Time-Complexity of Multilayered DNA Strand Displacement Circuits

Recently we have shown how molecular logic circuits with many components arranged in multiple layers can be built using DNA strand displacement reactions. The potential applications of this and similar technologies inspire the study of the computation time of multilayered molecular circuits. Using mass action kinetics to model DNA strand displacement-based circuits, we discuss how computation time scales with the number of layers. We show that depending on circuit architecture, the time-complexity does not necessarily scale linearly with the depth as is assumed in the usual study of circuit complexity. We compare circuits with catalytic and non-catalytic components, showing that catalysis fundamentally alters asymptotic time-complexity. Our results rely on simple asymptotic arguments that should be applicable to a wide class of chemical circuits. These results may help to improve circuit performance and may be useful for the construction of faster, larger and more reliable molecular circuitry. Circuit depth is the standard measure of time-complexity of feed-forward circuits [8]. While this is well justified in electronic digital circuits, in this paper we ask whether depth is the correct measure of time-complexity for chemical circuits. We provide a quantitative analysis of how computation time is related to circuit size and architecture. We compare two elementary mechanisms for the underlying components: in one case, the underlying chemical reactions are stoichiometric and one input molecule produces one output molecule. In the other case the underlying reactions are catalytic and a single input molecule can trigger an arbitrary number of output molecules. We show that for non-catalytic circuits, the time to half-completion does not always scale linearly with the depth of the circuit. Our analysis shows that for a tree of stoichiometric bimolecular reactions, the time to half-completion scales quadratically with the depth of the circuit — i.e. the additional time due to adding an extra layer increases linearly with the size of the circuit. In contrast, we find that for catalytic systems the time to half-completion is a linear function of the depth independently of the structure of the circuit. The latter results agrees with our intuition from electronics where all gates are amplifying. In this paper, for the physical model of molecular circuits we focus on DNAbased circuits implemented as cascades of strand displacement reactions. R. Deaton and A. Suyama (Eds.): DNA 15, LNCS 5877, pp. 144–153, 2009. c © Springer-Verlag Berlin Heidelberg 2009 Time-Complexity of Multilayered DNA Strand Displacement Circuits 145 Single-stranded nucleic acids serve as signals that are exchanged between multistranded gate complexes. We have previously shown that this technology allows us to build multi-component molecular circuits that incorporate all the main features of digital logic, such as Boolean logic gates like AND, NOT and OR, signal restoration and modularity [4]. More recent papers have implemented a variety of improvements including gates for fast catalytic amplification [11,9], and reversible logic gates based on a simple catalytic gate motif [3]. This technology provides a starting point for building large-scale molecular circuitry with quantitatively predictable behavior using standardized off-the-shelf components [6]. In our experiments on nucleic-acid logic circuits we noticed that often the measured time to half-completion did not seem to scale linearly with the depth of the circuit. Instead, every additional layer seemed to add more than a constant offset to the time to half-completion. The observed slowdown may in part be due to non-specific interactions between DNA species that compete with or hinder the desired interactions between DNA gates and their inputs. However, here we will argue that the observed slowdown is at least in part a consequence of the circuit layout used in these experiments and of the underlying reaction kinetics of the DNA components. In the next section we compare the time-complexity of linear cascades and converging trees for both non-catatylic and catalytic reactions. These circuit architecture represent extreme cases: A linear cascade of length N minimizes total fan-in while the converging tree of equal depth maximizes total fan-in. We describe specific implementations of these circuits with previously developed components and use numerical simulations to investigate how time complexity relates to circuit architecture and reaction mechanism. Then, we derive the asymptotic scaling of time-complexity of circuits using simplified kinetics. The simplified kinetics captures the essential features of the strand displacement circuits, but should be also applicable to alternative implementations of molecular circuits. Indeed the proofs are largely independent of the details of the underlying reaction mechanisms. 1 Comparing DNA Strand-Displacement Base Reaction Mechanisms and Circuit Architectures In this section we consider linear reaction cascades and converging trees based on DNA strand displacement chemistry previously described [4]. In the next section we consider more abstract and more generally applicable models of the same architectures that capture the essential behavior. 1.1 Cascades of Non-catalytic Reactions A strand displacement reaction (see Fig. 1(A)) can be intuitively described as a hybridization reaction between two complementary strands where one strand is initially fully single stranded (the “input”) while the other strand is already partially double stranded (the “gate”). The reaction is driven forward by the 146 G. Seelig and D. Soloveichik 1 2 3 4 2* 3* 4* 3 4 5 6 2* 3* 4* 1 2 3 4 4 6 5 3