Engineering tertiary chirality in helical biopolymers.

Tertiary chirality describes the handedness of supramolecular assemblies and relies not only on the primary and secondary structures of the building blocks but also on topological driving forces that have been sparsely characterized. Helical biopolymers, especially DNA, have been extensively investigated as they possess intrinsic chirality that determines the optical, mechanical, and physical properties of the ensuing material. Here, we employ the DNA tensegrity triangle as a model system to locate the tipping points in chirality inversion at the tertiary level by X-ray diffraction. We engineer tensegrity triangle crystals with incremental rotational steps between immobile junctions from 3 to 28 base pairs (bp). We construct a mathematical model that accurately predicts and explains the molecular configurations in both this work and previous studies. Our design framework is extendable to other supramolecular assemblies of helical biopolymers and can be used in the design of chiral nanomaterials, optically active molecules, and mesoporous frameworks, all of which are of interest to physical, biological, and chemical nanoscience.

DSBplot: Indels in DNA Double-strand Break Repair Experiments.

Double-strand breaks (DSBs) in DNA are naturally occurring destructive events in all organisms that may lead to genome instability. Cells employ various repair methods known as non-homologous end joining (NHEJ), microhomology mediated end joining (MMEJ), and homology-directed recombination (HDR). These repair processes may lead to DNA sequence variations (e.g., nucleotide insertions, deletions, and substitutions) at the location of the break. Studying DNA DSB repair processes often involves the use of high throughput sequencing assays to precisely quantify the sequence variations near the break with software tools. Often methods of assessing and visualizing these data have not taken into account the full complexity of the sequencing data, such as the frequency, type, and position of the sequence variations in a single comprehensive representation. Here we present a method that allows visualization of the overall variation pattern as well as comparison of these patterns among experimental setups.

SDRAP for annotating scrambled or rearranged genomes.

Genomes sometimes undergo large-scale rearrangements. Programmed genome rearrangements in ciliates offer an extreme example, making them a compelling model system to study DNA rearrangements. Currently, available methods for genome annotation are not adequate for highly scrambled genomes. We present a theoretical framework and software implementation for the systematic extraction and analysis of DNA rearrangement annotations from pairs of genome assemblies corresponding to precursor and product versions. The software makes no assumptions about the structure of the rearrangements, and permits the user to select parameters to suit the data. Compared to previous approaches, this work achieves more complete precursor-product mappings, allows for full transparency and reproducibility, and can be adapted to genomic data from different sources.

Environmentally Controlled Oscillator with Triplex Guided Displacement of DNA Duplexes.

The use of DNA triplex association is advantageous for the reconfiguration of dynamic DNA nanostructures through pH alteration and can provide environmental control for both structural changes and molecular signaling. The combination of pH-induced triplex-forming oligonucleotide (TFOs) binding with toehold-mediated strand displacement has recently garnered significant attention in the field of structural DNA nanotechnology. While most previous studies use single-stranded DNA to displace or replace TFOs within the triplex, here we demonstrate that pH alteration allows a DNA duplex, with a toehold assistance, to displace TFOs from the components of another DNA duplex. We examined the dependence of this process on toehold length and show that the pH changes allow for cyclic oscillations between two molecular formations. We implemented the duplex/triplex design onto the surface of 2D DNA origami in the form outlining binary digits 0 or 1 and verified the oscillatory conformational changes between the two formations with atomic force microscopy.

The Rule of Thirds: Controlling Junction Chirality and Polarity in 3D DNA Tiles.

The successful self-assembly of tensegrity triangle DNA crystals heralded the ability to programmably construct macroscopic crystalline nanomaterials from rationally-designed, nanoscale components. This 3D DNA tile owes its "tensegrity" nature to its three rotationally stacked double helices locked together by the tensile winding of a center strand segmented into 7 base pair (bp) inter-junction regions, corresponding to two-thirds of a helical turn of DNA. All reported tensegrity triangles to date have employed ( Z + 2 / 3 ) \[\left( {Z{\bm{ + }}2{\bf /}3} \right)\] turn inter-junction segments, yielding right-handed, antiparallel, "J1" junctions. Here a minimal DNA triangle motif consisting of 3-bp inter-junction segments, or one-third of a helical turn is reported. It is found that the minimal motif exhibits a reversed morphology with a left-handed tertiary structure mediated by a locally-parallel Holliday junction-the "L1" junction. This parallel junction yields a predicted helical groove matching pattern that breaks the pseudosymmetry between tile faces, and the junction morphology further suggests a folding mechanism. A Rule of Thirds by which supramolecular chirality can be programmed through inter-junction DNA segment length is identified. These results underscore the role that global topological forces play in determining local DNA architecture and ultimately point to an under-explored class of self-assembling, chiral nanomaterials for topological processes in biological systems.

