Diverse and robust molecular algorithms using reprogrammable DNA self-assembly

Molecular biology provides an inspiring proof-of-principle that chemical systems can store and process information to direct molecular activities such as the fabrication of complex structures from molecular components. To develop information-based chemistry as a technology for programming matter to function in ways not seen in biological systems, it is necessary to understand how molecular interactions can encode and execute algorithms. The self-assembly of relatively simple units into complex products is particularly well suited for such investigations. Theory that combines mathematical tiling and statistical–mechanical models of molecular crystallization has shown that algorithmic behaviour can be embedded within molecular self-assembly processes, and this has been experimentally demonstrated using DNA nanotechnology with up to 22 tile types. However, many information technologies exhibit a complexity threshold—such as the minimum transistor count needed for a general-purpose computer—beyond which the power of a reprogrammable system increases qualitatively, and it has been unclear whether the biophysics of DNA self-assembly allows that threshold to be exceeded. Here we report the design and experimental validation of a DNA tile set that contains 355 single-stranded tiles and can, through simple tile selection, be reprogrammed to implement a wide variety of 6-bit algorithms. We use this set to construct 21 circuits that execute algorithms including copying, sorting, recognizing palindromes and multiples of 3, random walking, obtaining an unbiased choice from a biased random source, electing a leader, simulating cellular automata, generating deterministic and randomized patterns, and counting to 63, with an overall per-tile error rate of less than 1 in 3,000. These findings suggest that molecular self-assembly could be a reliable algorithmic component within programmable chemical systems. The development of molecular machines that are reprogrammable—at a high level of abstraction and thus without requiring knowledge of the underlying physics—will establish a creative space in which molecular programmers can flourish

Planare Graphen und ihre Dualgraphen auf Zylinderoberflächen

In this thesis, we investigates plane drawings of undirected and directed graphs on cylinder surfaces. In the case of undirected graphs, the vertices are positioned on a line that is parallel to the cylinder’s axis and the edge curves must not intersect this line. We show that a plane drawing is possible if and only if the graph is a double-ended queue (deque) graph, i. e., the vertices of the graph can be processed according to a linear order and the edges correspond to items in the deque inserted and removed at their end vertices. A surprising consequence resulting from these observations is that the deque characterizes planar graphs with a Hamiltonian path. This result extends the known characterization of planar graphs with a Hamiltonian cycle by two stacks. By these insights, we also obtain a new characterization of queue graphs and their duals. We also consider the complexity of deciding whether a graph is a deque graph and prove that it is NP-complete. By introducing a split operation, we obtain the splittable deque and show that it characterizes planarity. For the proof, we devise an algorithm that uses the splittable deque to test whether a rotation system is planar. In the case of directed graphs, we study upward plane drawings where the edge curves follow the direction of the cylinder’s axis (standing upward planarity; SUP) or they wind around the axis (rolling upward planarity; RUP). We characterize RUP graphs by means of their duals and show that RUP and SUP swap their roles when considering a graph and its dual. There is a physical interpretation underlying this characterization: A SUP graph is to its RUP dual graph as electric current passing through a conductor to the magnetic field surrounding the conductor. Whereas testing whether a graph is RUP is NP-hard in general [Bra14], for directed graphs without sources and sink, we develop a linear-time recognition algorithm that is based on our dual graph characterization of RUP graphs.Die Arbeit beschäftigt sich mit planaren Zeichnungen ungerichteter und gerichteter Graphen auf Zylinderoberflächen. Im ungerichteten Fall werden Zeichnungen betrachtet, bei denen die Knoten auf einer Linie parallel zur Zylinderachse positioniert werden und die Kanten diese Linie nicht schneiden dürfen. Es kann gezeigt werden, dass eine planare Zeichnung genau dann möglich ist, wenn die Kanten des Graphen in einer double-ended queue (Deque) verarbeitet werden können. Ebenso lassen sich dadurch Queue, Stack und Doppelstack charakterisieren. Eine überraschende Konsequenz aus diesen Erkenntnissen ist, dass die Deque genau die planaren Graphen mit Hamiltonpfad charakterisiert. Dies erweitert die bereits bekannte Charakterisierung planarer Graphen mit Hamiltonkreis durch den Doppelstack. Im gerichteten Fall müssen die Kantenkurven entweder in Richtung der Zylinderachse verlaufen (SUP-Graphen) oder sich um die Achse herumbewegen (RUP-Graphen). Die Arbeit charakterisiert RUP-Graphen und zeigt, dass RUP und SUP ihre Rollen tauschen, wenn man Graph und Dualgraph betrachtet. Der SUP-Graph verhält sich dabei zum RUP-Graphen wie elektrischer Strom durch einen Leiter zum induzierten Magnetfeld. Ausgehend von dieser Charakterisierung ist es möglich einen Linearzeit-Algorithmus zu entwickeln, der entscheidet ob ein gerichteter Graph ohne Quellen und Senken ein RUP-Graph ist, während der allgemeine Fall NP-hart ist [Bra14]