Scholarly Work - Mathematical Sciences

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On the intrinsic dimensionality of chemical structure space
(1988-01) Veith, D.; Greenwood, B.; Hunter, R. S.; Niemi, G. J.; Regal, R. R.
An important expectation in chemistry and pharmacology is that similar chemical structures have similar properties and behavior. New industrial chemicals, pesticides, and therapeutics are often sublte modifications of "lead" structures with known chemical behavior. Chemical properties and recreation rates can be can be estimated from suitable homologs 1,2. Moreover, the safety of untested chemicals is often evaluated by comparing the chemical toanalogous structures for which toxicological data are available. Despite the widespread use of terms such as "homolog" and "analogs" in research, chemical similarity has evaded quantitative interpretation from a perspective where all chemicals are considered simultaneously. One reason is that chemical similarity is inherently a multivariate problem or, in other words, chemicals are simultaneously similar and different from many perspectives. We have approached chemical similarity by attempting to define a structure space in which all chemicals can be identified. Because there are so many potentially important variables, multivariate tools are necessary to reduce the dimensionality of this problem. When this is accomplished, we need to comprehend what this space means and what can be predicted from it . This paper is one of the first attempts to define chemical structure space for a large universe of chemicals.
Supporting Information for "Significance of trends toward earlier snowmelt runoff, Columbia and Missouri Basin headwaters, western United States"
(American Geophysical Union, 2007-08) Moore, Johnnie N.; Harper, Joel T.; Greenwood, Mark C.
This is auxiliary material for this article contains one text file, four tables, and three figures from the article "Significance of trends toward earlier snowmelt runoff, Columbia and Missouri Basin headwaters, western United States" from Geophysical Research Letters on the 28 August 2007. http://onlinelibrary.wiley.com/doi/10.1029/2007GL031022/full
Significance of trends toward earlier snowmelt runoff, Columbia and Missouri Basin headwaters, western United States
(American Geophysical Union, 2007-08) Moore, Johnnie N.; Harper, Joel T.; Greenwood, Mark C.
We assess changes in runoff timing over the last 55 years at 21 gages unaffected by human influences, in the headwaters of the Columbia-Missouri Rivers. Linear regression models and tests for significance that control for “false discoveries” of many tests, combined with a conceptual runoff response model, were used to examine the detailed structure of spring runoff timing. We conclude that only about one third of the gages exhibit significant trends with time but over half of the gages tested show significant relationships with discharge. Therefore, runoff timing is more significantly correlated with annual discharge than with time. This result differs from previous studies of runoff in the western USA that equate linear time trends to a response to global warming. Our results imply that predicting future snowmelt runoff in the northern Rockies will require linking climate mechanisms controlling precipitation, rather than projecting response to simple linear increases in temperature.
Percolation on Fitness Landscapes: Effects of Correlation, Phenotype, and Incompatibilities
(2007-10) Gravner, Janko; Pitman, Damien J.; Gavrilets, Sergey
We study how correlations in the random fitness assignment may affect the structure of fitness landscapes, in three classes of fitness models. The first is a phenotype space in which individuals are characterized by a large number n of continuously varying traits. In a simple model of random fitness assignment, viable phenotypes are likely to form a giant connected cluster percolating throughout the phenotype space provided the viability probability is larger than 1/2n. The second model explicitly describes genotype-to-phenotype and phenotype-to-fitness maps, allows for neutrality at both phenotype and fitness levels, and results in a fitness landscape with tunable correlation length. Here, phenotypic neutrality and correlation between fitnesses can reduce the percolation threshold, and correlations at the point of phase transition between local and global are most conducive to the formation of the giant cluster. In the third class of models, particular combinations of alleles or values of phenotypic characters are “incompatible” in the sense that the resulting genotypes or phenotypes have zero fitness. This setting can be viewed as a generalization of the canonical Bateson–Dobzhansky–Muller model of speciation and is related to K-SAT problems, prominent in computer science. We analyze the conditions for the existence of viable genotypes, their number, as well as the structure and the number of connected clusters of viable genotypes. We show that analysis based on expected values can easily lead to wrong conclusions, especially when fitness correlations are strong. We focus on pairwise incompatibilities between diallelic loci, but we also address multiple alleles, complex incompatibilities, and continuous phenotype spaces. In the case of diallelic loci, the number of clusters is stochastically bounded and each cluster contains a very large sub-cube. Finally, we demonstrate that the discrete NK model shares some signature properties of models with high correlations.
