Joseph C Watkins
 Director, Data Science Academy
 Professor, Mathematics
 Professor, BIO5 Institute
 Chair, StatisticsGIDP
 Professor, Applied Mathematics  GIDP
 Professor, Genetics  GIDP
 Professor, Public Health
Contact
 (520) 6215245
 ADMINISTRATION, Rm. 512A
 TUCSON, AZ 857210066
 jwatkins@math.arizona.edu
Degrees
 Ph.D. Mathematics
 University of Wisconsin, Madison, Wisconsin, United States
 A Central Limit Problem in Random Evolutions
 M.S. Mathematics
 University of Wisconsin, Madison, Wisconsin, United States
 none
 M.A. Mathematics
 University of Tennessee, Knoxville, Tennessee, United States
 Second Quantization
 B.A. Mathematics
 University of Tennessee, Knoxville, Tennessee, United States
Work Experience
 University of Arizona, Tucson, Arizona (2007  Ongoing)
 University of Arizona, Tucson, Arizona (1996  2007)
 University of Arizona, Tucson, Arizona (1992  1996)
 Northwestern University, Evanston, Illinois (1987)
 University of Southern California, Los Angeles, California (1986  1992)
 Institute for Mathematics and its Applications, University of Minnesota (1985)
 University of British Columbia, Vancouver, British Columbia (1982  1985)
 Freie Universität Berlin (1980)
Interests
Teaching
probability and statistics, stochastic processes, quantitative and mathematical biology
Research
probability theory and stochastic process, statistics, applications to the life sciences, especially to genetics and genomics
Courses
202021 Courses

Theory of Probability
MATH 564 (Fall 2020) 
Theory of Probability
STAT 564 (Fall 2020)
201920 Courses

Honors Thesis
DATA 498H (Spring 2020) 
Intro to Statistical Computing
DATA 375 (Spring 2020) 
Theory of Statistics
MATH 566 (Spring 2020) 
Theory of Statistics
STAT 566 (Spring 2020) 
Thesis
STAT 910 (Spring 2020) 
Topics in Math
MATH 596A (Spring 2020) 
Honors Thesis
MATH 498H (Fall 2019) 
Independent Study
STAT 599 (Fall 2019) 
Research
STAT 900 (Fall 2019) 
Theory of Probability
MATH 564 (Fall 2019) 
Theory of Probability
STAT 564 (Fall 2019) 
Thesis
STAT 910 (Fall 2019) 
Topics in Math
MATH 596A (Fall 2019)
201819 Courses

Intro Statistical Method
DATA 363 (Spring 2019) 
Research
STAT 900 (Spring 2019) 
Topics in Math
MATH 596A (Spring 2019) 
Intro Statistical Method
MATH 363 (Fall 2018) 
Research
STAT 900 (Fall 2018) 
Topics in Math
MATH 596A (Fall 2018)
201718 Courses

Thesis
STAT 910 (Summer I 2018) 
Directed Research
MATH 492 (Spring 2018) 
Dissertation
STAT 920 (Spring 2018) 
Independent Study
MATH 499 (Spring 2018) 
Intro Statistical Method
MATH 363 (Spring 2018) 
Thesis
STAT 910 (Spring 2018) 
Topics in Math
MATH 596A (Spring 2018) 
Dissertation
STAT 920 (Fall 2017) 
Intro Statistical Method
MATH 363 (Fall 2017) 
Thesis
STAT 910 (Fall 2017) 
Topics in Math
MATH 596A (Fall 2017)
201617 Courses

Thesis
STAT 910 (Summer I 2017) 
Dissertation
STAT 920 (Spring 2017) 
Honors Thesis
MATH 498H (Spring 2017) 
Intro Statistical Method
MATH 363 (Spring 2017) 
Thesis
STAT 910 (Spring 2017) 
Topics in Math
MATH 596A (Spring 2017) 
Topics in Undergrad Math
MATH 396T (Spring 2017) 
Directed Research
MATH 392 (Fall 2016) 
Dissertation
GENE 920 (Fall 2016) 
Dissertation
STAT 920 (Fall 2016) 
Honors Thesis
MATH 498H (Fall 2016) 
Independent Study
GENE 699 (Fall 2016) 
Intro Statistical Method
MATH 363 (Fall 2016) 
Thesis
STAT 910 (Fall 2016) 
Topics in Math
MATH 596A (Fall 2016)
201516 Courses

Intro Ord Diff Equations
MATH 254 (Summer I 2016) 
Directed Research
MATH 392 (Spring 2016) 
Dissertation
GENE 920 (Spring 2016) 
Dissertation
STAT 920 (Spring 2016) 
Independent Study
STAT 599 (Spring 2016) 
Research
STAT 900 (Spring 2016) 
Topics in Math
MATH 596A (Spring 2016)
Scholarly Contributions
Journals/Publications
 Encinas, A. C., Moore, I. K., Watkins, J. C., & Hammer, M. F. (2019). Influence of age at seizure onset on the acquisition of neurodevelopmental skills in an SCN8A cohort. Epilepsia, 60(8), 17111720.More infoTo characterize a cohort of patients with SCN8Arelated epilepsy and to perform analyses to identify correlations involving the acquisition of neurodevelopmental skills.
