This is a valuable and very well-written contribution to our understanding of biological information processing. The authors investigate (highly idealized) abstract biochemical reactions in which the chemical inputs and outputs determine the numerical inputs and outputs of functions.

A clear inductive presentation shows that reaction networks corresponding to algebraic functions can be constructed: constants k(x)=c, multiplication m(x,y)=x×y, division d(x,y)=x÷y, addition, subtraction, roots, powers, and the composition of functions. Independent variables have values that are concentrations of reactants in a steady state.

Reaction coefficients determine function coefficients and, therefore, are constant. The availability and temperature of reactants are assumed to be constant, and the possible accumulation of products is not considered. Undergraduate-level methods for differential equations are used to compute steady-state values. For example, the authors show how a biochemical reaction could exist whose reactants, in stable concentrations a, b, and c, produce an output in concentration d, where 0=a×d²+b×d+c. In other words, the quadratic formula is algebraic and, by the time the example is given, we know that algebraic functions can be computed.

Twenty years ago, I had planned to do something like this, but my plan collapsed under the weight of reality and a less-than-optimal strategy. Buisman et al.’s strategy is elegant and they have included just enough biochemical reality to make this a good starting point for further work. Unfortunately, I have read little of the work of others mentioned as references, so I am not able to spread credit as far and as fairly as I could have a decade ago. But I am sure that young researchers can find in this paper many interesting and important problems for further work.