Augmented DNA Nanoarchitectures: A Structural Library of 3D Self-Assembling Tensegrity Triangle Variants.

The DNA tensegrity triangle is known to reliably self-assemble into a 3D rhombohedral crystalline lattice via sticky-end cohesion. Here, the library of accessible motifs is expanded through covalent extensions of intertriangle regions and sticky-end-coordinated linkages of adjacent triangles with double helical segments using both geometrically symmetric and asymmetric configurations. The molecular structures of 18 self-assembled architectures at resolutions of 3.32-9.32 Å are reported; the observed cell dimensions, cavity sizes, and cross-sectional areas agree with theoretical expectations. These data demonstrate that fine control over triclinic and rhombohedral crystal parameters and the customizability of more complex 3D DNA lattices are attainable via rational design. It is anticipated that augmented DNA architectures may be fine-tuned for the self-assembly of designer nanocages, guest-host complexes, and proscriptive 3D nanomaterials, as originally envisioned. Finally, designer asymmetric crystalline building blocks can be seen as a first step toward controlling and encoding information in three dimensions.

Graph based analysis for gene segment organization In a scrambled genome.

DNA recombinant processes can involve gene segments that overlap or interleave with gene segments of another gene. Such gene segment appearances relative to each other are called here gene segment organization. We use graphs to represent the gene segment organization in a chromosome locus. Vertices of the graph represent contigs resulting after the recombination and the edges represent the gene segment organization prior to rearrangement. To each graph we associate a vector whose entries correspond to graph properties, and consider this vector as a point in a higher dimensional Euclidean space such that cluster formations and analysis can be performed with a hierarchical clustering method. The analysis is applied to a recently sequenced model organism Oxytricha trifallax, a species of ciliate with highly scrambled genome that undergoes massive rearrangement process after conjugation. The analysis shows some emerging star-like graph structures indicating that segments of a single gene can interleave, or even contain all of the segments from fifteen or more other genes in between its segments. We also observe that as many as six genes can have their segments mutually interleaving or overlapping.

Russian Doll Genes and Complex Chromosome Rearrangements in Oxytricha trifallax.

Ciliates have two different types of nuclei per cell, with one acting as a somatic, transcriptionally active nucleus (macronucleus; abbr. MAC) and another serving as a germline nucleus (micronucleus; abbr. MIC). Furthermore, Oxytricha trifallax undergoes extensive genome rearrangements during sexual conjugation and post-zygotic development of daughter cells. These rearrangements are necessary because the precursor MIC loci are often both fragmented and scrambled, with respect to the corresponding MAC loci. Such genome architectures are remarkably tolerant of encrypted MIC loci, because RNA-guided processes during MAC development reorganize the gene fragments in the correct order to resemble the parental MAC sequence. Here, we describe the germline organization of several nested and highly scrambled genes in Oxytricha trifallax These include cases with multiple layers of nesting, plus highly interleaved or tangled precursor loci that appear to deviate from previously described patterns. We present mathematical methods to measure the degree of nesting between precursor MIC loci, and revisit a method for a mathematical description of scrambling. After applying these methods to the chromosome rearrangement maps of O. trifallax we describe cases of nested arrangements with up to five layers of embedded genes, as well as the most scrambled loci in O. trifallax.

A Graph Isomorphism Condition and Equivalence of Reaction Systems.

We consider global dynamics of reaction systems as introduced by Ehrenfeucht and Rozenberg. The dynamics is represented by a directed graph, the so-called transition graph, and two reaction systems are considered equivalent if their corresponding transition graphs are isomorphic. We introduce the notion of a skeleton (a one-out graph) that uniquely determines a directed graph. We provide the necessary and sufficient conditions for two skeletons to define isomorphic graphs. This provides a necessary and sufficient condition for two reactions systems to be equivalent, as well as a characterization of the directed graphs that correspond to the global dynamics of reaction systems.

