### Abstract

Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: A type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.

Original language | English (US) |
---|---|

Pages (from-to) | 76-83 |

Number of pages | 8 |

Journal | ACS Synthetic Biology |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2017 |

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### Keywords

- DNA strand-displacement reaction
- Mass-Action kinetics
- Molecular computing
- Polynomials

### Cite this

*ACS Synthetic Biology*,

*6*(1), 76-83. https://doi.org/10.1021/acssynbio.5b00163

**Chemical reaction networks for computing polynomials.** / Salehi, Sayed Ahmad; Parhi, Keshab K; Riedel, Marc.

Research output: Contribution to journal › Article

*ACS Synthetic Biology*, vol. 6, no. 1, pp. 76-83. https://doi.org/10.1021/acssynbio.5b00163

}

TY - JOUR

T1 - Chemical reaction networks for computing polynomials

AU - Salehi, Sayed Ahmad

AU - Parhi, Keshab K

AU - Riedel, Marc

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: A type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.

AB - Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: A type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.

KW - DNA strand-displacement reaction

KW - Mass-Action kinetics

KW - Molecular computing

KW - Polynomials

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UR - http://www.scopus.com/inward/citedby.url?scp=85012815333&partnerID=8YFLogxK

U2 - 10.1021/acssynbio.5b00163

DO - 10.1021/acssynbio.5b00163

M3 - Article

VL - 6

SP - 76

EP - 83

JO - ACS Synthetic Biology

JF - ACS Synthetic Biology

SN - 2161-5063

IS - 1

ER -