Composable computation in discrete chemical reaction networks

We study the composability of discrete chemical reaction networks (CRNs) that stably compute (i.e., with probability 0 of error) integer-valued functions $f:\mathbb{N}^d\to\mathbb{N}$. We consider output-oblivious CRNs in which the output species is never a reactant (input) to any reaction. The class of output-oblivious CRNs is fundamental, appearing in earlier studies of CRN computation, because it is precisely the class of CRNs that can be composed by simply renaming the output of the upstream CRN to match the input of the downstream CRN. Our main theorem precisely characterizes the functions $f$ stably computable by output-oblivious CRNs with an initial leader. The key necessary condition is that for sufficiently large inputs, $f$ is the minimum of a finite number of nondecreasing quilt-affine functions. (An affine function is linear with a constant offset; a quilt-affine function is linear with a periodic offset)

Rate-Independent Computation in Continuous Chemical Reaction Networks

Coupled chemical interactions in a well-mixed solution are commonly formalized as chemical reaction networks (CRNs). However, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. Here we study the following problem: what functions f:R^k --> R can be computed by a CRN, in which the CRN eventually produces the correct amount of the "output" molecule, no matter the rate at which reactions proceed? This captures a previously unexplored, but very natural class of computations: for example, the reaction X1 + X2 --> Y can be thought to compute the function y = min(x1, x2). Such a CRN is robust in the sense that it is correct no matter the kinetic model of chemistry, so long as it respects the stoichiometric constraints. We develop a reachability relation based on "what could happen" if reaction rates can vary arbitrarily over time. We define *stable computation* analogously to probability 1 computation in distributed computing, and connect it with a seemingly stronger notion of rate-independent computation based on convergence under a wide class of generalized rate laws. We also consider the "dual-rail representation" that can represent negative values as the difference of two concentrations and allows the composition of CRN modules. We prove that a function is rate-independently computable if and only if it is piecewise linear (with rational coefficients) and continuous (dual-rail representation), or non-negative with discontinuities occurring only when some inputs switch from zero to positive (direct representation). The many contexts where continuous piecewise linear functions are powerful targets for implementation, combined with the systematic construction we develop for computing these functions, demonstrate the potential of rate-independent chemical computation.Comment: preliminary version appeared in ITCS 2014: http://doi.org/10.1145/2554797.255482