAbsolute Functional Robustness (AFR), for the CRNs that stabilize theconcentration values of some output species to the result of one function ofthe input species concentrations, in a perfectly robust manner with respect toperturbations of both intermediate and output species. We prove that the set ofreal functions stabilized by a CRN with mass action law kinetics is preciselythe set of real algebraic functions. Based on this result, we present acompiler which takes as input any algebraic function (defined by one polynomialand one point for selecting one branch of the algebraic curve defined by thepolynomial) and generates an abstract CRN to stabilize it. Furthermore, weprovide error bounds to estimate and control the error of an unperturbedsystem, under the assumption that the environment inputs are driven byk-Lipschitz functions.

On a model of online analog computation in the cell with absolute  functional robustness: algebraic characterization, function compiler and  error control