Deterministic Function Computation with Chemical Reaction Networks

Abstract

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising language for the design of artificial molecular control circuitry. Nonetheless, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood.

CRNs have been shown to be efficiently Turing-universal when allowing for a small probability of error. CRNs that are guaranteed to converge on a correct answer, on the other hand, have been shown to decide only the semilinear predicates. We introduce the notion of function, rather than predicate, computation by representing the output of a function f:ℕ^k → ℕ^l by a count of some molecular species, i.e., if the CRN starts with x ₁,…,x _k molecules of some “input” species X ₁,…,X _k , the CRN is guaranteed to converge to having f(x ₁,…,x _k ) molecules of the “output” species Y ₁,…,Y _l . We show that a function f: ℕ^k → ℕ^l is deterministically computed by a CRN if and only if its graph { (x,y) $\in{\mathbb N}^k \times{\mathbb N}^l\ |$ f(x) = y } is a semilinear set.

Finally, we show that each semilinear function f can be computed on input x in expected time $O(\mathrm{polylog}\ \|{\rm x}\|_1)$.

The first author was supported by the Molecular Programming Project under NSF grant 0832824, the second and third authors were supported by a Computing Innovation Fellowship under NSF grant 1019343. The third author was supported by NIGMS Systems Biology Center grant P50 GM081879.