Something old, something new, something borrowed; how the thermoacidophilic archaeon Sulfolobus solfataricus responds to oxidative stress
(2009-09) Maaty, Walid S.; Wiedenheft, Blake A.; Tarlykov, Pavel V.; Schaff, Nathan; Heinemann, Joshua V.; Robison-Cox, James; Dougherty, Amanda; Blum, Paul; Lawrence, C. Martin; Douglas, Trevor; Young, Mark J.; Bothner, Brian
To avoid molecular damage of biomolecules due to oxidation, all cells have evolved constitutive and responsive systems to mitigate and repair chemical modifications. Archaea have adapted to some of the most extreme environments known to support life, including highly oxidizing conditions. However, in comparison to bacteria and eukaryotes, relatively little is known about the biology and biochemistry of archaea in response to changing conditions and repair of oxidative damage. In this study transcriptome, proteome, and chemical reactivity analyses of hydrogen peroxide (H2O2) induced oxidative stress in Sulfolobus solfataricus (P2) were conducted. Microarray analysis of mRNA expression showed that 102 transcripts were regulated by at least 1.5 fold, 30 minutes after exposure to 30 µM H2O2. Parallel proteomic analyses using two-dimensional differential gel electrophoresis (2D-DIGE), monitored more than 800 proteins 30 and 105 minutes after exposure and found that 18 had significant changes in abundance. A recently characterized ferritin-like antioxidant protein, DPSL, was the most highly regulated species of mRNA and protein, in addition to being post-translationally modified. As expected, a number of antioxidant related mRNAs and proteins were differentially regulated. Three of these, DPSL, superoxide dismutase, and peroxiredoxin were shown to interact and likely form a novel supramolecular complex for mitigating oxidative damage. A scheme for the ability of this complex to perform multi-step reactions is presented. Despite the central role played by DPSL, cells maintained a lower level of protection after disruption of the dpsl gene, indicating a level of redundancy in the oxidative stress pathways of S. solfataricus. This work provides the first “omics” scale assessment of the oxidative stress response for an archeal organism and together with a network analysis using data from previous studies on bacteria and eukaryotes reveals evolutionarily conserved pathways where complex and overlapping defense mechanisms protect against oxygen toxicity.
Oscillations in Monotone Systems with a Negative Feedback
(2010-01) Gedeon, Tomas
We study a finite-dimensional monotone system coupled to a slowly evolving scalar differential equation which provides a negative feedback to the monotone system. We use a theory of multivalued characteristics to show that this system admits a relaxation periodic orbit if a simple model system in $\mathbf{R}^2$ does. Our construction can be used to prove the existence of periodic orbits in slow-fast systems of arbitrary dimension. We apply our theory to a model of a cell cycle in Xenopus embryos. Abrupt changes in signals upon entry to mitosis suggests that the cell cycle is generated by a relaxation oscillation. Our results show that the cell cycle orbit is not a relaxation oscillator. However, we construct a closely related system that exhibits relaxation oscillations and that approximates the cell cycle oscillator for an intermediate range of negative feedback strengths. We show that the cell cycle oscillation disappears if the negative feedback is too weak or too strong.
Symmetry breaking clusters in soft clustering decoding of neural codes
(2010-02) Parker, Albert E.; Dimitrov, Alexander G.; Gedeon, Tomas
Information-based distortion methods have been used successfully in the analysis of neural coding problems. These approaches allow the discovery of neural symbols and the corresponding stimulus space of a neuron or neural ensemble quantitatively, while making few assumptions about the nature of either the code or of relevant stimulus features. The neural codebook is derived by quantizing sensory stimuli and neural responses into a small set of clusters, and optimizing the quantization to minimize an information distortion function. The method of annealing has been used to solve the corresponding high-dimensional nonlinear optimization problem. The annealing solutions undergo a series of bifurcations, which we study using bifurcation theory in the presence of symmetries. In this contribution we describe these symmetry breaking bifurcations in detail, and indicate some of the consequences of the form of the bifurcations. In particular, we show that the annealing solutions break symmetry at pitchfork bifurcations, and that subcritical branches can exist. Thus, at a subcritical bifurcation, there are local information distortion solutions which are not found by the method of annealing. Since the annealing procedure is guaranteed to converge to a local solution eventually, the subcritical branch must turn and become optimal at some later saddle-node bifurcation, which we have shown occur generically for this class of problems. This implies that the rate distortion curve, while convex for noninformation-based distortion measures, is not convex for information-based distortion methods.