 Hallmark, B., Karafet, T. M., Hsieh, P., Osipova, L. P., Watkins, J. C., & Hammer, M. F. (2018). Genomic Evidence of Local Adaptation to Climate and Diet in Indigenous Siberians. Molecular biology and evolution.More infoThe indigenous inhabitants of Siberia live in some of the harshest environments on earth, experiencing extended periods of severe cold temperatures, dramatic variation in photoperiod, and limited and highly variable food resources. While the successful longterm settlement of this area by humans required multiple behavioral and cultural innovations, the nature of the underlying genetic changes has generally remained elusive. In this study, we used a threepart approach to identify putative targets of positive natural selection in Siberians. We first performed selection scans on whole exome and genomewide SNP array data from multiple Siberian populations. We then annotated candidates in the tails of the empirical distributions, focusing on candidates with evidence linking them to biological processes and phenotypes previously identified as relevant to adaptation in circumpolar groups. The top candidates were then genotyped in additional populations to determine their spatial allele frequency distributions and associations with climate variables. Our analysis reveals missense mutations in three genes involved in lipid metabolism (PLA2G2A, PLIN1, ANGPTL8) that exhibit genomic and spatial patterns consistent with selection for cold climate and/or diet. These variants are unified by their connection to brown adipose tissue, and may help to explain previously observed physiological differences in Siberians such as low serum lipid levels and increased basal metabolic rate. These results support the hypothesis that indigenous Siberians have genetically adapted to their local environment by selection on multiple genes.
 QuintoCortés, C. D., Woerner, A. E., Watkins, J. C., & Hammer, M. F. (2018). Modeling SNP array ascertainment with Approximate Bayesian Computation for demographic inference. Scientific reports, 8(1), 10209.More infoSingle nucleotide polymorphisms (SNPs) in commercial arrays have often been discovered in a small number of samples from selected populations. This ascertainment skews patterns of nucleotide diversity and affects population genetic inferences. We propose a demographic inference pipeline that explicitly models the SNP discovery protocol in an Approximate Bayesian Computation (ABC) framework. We simulated genomic regions according to a demographic model incorporating parameters for the divergence of three wellcharacterized HapMap populations and recreated the SNP distribution of a commercial array by varying the number of haploid samples and the allele frequency cutoff in the given regions. We then calculated summary statistics obtained from both the ascertained and genomic data and inferred ascertainment and demographic parameters. We implemented our pipeline to study the admixture process that gave rise to the presentday Mexican population. Our estimate of the time of admixture is closer to the historical dates than those in previous works which did not consider ascertainment bias. Although the use of whole genome sequences for demographic inference is becoming the norm, there are still underrepresented areas of the world from where only SNP array data are available. Our inference framework is applicable to those cases and will help with the demographic inference.
 Woerner, A. E., Veeramah, K. R., Watkins, J. C., & Hammer, M. F. (2018). The role of phylogenetically conserved elements in shaping patterns of human genomic diversity. Molecular biology and evolution.More infoEvolutionary genetic studies have shown a positive correlation between levels of nucleotide diversity and either rates of recombination or genetic distance to genes. Both positivedirectional and purifying selection have been offered as the source of these correlations via genetic hitchhiking and background selection, respectively. Phylogenetically conserved elements (CEs) are short (∼100bp), widely distributed (comprising ∼5% of genome), sequences that are often found far from genes. While the function of many CEs is unknown, CEs also are associated with reduced diversity at linked sites. Using high coverage (>80x) whole genome data from two human populations, the Yoruba and the CEU, we perform fine scale evaluations of diversity, rates of recombination, and linkage to genes. We find that the local rate of recombination has a stronger effect on levels of diversity than linkage to genes, and that these effects of recombination persist even in regions far from genes. Our whole genome modeling demonstrates that, rather than recombination or GCbiased gene conversion, selection on sites within or linked to CEs better explains the observed genomic diversity patterns. A major implication is that very few sites in the human genome are predicted to be free of the effects of selection. These sites, which we refer to as the human "neutralome", comprise only 1.2% of the autosomes and 5.1% of the X chromosome. Demographic analysis of the neutralome reveals larger population sizes and lower rates of growth for ancestral human populations than inferred by previous analyses.
