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.

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This paper deals with the question of whether uncertainty regarding model structure, especially in climate modeling, exhibits a kind of "chaos." Do small changes in model structure, in other words, lead to large variations in ensemble predictions? More specifically, does model error destroy forecast skill faster than the ordinary or "classical" chaos inherent in the real-world attractor? In some cases, the answer to this question seems to be "yes." But how common is this state of affairs? And are there precise mathematical results that can help us answer this question? We examine some efforts in the literature to answer this last question in the affirmative and find them to be unconvincing. * We would like to thank Mathias Frisch, Blaine Lawson, Seung-Yeop Lee, Connor Mayo-Wilson, Jessica Williams, the audience members at talks in London, Ontario and Chicago, and our anonymous reviewers for helpful comments on earlier drafts as well as many useful discussions of mathematical questions. Responsibility for any remaining errors is of course our own!

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 double occurrence word (DOW) is a word in which every symbol appears exactly twice; two DOWs are equivalent if one is a symbol-to-symbol image of the other. We consider the so called repeat pattern (αα) and the return pattern (αα R ), with gaps allowed between the α's. These patterns generalize square and palindromic factors of DOWs, respectively. We introduce a notion of inserting repeat/return words into DOWs and study how two distinct insertions into the same word can produce equivalent DOWs. Given a DOW w, we characterize the structure of w which allows two distinct insertions to yield equivalent DOWs. This characterization depends on the locations of the insertions and on the length of the inserted repeat/return words and implies that when one inserted word is a repeat word and the other is a return word, then both words must be trivial (i.e., have only one symbol). The characterization also introduces a method to generate families of words recursively.