Abstract
Biological regulatory networks depend upon chemical interactions to process information. Engineering such molecular computing systems is a major challenge for synthetic biology and related fields. The chemical reaction network (CRN) model idealizes chemical interactions, abstracting away specifics of the molecular implementation, and allowing rigorous reasoning about the computational power of chemical kinetics. Here we focus on function computation with CRNs, where we think of the initial concentrations of some species as the input and the eventual steady-state concentration of another species as the output. Specifically, we are concerned with CRNs that are rate-independent (the computation must be correct independent of the reaction rate law) and composable (\(f \circ g\) can be computed by concatenating the CRNs computing f and g). Rate independence and composability are important engineering desiderata, permitting implementations that violate mass-action kinetics, or even “well-mixedness”, and allowing the systematic construction of complex computation via modular design. We show that to construct composable rate-independent CRNs, it is necessary and sufficient to ensure that the output species of a module is not a reactant in any reaction within the module. We then exactly characterize the functions computable by such CRNs as superadditive, positive-continuous, and piecewise rational linear. Our results show that composability severely limits rate-independent computation unless more sophisticated input/output encodings are used.
These authors’ work was supported in part by National Science Foundation grants CCF-1618895 and CCF-1652824.
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Notes
- 1.
To approximate arbitrary continuous non-linear functions, piecewise linear functions are not sufficient, but rather we need piecewise affine functions (linear functions with offset). However, affine functions can be computed if we use an additional input fixed at 1.
- 2.
As we are studying CRNs whose output is independent of the reaction rates, we leave the rate constants out of the definition.
- 3.
Although the formal definition of mass-action kinetics is outside the scope of this paper, we remind the reader that a CRN with rate constants on each reaction define a system of ODEs under mass-action kinetics. For example, the two reactions \(A + B \rightarrow A + C\) and \(C + C \rightarrow B\) correspond to the following ODEs: \(\dot{a} = 0\), \(\dot{b} = k_2c^2 - k_1ab\), and \(\dot{c} = k_1ab - 2k_2c^2\), where a, b, and c are the concentrations of species A, B, and C over time and \(k_1\), \(k_2\) are the rate constants of the reactions.
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Chalk, C., Kornerup, N., Reeves, W., Soloveichik, D. (2018). Composable Rate-Independent Computation in Continuous Chemical Reaction Networks. In: Češka, M., Šafránek, D. (eds) Computational Methods in Systems Biology. CMSB 2018. Lecture Notes in Computer Science(), vol 11095. Springer, Cham. https://doi.org/10.1007/978-3-319-99429-1_15
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