 Hammer, M. F., Ishii, A., Johnstone, L., Tchourbanov, A., Lau, B., Sprissler, R., Hallmark, B., Zhang, M., Zhou, J., Watkins, J., & Hirose, S. (2017). Rare variants of small effect size in neuronal excitability genes influence clinical outcome in Japanese cases of SCN1A truncationpositive Dravet syndrome. PloS one, 12(7), e0180485.More infoDravet syndrome (DS) is a rare, devastating form of childhood epilepsy that is often associated with mutations in the voltagegated sodium channel gene, SCN1A. There is considerable variability in expressivity within families, as well as among individuals carrying the same primary mutation, suggesting that clinical outcome is modulated by variants at other genes. To identify modifier gene variants that contribute to clinical outcome, we sequenced the exomes of 22 individuals at both ends of a phenotype distribution (i.e., mild and severe cognitive condition). We controlled for variation associated with different mutation types by limiting inclusion to individuals with a de novo truncation mutation resulting in SCN1A haploinsufficiency. We performed tests aimed at identifying 1) single common variants that are enriched in either phenotypic group, 2) sets of common or rare variants aggregated in and around genes associated with clinical outcome, and 3) rare variants in 237 candidate genes associated with neuronal excitability. While our power to identify enrichment of a common variant in either phenotypic group is limited as a result of the rarity of mild phenotypes in individuals with SCN1A truncation variants, our top candidates did not map to functional regions of genes, or in genes that are known to be associated with neurological pathways. In contrast, we found a statisticallysignificant excess of rare variants predicted to be damaging and of small effect size in genes associated with neuronal excitability in severely affected individuals. A KCNQ2 variant previously associated with benign neonatal seizures is present in 3 of 12 individuals in the severe category. To compare our results with the healthy population, we performed a similar analysis on whole exome sequencing data from 70 Japanese individuals in the 1000 genomes project. Interestingly, the frequency of rare damaging variants in the same set of neuronal excitability genes in healthy individuals is nearly as high as in severely affected individuals. Rather than a single common gene/variant modifying clinical outcome in SCN1Arelated epilepsies, our results point to the cumulative effect of rare variants with little to no measurable phenotypic effect (i.e., typical genetic background) unless present in combination with a diseasecausing truncation mutation in SCN1A.
 Hsieh, P., Hallmark, B., Watkins, J., Karafet, T. M., Osipova, L. P., Gutenkunst, R. N., & Hammer, M. F. (2017). Exome Sequencing Provides Evidence of Polygenic Adaptation to a FatRich Animal Diet in Indigenous Siberian Populations. Molecular biology and evolution, 34(11), 29132926.More infoSiberia is one of the coldest environments on Earth and has great seasonal temperature variation. Longterm settlement in northern Siberia undoubtedly required biological adaptation to severe cold stress, dramatic variation in photoperiod, and limited food resources. In addition, recent archeological studies show that humans first occupied Siberia at least 45,000 years ago; yet our understanding of the demographic history of modern indigenous Siberians remains incomplete. In this study, we use wholeexome sequencing data from the Nganasans and Yakuts to infer the evolutionary history of these two indigenous Siberian populations. Recognizing the complexity of the adaptive process, we designed a modelbased test to systematically search for signatures of polygenic selection. Our approach accounts for stochasticity in the demographic process and the hitchhiking effect of classic selective sweeps, as well as potential biases resulting from recombination rate and mutation rate heterogeneity. Our demographic inference shows that the Nganasans and Yakuts diverged ∼12,00013,000 years ago from EastAsian ancestors in a process involving continuous gene flow. Our polygenic selection scan identifies seven candidate gene sets with Siberianspecific signals. Three of these gene sets are related to diet, especially to fat metabolism, consistent with the hypothesis of adaptation to a fatrich animal diet. Additional testing rejects the effect of hitchhiking and favors a model in which selection yields small allele frequency changes at multiple unlinked genes.
