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.

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.

Computations of quandle 2-cocycle knot invariants without explicit 2-cocycles.

We explore a knot invariant derived from colorings of corresponding 1-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle 2-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding 2-cocycles. This permits the construction of many 2-cocycle invariants without exhibiting explicit 2-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann's knot coloring polynomial. Computations using this technique show that the 2-cocycle invariant distinguishes all of the oriented prime knots up to 11 crossings and most oriented prime knots with 12 crossings including classification by symmetry: mirror images, reversals, and reversed mirrors.

HOMOLOGY FOR QUANDLES WITH PARTIAL GROUP OPERATIONS.

A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group and quandle homology theories. A quandle that is a union of groups with the operation restricting to conjugation on each group component is called a multiple conjugation quandle (MCQ, defined rigorously within). In this definition, compatibilities between the group and quandle operations are imposed which are motivated by considerations on colorings of handlebody-links. The homology theory defined here for MCQs takes into consideration both group and quandle operations, as well as their compatibility. The first homology group is characterized, and the notion of extensions by 2-cocycles is provided. Degenerate subcomplexes are defined in relation to simplicial decompositions of prismatic (products of simplices) complexes and group inverses. Cocycle invariants are also defined for handlebody-links.

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.

Algebraic properties of quandle extensions and values of cocycle knot invariants.

Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial 2-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also examined.

Quandle coloring and cocycle invariants of composite knots and abelian extensions.

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.

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.

Quandle colorings of knots and applications.

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q1 → Q0. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin's list. Links to the data computed and various programs in C, GAP and Maple are provided.

TWIST REGIONS AND COEFFICIENTS STABILITY OF THE COLORED JONES POLYNOMIAL.

We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of q-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of the alternating link diagram.

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.

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.

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.