Improved Lower Bounds on the Compatibility of Multi-State Characters

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function $f(r)$ such that, for any set $C$ of $r$-state characters, $C$ is compatible if and only if every subset of $f(r)$ characters of $C$ is compatible. We show that for every $r \ge 2$, there exists an incompatible set $C$ of $\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1$ $r$-state characters such that every proper subset of $C$ is compatible. Thus, $f(r) \ge \lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1$ for every $r \ge 2$. This improves the previous lower bound of $f(r) \ge r$ given by Meacham (1983), and generalizes the construction showing that $f(4) \ge 5$ given by Habib and To (2011). We prove our result via a result on quartet compatibility that may be of independent interest: For every integer $n \ge 4$, there exists an incompatible set $Q$ of $\lfloor\frac{n-2}{2}\rfloor\cdot\lceil\frac{n-2}{2}\rceil + 1$ quartets over $n$ labels such that every proper subset of $Q$ is compatible. We contrast this with a result on the compatibility of triplets: For every $n \ge 3$, if $R$ is an incompatible set of more than $n-1$ triplets over $n$ labels, then some proper subset of $R$ is incompatible. We show this upper bound is tight by exhibiting, for every $n \ge 3$, a set of $n-1$ triplets over $n$ taxa such that $R$ is incompatible, but every proper subset of $R$ is compatible

A Simple Characterization of the Minimal Obstruction Sets for Three-State Perfect Phylogenies

Lam, Gusfield, and Sridhar (2009) showed that a set of three-state characters has a perfect phylogeny if and only if every subset of three characters has a perfect phylogeny. They also gave a complete characterization of the sets of three three-state characters that do not have a perfect phylogeny. However, it is not clear from their characterization how to find a subset of three characters that does not have a perfect phylogeny without testing all triples of characters. In this note, we build upon their result by giving a simple characterization of when a set of three-state characters does not have a perfect phylogeny that can be inferred from testing all pairs of characters

Contributions to computational phylogenetics and algorithmic self-assembly

This dissertation addresses some of the algorithmic and combinatorial problems at the interface between biology and computation. In particular, it focuses on problems in both computational phylogenetics, an area of study in which computation is used to better understand evolutionary relationships, and algorithmic self-assembly, an area of study in which biological processes are used to perform computation. The first set of results investigate inferring phylogenetic trees from multi-state character data. We give a novel characterization of when a set of three-state characters has a perfect phylogeny and make progress on a long-standing conjecture regarding the compatibility of multi-state characters. The next set of results investigate inferring phylogenetic supertrees from collections of smaller input trees when the input trees do not fully agree on the relative positions of the taxa. Two approaches to dealing with such conflicting input trees are considered. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. The second is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. We give fixed-parameter tractable algorithms for both approaches. We then turn to the algorithmic self-assembly of fractal structures from DNA tiles and investigate approximating the Sierpinski triangle and the Sierpinski carpet with strict self-assembly. We prove tight bounds on approximating the Sierpinski triangle and exhibit a class of fractals that are generalizations of the Sierpinski carpet that can approximately self-assemble. We conclude by discussing some ideas for further research.</p

Statistical and Biological Study Design Options to Evaluate New Turbine Runners Designed for Safer Fish Passage at Ice Harbor Dam on the Lower Snake River, Washington

New turbine runners designed for safer fish passage will be installed in Units 1, 2 and 3 at Ice Harbor Lock and Dam. Installation of the new runners will begin in 2015 with completion anticipated in 2018. Unit 2 will receive a fixed blade runner and will be the first installed. Unit 1 and Unit 3 will receive identical adjustable blade (Kaplan) runners. Installation of Unit 2 will be complete in early 2016; Units 3 and 1 will be complete in 2017 and 2018 respectively. These new turbine runners are designed to reduce risk of injury to juvenile fish caused by mechanisms such as blade strike and shear, as well as pressure injuries known as barotrauma. Extensive computational fluid dynamics (CFD) and physical hydraulic modeling efforts have focused on achieving good hydraulic conditions with minimum pressures of 83kPa to 103kPa. Maintaining nadir pressures ≥ 83kPa will greatly reduce risk of pressure related injury and mortality experienced by turbine passed juvenile salmonids. Once installed the new turbine runners require a biological evaluation to establish the biological performance and to validate the design process and criteria. The biological study design should produce precise, accurate survival estimates with the least associated bias possible. A combination of balloon tag and acoustic telemetry study methods will provide turbine survival estimates encompassing both direct and indirect effects of turbine passage. These study methods have been Regionally accepted and used to establish the most recent estimates of turbine passage survival at the lower Snake and Columbia River dams. Turbine pressure and acceleration data will be collected. These data will be used correlate conditions fish experience as they pass through the turbines with injury and mortality. These data provide evidence of blade strike, shear, and pressure and will also be used to determine how well the new turbine designs meet the biological criteria, and for comparison with and validation of CFD model results. Pertinent factors are discussed for consideration in choosing and implementing each study design. Statistical analysis methods and preliminary sample sizes associated with these study designs are presented, as well as design assumptions and biases, project operations, and sample season considerations. Internal and external acoustic and balloon tagging methods, release locations for upstream releases and direct turbine intake releases, and acoustic detection array locations are discussed as well