## Novel Approaches to DNA Computing

This dissertation presents several novel architectures for DNA computing from different perspectives including analog DNA circuits, polymerase-based DNA logic circuits, and localized DNA-based biomolecular reaction networks on cancer cell membranes.

Firstly, we present an architecture for the systematic construction of DNA circuits for analog arithmetic computation based on DNA strand displacement. The elementary gates in our architecture include addition, subtraction, and multiplication gates. The input and output of these gates are analog, which means that they are directly represented by the concentrations of the input and output DNA strands respectively, without requiring a threshold for converting to Boolean signals. We provide detailed domain designs and kinetic simulations of the gates to demonstrate their expected performance. Based on these gates, we describe how DNA circuits to compute polynomial functions of inputs can be built. Using Taylor Series and Newton Iteration methods, functions beyond the scope of polynomials can also be computed by DNA circuits built upon our architecture.

Secondly, we present an architecture to build compact DNA strand displacement circuits to compute a broad scope of functions in an analog fashion. A circuit by this architecture is composed of three autocatalytic amplifiers, and the amplifiers interact to perform computation. We show DNA circuits to compute functions sqrt(x), ln(x) and exp(x) for x in tunable ranges with simulation results. A key innovation in our architecture, inspired by Napier's use of logarithm transforms to compute square roots on a slide rule, is to make use of autocatalytic amplifiers to do logarithmic and exponential transforms in concentration and time. In particular, we convert from the input that is encoded by the initial concentration of the input DNA strand, to time, and then back again to the output encoded by the concentration of the output DNA strand at equilibrium. This combined use of strand-concentration and time encoding of computational values may have impact on other forms of molecular computation.

Thirdly, we introduce an architecture for fast diffusion-based DNA logic circuits based on Bst 2.0 DNA polymerase and single-stranded logic gates. Each gate consists of only single-stranded DNAs that are easy to design and robust to environmental changes. Large-scale logic circuits can be constructed from the gates by simple cascading strategies. The logic gates and circuits respond in minutes (in terms of half-completion time) compared to hours in prior architectures, providing a very substantial speed-up over prior DNA computing architectures. In particular, we have demonstrated a large-scale DNA logic circuit that computes (the floor of) the square root of 4-bit input numbers. The scale of this circuit is comparable to the largest DNA logic circuit to date (that circuit computes the same function as ours) in terms of the number of gates, and the half-completion time of computing by our circuit is only several minutes, compared to hours by the prior circuit.

Finally, we propose an architecture to program localized DNA-based biomolecular reaction networks on cancer cell membranes. Each node in a network targets a designated cancer cell membrane receptor via aptamer-receptor binding. If all nodes find their corresponding receptors on a cancer cell, the network can start to function by adding initiator DNA strands. Various types of circuits have been experimentally demonstrated from simple linear cascades to reaction networks of more complex structures. These localized reaction networks can be used for medical applications such as cancer detection and therapies.