Mathematical Manipulative Models: In Defense of Beanbag Biology

Mathematical manipulative models have had a long history of influence in biological research and in secondary school education, but they are frequently neglected in undergraduate biology education. By linking mathematical manipulative models in a four-step process-1) use of physical manipulatives, 2) interactive exploration of computer simulations, 3) derivation of mathematical relationships from core principles, and 4) analysis of real data sets-we demonstrate a process that we have shared in biological faculty development workshops led by staff from the BioQUEST Curriculum Consortium over the past 24 yr. We built this approach based upon a broad survey of literature in mathematical educational research that has convincingly demonstrated the utility of multiple models that involve physical, kinesthetic learning to actual data and interactive simulations. Two projects that use this approach are introduced: The Biological Excel Simulations and Tools in Exploratory, Experiential Mathematics (ESTEEM) Project (http://bioquest.org/esteem) and Numerical Undergraduate Mathematical Biology Education (NUMB3R5 COUNT; http://bioquest.org/numberscount). Examples here emphasize genetics, ecology, population biology, photosynthesis, cancer, and epidemiology. Mathematical manipulative models help learners break through prior fears to develop an appreciation for how mathematical reasoning informs problem solving, inference, and precise communication in biology and enhance the diversity of quantitative biology education

Regression analysis with categorized regression calibrated exposure: some interesting findings

BACKGROUND: Regression calibration as a method for handling measurement error is becoming increasingly well-known and used in epidemiologic research. However, the standard version of the method is not appropriate for exposure analyzed on a categorical (e.g. quintile) scale, an approach commonly used in epidemiologic studies. A tempting solution could then be to use the predicted continuous exposure obtained through the regression calibration method and treat it as an approximation to the true exposure, that is, include the categorized calibrated exposure in the main regression analysis. METHODS: We use semi-analytical calculations and simulations to evaluate the performance of the proposed approach compared to the naive approach of not correcting for measurement error, in situations where analyses are performed on quintile scale and when incorporating the original scale into the categorical variables, respectively. We also present analyses of real data, containing measures of folate intake and depression, from the Norwegian Women and Cancer study (NOWAC). RESULTS: In cases where extra information is available through replicated measurements and not validation data, regression calibration does not maintain important qualities of the true exposure distribution, thus estimates of variance and percentiles can be severely biased. We show that the outlined approach maintains much, in some cases all, of the misclassification found in the observed exposure. For that reason, regression analysis with the corrected variable included on a categorical scale is still biased. In some cases the corrected estimates are analytically equal to those obtained by the naive approach. Regression calibration is however vastly superior to the naive method when applying the medians of each category in the analysis. CONCLUSION: Regression calibration in its most well-known form is not appropriate for measurement error correction when the exposure is analyzed on a percentile scale. Relating back to the original scale of the exposure solves the problem. The conclusion regards all regression models

Multilocus Bayesian Estimates of Intra-Oceanic Genetic Differentiation, Connectivity, and Admixture in Atlantic Swordfish (Xiphias gladius L.)

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On the Polynomial Measurement Error Model

This paper discusses point estimation of the coefficients of polynomial measurement error (errors-in-variables) models. This includes functional and structural models. The connection between these models and total least squares (TLS) is also examined. A compendium of existing as well as new results is presented

Oceanographic barriers, divergence, and admixture : phylogeography and taxonomy of two putative subspecies of short-finned pilot whale

Funding:Commander, U.S. Pacific Fleet Environmental Readiness Division and NMFS Pacific Islands Fisheries Science Center; NMFS West Coast Region; Scripps Institution of Oceanography Edna Bailey Sussman Research Fellowship; and Woods Hole Oceanographic Institution.Genomic phylogeography plays an important role in describing evolutionary processes and their geographic, ecological, or cultural drivers. These drivers are often poorly understood in marine environments, which have fewer obvious barriers to mixing than terrestrial environments. Taxonomic uncertainty of some taxa (e.g., cetaceans), due to the difficulty in obtaining morphological data, can hamper our understanding of these processes. One such taxon, the short‐finned pilot whale, is recognized as a single global species but includes at least two distinct morphological forms described from stranding and drive hunting in Japan, the “Naisa” and “Shiho” forms. Using samples (n = 735) collected throughout their global range, we examine phylogeographic patterns of divergence by comparing mitogenomes and nuclear SNP loci. Our results suggest three types within the species: an Atlantic Ocean type, a western/central Pacific and Indian Ocean (Naisa) type, and an eastern Pacific Ocean and northern Japan (Shiho) type. mtDNA control region differentiation indicates these three types form two subspecies, separated by the East Pacific Barrier: Shiho short‐finned pilot whale, in the eastern Pacific Ocean and northern Japan, and Naisa short‐finned pilot whale, throughout the remainder of the species' distribution. Our data further indicate two diverging populations within the Naisa subspecies, in the Atlantic Ocean and western/central Pacific and Indian Oceans, separated by the Benguela Barrier off South Africa. This study reveals a process of divergence and speciation within a globally‐distributed, mobile marine predator, and indicates the importance of the East Pacific Barrier to this evolutionary process.PostprintPeer reviewe

Genetic Structure Among 50 Species of the Northeastern Pacific Rocky Intertidal Community

Comparing many species' population genetic patterns across the same seascape can identify species with different levels of structure, and suggest hypotheses about the processes that cause such variation for species in the same ecosystem. This comparative approach helps focus on geographic barriers and selective or demographic processes that define genetic connectivity on an ecosystem scale, the understanding of which is particularly important for large-scale management efforts. Moreover, a multispecies dataset has great statistical advantages over single-species studies, lending explanatory power in an effort to uncover the mechanisms driving population structure. Here, we analyze a 50-species dataset of Pacific nearshore invertebrates with the aim of discovering the most influential structuring factors along the Pacific coast of North America. We collected cytochrome c oxidase I (COI) mtDNA data from populations of 34 species of marine invertebrates sampled coarsely at four coastal locations in California, Oregon, and Alaska, and added published data from 16 additional species. All nine species with non-pelagic development have strong genetic structure. For the 41 species with pelagic development, 13 show significant genetic differentiation, nine of which show striking FST levels of 0.1–0.6. Finer scale geographic investigations show unexpected regional patterns of genetic change near Cape Mendocino in northern California for five of the six species tested. The region between Oregon and Alaska is a second focus of intraspecific genetic change, showing differentiation in half the species tested. Across regions, strong genetic subdivision occurs more often than expected in mid-to-high intertidal species, a result that may reflect reduced gene flow due to natural selection along coastal environmental gradients. Finally, the results highlight the importance of making primary research accessible to policymakers, as unexpected barriers to marine dispersal break the coast into separate demographic zones that may require their own management plans