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We can often think of cells as a sort of circuit on macromolecules, and can show that they can accurately and robustly implement functions like $\text{MAJ}(x_1,...,x_n)$ (return $1$ if more than half of $x_1,...,x_n$ is $1$ and $0$ otherwise) under natural conditions (Cardell & Csikász-Nagy, 2012).

Not all functions are created equal. $\text{XOR}(x,y)$ is a boolean function that is $1$ if exactly one of $x$ or $y$ is $1$ and $0$ otherwise (i.e, if $x = y$). It is often generalized to a multi-input function parity (or odd) as $\text{ODD}(x_1,...,x_n)$ which returns $1$ if an odd number of $x_1,...,x_n$ is $1$. Notice that this is superficially similar to $\text{MAJ}$ but is doing its counting in base 2, which one would expect to be easier for say finite state machines.

However, in some ways, this sensitivity makes the function much more difficult to the point that Valiant (2009) writes:

[Parity] appears to be biologically unnatural. That is exactly the prediction of our theory, which asserts that evolution cannot be based on the evolvability of [the class of functions parity belongs to]

Synthetic biologists used to agree to some extent, with Tamsir et al. (2011) writing that the closely related XNOR (i.e. $\text{XNOR}(x,y) = 1 - \text{XOR}(x,y)$) is empirically impossible to implement. However, they were proven wrong when Bonnet et al. (2012) implemented XNOR among many other amplifying gates by exploiting the structure and process of transcription. Unfortunately, this is an example from synthetic biology, and I really want to learn ones (if any) that occur naturally.

Are there any examples of XOR or related gates implemented (without human engineering) at the cellular level? Are there any naturally occurring molecular pathways that compute something similar to parity? (i.e. something similar to the majority result, but for a more 'sensitive' function) I am primarily interested in empirical demonstrations, but I would be satisfied with theoretical examples that are appealing to biologists (i.e. not something purely comp. sci. like P-systems or something).

References

Bonnet J, Yin P, Ortiz ME, Subsoontorn P, & Endy D (2013). Amplifying Genetic Logic Gates. Science, 340(6132): 599-603. PMID: 23539178

Cardelli L, & Csikász-Nagy A (2012). The cell cycle switch computes approximate majority. Scientific Reports, 2: 656. PMID: 22977731

Tamsir, A., Tabor, J. J., & Voigt, C. A. (2011). Robust multicellular computing using genetically encoded NOR gates and chemical wires. Nature, 469(7329), 212-215.

Valiant, L.G. (2009) Evolvability. Journal of the ACM, 56(1): 3-22.