Recurring patterns among scrambled genes in the encrypted genome of the ciliate Oxytricha trifallax.

Some genera of ciliates, such as Oxytricha and Stylonychia, undergo massive genome reorganization during development and provide model organisms to study DNA rearrangement. A common feature of these ciliates is the presence of two types of nuclei: a germline micronucleus and a transcriptionally-active somatic macronucleus containing over 16,000 gene sized "nano-chromosomes". During conjugation the old parental macronucleus disintegrates and a new macronucleus forms from a copy of the zygotic micronucleus. During this process, macronuclear chromosomes assemble through DNA processing events that delete 90-98% of the DNA content of the micronucleus. This includes the deletion of noncoding DNA segments that interrupt precursor DNA regions in the micronucleus, as well as transposons and other germline-limited DNA. Each macronuclear locus may be present in the micronucleus as several nonconsecutive, permuted, and/or inverted DNA segments. Here we investigate the genome-wide range of scrambled gene architectures that describe all precursor-product relationships in Oxytricha trifallax, the first completely sequenced scrambled genome. We find that five general, recurrent patterns in the sets of scrambled micronuclear precursor pieces can describe over 80% of Oxytricha's scrambled genes. These include instances of translocations and inversions, and other specific patterns characterized by alternating stretches of consecutive odd and even DNA segments. Moreover, we find that iterating patterns of alternating odd-even segments up to four times can describe over 96% of the scrambled precursor loci. Recurrence of these highly structured genetic architectures within scrambled genes presumably reflects recurrent evolutionary events that gave rise to over 3000 of scrambled loci in the germline genome.

: a database of ciliate genome rearrangements.

Ciliated protists exhibit nuclear dimorphism through the presence of somatic macronuclei (MAC) and germline micronuclei (MIC). In some ciliates, DNA from precursor segments in the MIC genome rearranges to form transcriptionally active genes in the mature MAC genome, making these ciliates model organisms to study the process of somatic genome rearrangement. Similar broad scale, somatic rearrangement events occur in many eukaryotic cells and tumors. The (http://oxytricha.princeton.edu/mds_ies_db) is a database of genome recombination and rearrangement annotations, and it provides tools for visualization and comparative analysis of precursor and product genomes. The database currently contains annotations for two completely sequenced ciliate genomes: Oxytricha trifallax and Tetrahymena thermophila.

Genus Ranges of Chord Diagrams.

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can be, and those that cannot be, realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.

Dynamic Simulation of 1D Cellular Automata in the Active aTAM.

The Active aTAM is a tile based model for self-assembly where tiles are able to transfer signals and change identities according to the signals received. We extend Active aTAM to include deactivation signals and thereby allow detachment of tiles. We show that the model allows a dynamic simulation of cellular automata with assemblies that do not record the entire computational history but only the current updates of the states, and thus provide a way for (a) algorithmic dynamical structural changes in the assembly and (b) reusable space in self-assembly. The simulation is such that at a given location the sequence of tiles that attach and detach corresponds precisely to the sequence of states the synchronous cellular automaton generates at that location.

Genus Ranges of 4-Regular Rigid Vertex Graphs.

A rigid vertex of a graph is one that has a prescribed cyclic order of its incident edges. We study orientable genus ranges of 4-regular rigid vertex graphs. The (orientable) genus range is a set of genera values over all orientable surfaces into which a graph is embedded cellularly, and the embeddings of rigid vertex graphs are required to preserve the prescribed cyclic order of incident edges at every vertex. The genus ranges of 4-regular rigid vertex graphs are sets of consecutive integers, and we address two questions: which intervals of integers appear as genus ranges of such graphs, and what types of graphs realize a given genus range. For graphs with 2n vertices (n > 1), we prove that all intervals [a, b] for all a < b ≤ n, and singletons [h, h] for some h ≤ n, are realized as genus ranges. For graphs with 2n - 1 vertices (n ≥ 1), we prove that all intervals [a, b] for all a < b ≤ n except [0, n], and [h, h] for some h ≤ n, are realized as genus ranges. We also provide constructions of graphs that realize these ranges.