Hantavirus transmission in sylvan and peridomestic environment
(2010-04) Gedeon, Tomas; Bodelon, Clara; Kuenzi, Amy
We developed a compartmental model for hantavirus infection in deer mice (Peromyscus maniculatus) with the goal of comparing relative importance of direct and indirect transmission in sylvan and peridomestic environments. A direct transmission occurs when the infection is mediated by the contact of an infected and an uninfected mouse, while an indirect transmission occurs when the infection is mediated by the contact of an uninfected mouse with, for instance, infected soil. Based on population dynamics data and estimates of hantavirus decay in the two types of environments, our model predicts that direct transmission dominates in the sylvan environment, while both pathways are important in peridomestic environments. The model allows us to compute a basic reproduction number R0, which indicates whether the virus will be endemic or eradicated from the mouse population, in both an autonomous and a time-periodic model. Our analysis can be used to evaluate various eradication strategies.
Somitogenesis clock-wave initiation requires differential decay and multiple binding sites for clock protein
(2010-04) Campanelli, Mark; Gedeon, Tomas
Somitogenesis is a process common to all vertebrate embryos in which repeated blocks of cells arise from the presomitic mesoderm (PSM) to lay a foundational pattern for trunk and tail development. Somites form in the wake of passing waves of periodic gene expression that originate in the tailbud and sweep posteriorly across the PSM. Previous work has suggested that the waves result from a spatiotemporally graded control protein that affects the oscillation rate of clock-gene expression. With a minimally constructed mathematical model, we study the contribution of two control mechanisms to the initial formation of this gene-expression wave. We test four biologically motivated model scenarios with either one or two clock protein transcription binding sites, and with or without differential decay rates for clock protein monomers and dimers. We examine the sensitivity of wave formation with respect to multiple model parameters and robustness to heterogeneity in cell population. We find that only a model with both multiple binding sites and differential decay rates is able to reproduce experimentally observed waveforms. Our results show that the experimentally observed characteristics of somitogenesis wave initiation constrain the underlying genetic control mechanisms.
Hyperbolicity of the fixed point set for the simple genetic algorithm
(2010-05) Hayes, Christina; Gedeon, Tomas
We study an infinite population model for the genetic algorithm, where the iteration of the algorithm corresponds to an iteration of a map G. The map G is a composition of a selection operator and a mixing operator, where the latter models effects of both mutation and crossover. We examine the hyperbolicity of fixed points of this model. We show that for a typical mixing operator all the fixed points are hyperbolic.
ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast
(2010-07) Boczko, E.; Gedeon, Tomas; Stowers, C.
Biologists have long observed periodic-like oxygen consumption oscillations in yeast populations under certain conditions, and several unsatisfactory explanations for this phenomenon have been proposed. These ‘autonomous oscillations’ have often appeared with periods that are nearly integer divisors of the calculated doubling time of the culture. We hypothesize that these oscillations could be caused by a form of cell cycle synchronization that we call clustering. We develop some novel ordinary differential equation models of the cell cycle. For these models, and for random and stochastic perturbations, we give both rigorous proofs and simulations showing that both positive and negative growth rate feedback within the cell cycle are possible agents that can cause clustering of populations within the cell cycle. It occurs for a variety of models and for a broad selection of parameter values. These results suggest that the clustering phenomenon is robust and is likely to be observed in nature. Since there are necessarily an integer number of clusters, clustering would lead to periodic-like behaviour with periods that are nearly integer divisors of the period of the cell cycle. Related experiments have shown conclusively that cell cycle clustering occurs in some oscillating yeast cultures.
