Genus Ranges of 4-Regular Rigid Vertex Graphs

We introduce a notion of genus range as a set of values of genera over all surfaces into which a graph is embedded cellularly, and we study the genus ranges of a special family of four-regular graphs with rigid vertices that has been used in modeling homologous DNA recombination. We show that the genus ranges are sets of consecutive integers. For any positive integer $n$, there are graphs with $2n$ vertices that have genus range ${m,m+1,...,m'}$ for all $0\le m<m'\le n$, and there are graphs with $2n-1$ vertices with genus range ${m,m+1,...,m'}$ for all $0\le m<m' <n$ or $0<m<m'\le n$. Further, we show that for every $n$ there is $k<n$ such that ${h}$ is a genus range for graphs with $2n-1$ and $2n$ vertices for all $h\le k$. It is also shown that for every $n$, there is a graph with $2n$ vertices with genus range ${0,1,...,n}$, but there is no such a graph with $2n-1$ vertices

A Signal-Passing DNA-Strand-Exchange Mechanism for Active Self-Assembly of DNA Nanostructures

DNA nanostructured tiles play an active role in their own self-assembly in the system described herein whereby they initiate a binding event that produces a cascading assembly process. We present DNA tiles that have a simple but powerful property: they respond to a binding event at one end of the tile by passing a signal across the tile to activate a binding site at the other end. This action allows sequential, virtually irreversible self-assembly of tiles and enables local communication during the self-assembly process. This localized signal-passing mechanism provides a new element of control for autonomous self-assembly of DNA nanostructures

DNA Splicing: Computing by Observing

Motivated by several techniques for observing molecular processes in real-time we introduce a computing device that stresses the role of the observer in biological computations and that is based on the observed behavior of a splicing system. The basic idea is to introduce a marked DNA strand into a test tube with other DNA strands and restriction enzymes. Under the action of these enzymes the DNA starts to splice. An external observer monitors and registers the evolution of the marked DNA strand. The input marked DNA strand is then “accepted” if its observed evolution follows a certain expected pattern. We prove that using simple observers (finite automata), applied on finite splicing systems (finite set of rules, i.e., enzymes and finite set of axioms, i.e., initial strands), the class of recursively enumerable languages can be recognized. This is the preliminary version of a paper that was published in Natural Computing, 8,1, 2009. The original publication is available at http://www.springerlink.co

Counter Machines and Crystallographic Structures

One way to depict a crystallographic structure is by a periodic (di)graph, i.e., a graph whose group of automorphisms has a translational subgroup of finite index acting freely on the structure. We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages DCLd,d=0,1,2,…, within the intersection classes of deterministic context-free languages. We introduce a class of counter machines that accept these languages, where the machines with d counters recognize the class DCLd. An intersection of d languages in DCL1 defines DCLd. We prove that there is a one-to-one correspondence between sets of walks starting and ending in the same unit of a d-dimensional periodic (di)graph and the class of languages in DCLd. The proof uses the following result: given a digraph Δ and a group G, there is a unique diagraph Γ such that G ≤ AutΓ,G acts freely on the structure, and Γ/G ≅ Δ

Biologically Relevant Molecular Transducer with Increased Computing Power and Iterative Abilities

SummaryAs computing devices, which process data and interconvert information, transducers can encode new information and use their output for subsequent computing, offering high computational power that may be equivalent to a universal Turing machine. We report on an experimental DNA-based molecular transducer that computes iteratively and produces biologically relevant outputs. As a proof of concept, the transducer accomplished division of numbers by 3. The iterative power was demonstrated by a recursive application on an obtained output. This device reads plasmids as input and processes the information according to a predetermined algorithm, which is represented by molecular software. The device writes new information on the plasmid using hardware that comprises DNA-manipulating enzymes. The computation produces dual output: a quotient, represented by newly encoded DNA, and a remainder, represented by E. coli phenotypes. This device algorithmically manipulates genetic codes