Existence of constants in regular splicing languages.

In spite of wide investigations of finite splicing systems in formal language theory, basic questions, such as their characterization, remain unsolved. It has been conjectured that a necessary condition for a regular language L to be a splicing language is that L must have a constant in the Schutzenberger sense. We prove this longstanding conjecture to be true. The result is based on properties of strongly connected components of the minimal deterministic finite state automaton for a regular splicing language. Using constants of the corresponding languages, we also provide properties of transitive automata and pathautomata.

Computing by molecular self-assembly.

The paper reviews two computing models by DNA self-assembly whose proof of principal have recently been experimentally confirmed. The first model incorporates DNA nano-devices and triple crossover DNA molecules to algorithmically arrange non-DNA species. This is achieved by simulating a finite-state automaton with output where golden nanoparticles are assembled to read-out the result. In the second model, a complex DNA molecule representing a graph emerges as a solution of a computational problem. This supports the idea that in molecular self-assembly computing, it may be necessary to develop the notion of shape processing besides the classical approach through symbol processing.

TRANSDUCER GENERATED ARRAYS OF ROBOTIC NANO-ARMS.

We consider sets of two-dimensional arrays, called here transducer generated languages, obtained by iterative applications of transducers (finite state automata with output). Each transducer generates a set of blocks of symbols such that the bottom row of a block is an input string accepted by the transducer and, by iterative application of the transducer, each row of the block is an output of the transducer on the preceding row. We show how these arrays can be implemented through molecular assembly of triple crossover DNA molecules. Such assembly could serve as a scaffold for arranging molecular robotic arms capable for simultaneous movements. We observe that transducer generated languages define a class of languages which is a proper subclass of recognizable picture languages, but it containing the class of all factorial local two-dimensional languages. By taking the average growth rate of the number of blocks in the language as a measure of its complexity, we further observe that arrays with high complexity patterns can be generated in this way.

On existence of reporter strands in DNA-based graph structures.

Through self-assembly of branched junction molecules many different DNA structures (graphs) can be assembled. We show that every multigraph can be assembled by DNA such that there is a single strand that traces each edge in the graph at least once. This strand corresponds to a boundary component of a two-dimensional orientable surface that has the given graph as a deformation retract. This boundary component traverses every edge at least once, and it defines a circular path in the graph that "preserves the graph structure" and traverses each edge.

Knee joint injury and repair modeled by membrane systems.

Tissue reparative processes following tissue injury are modeled by a basic membrane system, dealing only with objects, non-active membranes, and non-deterministic evolution rules. At the biological level, tissue repair is regulated by multiple interactions between cells and macromolecules, the latter acting as signals. Such signals modify cell behavior including proliferation, migration, differentiation, and phagocytosis. The signaling components themselves are produced and removed by the resident cell population, and this set of events may provide additional stimuli for altering cell activities. In this paper we have focused on modeling the biology of events following an injury to the knee joint, and have used hyaluronan (a polymer produced by cartilage and synovial cells) as an example for a signaling component in the healing process. The intrinsic non-determinism of the model is a key feature, which allows a mathematical description of the repair responses as well as a possibility for either functional restoration or chronic degeneration, leading to arthritis.

RNA-guided DNA assembly.

We propose molecular models for homologous DNA recombination events that are guided by either double-stranded RNA (dsRNA) or single-stranded RNA (ssRNA) templates. The models are applied to explain DNA rearrangements in some groups of ciliates, such as Stylonychia or Oxytricha, where extensive gene rearrangement occurs during differentiation of a somatic macronucleus from a germline micronucleus. We describe a model for RNA template guided DNA recombination, such that the template serves as a catalyst that remains unchanged after DNA recombination. This recombination can be seen as topological braiding of the DNA, with the template-guided alignment proceeding through DNA branch migration. We show that a virtual knot diagram can provide a physical representation of the DNA at the time of recombination. Schematically, the braiding process can be represented as a crossing in the virtual knot diagram. The homologous recombination corresponds to removal of the crossings in the knot diagram (called smoothing). We show that if all recombinations are performed at the same time (i.e., simultaneous smoothings of the crossings) then one of the resulting DNA molecules will always contain all of the gene segments in their correct, linear order, which produces the mature DNA sequence.