Search for: All records

Ciliates are microbial eukaryotes that undergo extensive programmed genome rearrangement, a natural genome editing process that converts long germline chromosomes into smaller gene-rich somatic chromosomes. Three well-studied ciliates include Oxytricha trifallax , Tetrahymena thermophila and Paramecium tetraurelia , but only the Oxytricha lineage has a massively scrambled genome, whose assembly during development requires hundreds of thousands of precise programmed DNA joining events, representing the most complex genome dynamics of any known organism. Here we study the emergence of such complex genomes by examining the origin and evolution of discontinuous and scrambled genes in the Oxytricha lineage. This study compares six genomes from three species, the germline and somatic genomes for Euplotes woodruffi , Tetmemena sp. , and the model ciliate Oxytricha trifallax . To complement existing data, we sequenced, assembled and annotated the germline and somatic genomes of Euplotes woodruffi , which provides an outgroup, and the germline genome of Tetmemena sp.. We find that the germline genome of Tetmemena is as massively scrambled and interrupted as Oxytricha's : 13.6% of its gene loci require programmed translocations and/or inversions, with some genes requiring hundreds of precise gene editing events during development. This study revealed that the earlier-diverged spirotrich, E. woodruffi , also has a scrambled genome, but only roughly half as many loci (7.3%) are scrambled. Furthermore, its scrambled genes are less complex, together supporting the position of Euplotes as a possible evolutionary intermediate in this lineage, in the process of accumulating complex evolutionary genome rearrangements, all of which require extensive repair to assemble functional coding regions. Comparative analysis also reveals that scrambled loci are often associated with local duplications, supporting a gradual model for the origin of complex, scrambled genomes via many small events of DNA duplication and decay. 

A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations. 

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive [Formula: see text]-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing [Formula: see text]-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given. 

A double occurrence word (DOW) is a word in which every symbol appears exactly twice. We define the symbol separation of a DOW [Formula: see text] to be the number of letters between the two copies of a symbol, and the separation of [Formula: see text] to be the sum of separations over all symbols in [Formula: see text]. We then analyze relationship among size, reducibility and separation of DOWs. Specifically, we provide tight bounds of separations of DOWs with a given size and characterize the words that attain those bounds. We show that all separation numbers within the bounds can be realized. We present recursive formulas for counting the numbers of DOWs with a given separation under various restrictions, such as the number of irreducible factors. These formulas can be obtained by inductive construction of all DOWs with the given separation. 

For a family of sets we consider elements that belong to the same sets within the family as companions. The global dynamics of a reactions system (as introduced by Ehrenfeucht and Rozenberg) can be represented by a directed graph, called a transition graph, which is uniquely determined by a one-out subgraph, called the 0-context graph. We consider the companion classes of the outsets of a transition graph and introduce a directed multigraph, called an essential motion, whose vertices are such companion classes. We show that all one-out graphs obtained from an essential motion represent 0-context graphs of reactions systems with isomorphic transition graphs. All such 0-context graphs are obtained from one another by swapping the outgoing edges of companion vertices.