Abstract

Chemical reaction networks (CRNs) and DNA strand displacement systems (DSDs) are widely-studied and useful models of molecular programming. However, in order for some DSDs in the literature to behave in an expected manner, the initial number of copies of some reagents is required to be fixed. In this paper we show that, when multiple copies of all initial molecules are present, general types of CRNs and DSDs fail to work correctly if the length of the shortest sequence of reactions needed to produce any given molecule exceeds a threshold that grows polynomially with attributes of the system.

Highlights

  • DNA strand displacement systems (DSDs) (Yurke and Mills 2003; Zhang et al 2007) and chemical reaction networks (CRNs) (Cook et al 2009; Soloveichik 2009, 2008) are important molecular programming models

  • We previously considered the conditions for a class of CRNs to work correctly when multiple copies of all initial molecules are present and showed that the length of the shortest trace needed to ‘‘reach’’, i.e., produce, any given molecule is bounded by a polynomial function of some attributes of a CRN in this class (Condon et al 2012)

  • We use the fact that a Uncooperative DSDs (UDSDs) with simple signals can be simulated by a tagged CRN with volume that is polynomial in the volume of the UDSD, and we can use the bound in Theorem 1 to obtain the following result

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Summary

Introduction

DNA strand displacement systems (DSDs) (Yurke and Mills 2003; Zhang et al 2007) and chemical reaction networks (CRNs) (Cook et al 2009; Soloveichik 2009, 2008) are important molecular programming models. Another very nice feature of this CRN is that it works correctly even if multiple copies of the initial species are present, in the sense of being copy-tolerant and in the sense that the trace of the multi-copy system is an interleaving of traces of the single-copy system, even in the presence of cross-talk This follows from the fact that for every i, to produce each copy of molecule 1i, reactions (i) and (i - 1) have to be executed at least once in forward direction, (i - 2) at least twice, ..., (1) at least 2i-2 times, which can be proved by induction. Since we know that it takes 2n-1 steps to produce 1n even in the multi-copy setting, we have a contradiction

Definition of tagged CRNs
The main upper bound
Result for 1-proper tagged CRNs
Comparison with the previous result
Reachability upper bound for uncooperative DSDs
Definition of uncooperative DSDs
The upper bounds
Marked PDs
Connector sequences
Concatenated templates
Irreversible reactions
Conclusions and open questions
Full Text
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