Senescence and antibiotic resistance in an age-structured population model
(2010-10) De Leenheer, Patrick; Dockery, Jack D.; Gedeon, Tomas; Pilyugin, Sergei S.
Different theories have been proposed to understand the growing problem of antibiotic resistance of microbial populations. Here we investigate a model that is based on the hypothesis that senescence is a possible explanation for the existence of so-called persister cells which are resistant to antibiotic treatment. We study a chemostat model with a microbial population which is age-structured and show that if the growth rates of cells in different age classes are sufficiently close to a scalar multiple of a common growth rate, then the population will globally stabilize at a coexistence steady state. This steady state persists under an antibiotic treatment if the level of antibiotics is below a certain threshold; if the level exceeds this threshold, the washout state becomes a globally attracting equilibrium.
The chemostat with lateral gene transfer
(2010-11) De Leenheer, Patrick; Dockery, Jack D.; Gedeon, Tomas; Young, T.
We investigate the standard chemostat model when lateral gene transfer is taken into account. We will show that when the different genotypes have growth rate functions that are sufficiently close to a common growth rate function, and when the yields of the genotypes are sufficiently close to a common value, then the population evolves to a globally stable steady state, at which all genotypes coexist. These results can explain why the antibiotic-resistant strains persist in the pathogen population.
Teaching highs and lows: Exploring university teaching assistants’ experiences
(2010-11) Green, Jennifer L.
Recent reforms in statistics education have initiated the need to prepare graduate teaching assistants (TAs) for these changes. A focus group study explored the experiences and perceptions of University of Nebraska-Lincoln TAs. The results reinforced the idea that content, pedagogy, and technology are central aspects for teaching an introductory statistics course. The TAs addressed the need for clear, specific guidelines and examples, as well as collaboration between colleagues. The TAs also sought opportunities to enrich their teaching skills and, ultimately, their impact on students’ learning. These findings support previous research on graduate TAs and highlight the need for additional exploration of the role graduate statistics TAs play in introductory statistics education.
Random 2-SAT Solution Components and a Fitness Landscape
(2011) Pitman, Damien J.
We describe a limiting distribution for the number of connected components in the subgraph of the discrete cube induced by the satisfying assignments to a random 2-SAT formula. We show that, for the probability range where formulas are likely to be satisfied, the random number of components converges weakly (in the number of variables) to a distribution determined by a Poisson random variable. The number of satisfying assignments or solutions is known to grow exponentially in the number of variables. Thus, our result implies that exponentially many solutions are organized into a stochastically bounded number of components. We also describe an application to biological evolution; in particular, to a type of fitness landscape where satisfying assignments represent viable genotypes and connectivity of genotypes is limited by single site mutations. The biological result is that, with probability approaching 1, each viable genotype is connected by single site mutations to an exponential number of other viable genotypes while the number of viable clusters is finite.
Multi-scale clustering of functional data with application to hydraulic gradients in wetlands
(Columbia University, New York, 2011) Greenwood, Mark C.; Soida, Richard S.; Sharp, Julia L.; Peck, Rory G.; Rosenberry, Donald O.
A new set of methods are developed to perform cluster analysis of functions, motivated by a data set consisting of hydraulic gradients at several locations distributed across a wetland complex. The methods build on previous work on clustering of functions, such as Tarpey and Kinateder (2003) and Hitchcock et al. (2007), but explore functions generated from an additive model decomposition (Wood, 2006) of the original time se- ries. Our decomposition targets two aspects of the series, using an adaptive smoother for the trend and circular spline for the diurnal variation in the series. Different measures for comparing locations are discussed, including a method for efficiently clustering time series that are of different lengths using a functional data approach. The complicated nature of these wetlands are highlighted by the shifting group memberships depending on which scale of variation and year of the study are considered.
When a habitat freezes solid: Microorganisms over-winter within the ice column of a coastal Antarctic lake
(2011-03) Foreman, Christine M.; Dieser, Markus; Greenwood, Mark C.; Cory, R. M.; Laybourn-Parry, Johanna; Lisle, John T.; Jaros, C.; Miller, P. L.; Chin, Yu-Ping; McKnight, Diane M.
A major impediment to understanding the biology of microorganisms inhabiting Antarctic environments is the logistical constraint of conducting field work primarily during the summer season. However, organisms that persist throughout the year encounter severe environmental changes between seasons. In an attempt to bridge this gap, we collected ice core samples from Pony Lake in early November 2004 when the lake was frozen solid to its base, providing an archive for the biological and chemical processes that occurred during winter freezeup. The ice contained bacteria and virus-like particles, while flagellated algae and ciliates over-wintered in the form of inactive cysts and spores. Both bacteria and algae were metabolically active in the ice core melt water. Bacterial production ranged from 1.8 to 37.9 μg C L−1 day−1. Upon encountering favorable growth conditions in the melt water, primary production ranged from 51 to 931 μg C L−1 day−1. Because of the strong H2S odor and the presence of closely related anaerobic organisms assigned to Pony Lake bacterial 16S rRNA gene clones, we hypothesize that the microbial assemblage was strongly affected by oxygen gradients, which ultimately restricted the majority of phylotypes to distinct strata within the ice column. This study provides evidence that the microbial community over-winters in the ice column of Pony Lake and returns to a highly active metabolic state when spring melt is initiated.
Temporal coding in a nervous system
(2011-05) Aldworth, Zane N.; Dimitrov, Alexander G.; Cummins, G.; Gedeon, Tomas; Miller, J. P.
We examined the extent to which temporal encoding may be implemented by single neurons in the cercal sensory system of the house cricket Acheta domesticus. We found that these neurons exhibit a greater-than-expected coding capacity, due in part to an increased precision in brief patterns of action potentials. We developed linear and non-linear models for decoding the activity of these neurons. We found that the stimuli associated with short-interval patterns of spikes (ISIs of 8 ms or less) could be predicted better by second-order models as compared to linear models. Finally, we characterized the difference between these linear and second-order models in a low-dimensional subspace, and showed that modification of the linear models along only a few dimensions improved their predictive power to parity with the second order models. Together these results show that single neurons are capable of using temporal patterns of spikes as fundamental symbols in their neural code, and that they communicate specific stimulus distributions to subsequent neural structures.
Quantitative characterization of the filiform mechanosensory hair array on the cricket cercus
(2011-11) Miller, J. P.; Krueger, S.; Heys, Jeffrey J.; Gedeon, Tomas
Background: Crickets and other orthopteran insects sense air currents with a pair of abdominal appendages resembling antennae, called cerci. Each cercus in the common house cricket Acheta domesticus is approximately 1 cm long, and is covered with 500 to 750 filiform mechanosensory hairs. The distribution of the hairs on the cerci, as well as the global patterns of their movement vectors, have been characterized semi-quantitatively in studies over the last 40 years, and have been shown to be very stereotypical across different animals in this species. Although the cercal sensory system has been the focus of many studies in the areas of neuroethology, development, biomechanics, sensory function and neural coding, there has not yet been a quantitative study of the functional morphology of the receptor array of this important model system. Methodology/Principal Findings: We present a quantitative characterization of the structural characteristics and functional morphology of the cercal filiform hair array. We demonstrate that the excitatory direction along each hair's movement plane can be identified by features of its socket that are visible at the light-microscopic level, and that the length of the hair associated with each socket can also be estimated accurately from a structural parameter of the socket. We characterize the length and directionality of all hairs on the basal half of a sample of three cerci, and present statistical analyses of the distributions. Conclusions/Significance: The inter-animal variation of several global organizational features is low, consistent with constraints imposed by functional effectiveness and/or developmental processes. Contrary to previous reports, however, we show that the filiform hairs are not re-identifiable in the strict sense.
Advanced college-level students' categorization and use of mathematical definitions
(2012) Dickerson, David S.; Pitman, Damien J.
This qualitative study of five undergraduate mathematics majors found that some students, (even students at an advanced level of undergraduate mathematical study) have a mathematician’s perspective neither on the concept of mathematical definition nor on the structure of mathematics as a whole. Participants in this study were likely to reason from incomplete concept images rather than from concept definitions and were likely to perceive that definitions (like theorems) need to be verified. The results of this study have implications for college-level mathematics instruction.