 Hsieh, P., Hsieh, P., Hallmark, B., Hallmark, B., Watkins, J. C., Watkins, J. C., Karafet, T. C., Karafet, T. C., Osipova, L. P., Osipova, L. P., Gutenkunst, R. N., Gutenkunst, R. N., Hammer, M. F., & Hammer, M. F. (2017). Exome sequencing provides evidence of polygenic adaptation to a fatrich animal diet in indigenous Siberian populations. Molecular Biology and Evolution, 34, 2914.
 Ishii, A., Kang, J. Q., Schornak, C. C., Hernandez, C. C., Shen, W., Watkins, J. C., Macdonald, R. L., & Hirose, S. (2017). A de novo missense mutation of GABRB2 causes early myoclonic encephalopathy. Journal of medical genetics, 54(3), 202211.More infoEarly myoclonic encephalopathy (EME), a disease with a devastating prognosis, is characterised by neonatal onset of seizures and massive myoclonus accompanied by a continuous suppressionburst EEG pattern. Three genes are associated with EMEs that have metabolic features. Here, we report a pathogenic mutation of an ion channel as a cause of EME for the first time.
 Ishii, A., Watkins, J. C., Chen, D., Hirose, S., & Hammer, M. F. (2017). Clinical implications of SCN1A missense and truncation variants in a large Japanese cohort with Dravet syndrome. Epilepsia, 58(2), 282290.More infoTwo major classes of SCN1A variants are associated with Dravet syndrome (DS): those that result in haploinsufficiency (truncating) and those that result in an amino acid substitution (missense). The aim of this retrospective study was to describe the first large cohort of Japanese patients with SCN1A mutationpositive DS (n = 285), and investigate the relationship between variant (type and position) and clinical expression and response to treatment.
 Glazer, E. S., Zhang, H. H., Hill, K. A., Patel, C., Kha, S. T., Yozwiak, M. L., Bartels, H., Nafissi, N. N., Watkins, J. C., Alberts, D. S., & Krouse, R. S. (2016). Evaluating IPMN and pancreatic carcinoma utilizing quantitative histopathology. Cancer medicine, 5(10), 28412847.More infoIntraductal papillary mucinous neoplasms (IPMN) are pancreatic lesions with uncertain biologic behavior. This study sought objective, accurate prediction tools, through the use of quantitative histopathological signatures of nuclear images, for classifying lesions as chronic pancreatitis (CP), IPMN, or pancreatic carcinoma (PC). Fortyfour pancreatic resection patients were retrospectively identified for this study (12 CP; 16 IPMN; 16 PC). Regularized multinomial regression quantitatively classified each specimen as CP, IPMN, or PC in an automated, blinded fashion. Classification certainty was determined by subtracting the smallest classification probability from the largest probability (of the three groups). The certainty function varied from 1.0 (perfectly classified) to 0.0 (random). From each lesion, 180 ± 22 nuclei were imaged. Overall classification accuracy was 89.6% with six unique nuclear features. No CP cases were misclassified, 1/16 IPMN cases were misclassified, and 4/16 PC cases were misclassified. Certainty function was 0.75 ± 0.16 for correctly classified lesions and 0.47 ± 0.10 for incorrectly classified lesions (P = 0.0005). Uncertainty was identified in four of the five misclassified lesions. Quantitative histopathology provides a robust, novel method to distinguish among CP, IPMN, and PC with a quantitative measure of uncertainty. This may be useful when there is uncertainty in diagnosis.
 Bailey, B. L., Visscher, K., & Watkins, J. (2014). A stochastic model of translation with 1 programmed ribosomal frameshifting. Physical biology, 11(1), 016009.More infoMany viruses produce multiple proteins from a single mRNA sequence by encoding overlapping genes. One mechanism to decode both genes, which reside in alternate reading frames, is 1 programmed ribosomal frameshifting. Although recognized for over 25 years, the molecular and physical mechanism of 1 frameshifting remains poorly understood. We have developed a mathematical model that treats mRNA translation and associated 1 frameshifting as a stochastic process in which the transition probabilities are based on the energetics of local molecular interactions. The model predicts both the location and efficiency of 1 frameshift events in HIV1. Moreover, we compute 1 frameshift efficiencies upon mutations in the viral mRNA sequence and variations in relative tRNA abundances, predictions that are directly testable in experiment.
 Veeramah, K. R., Gutenkunst, R. N., Woerner, A. E., Watkins, J. C., & Hammer, M. F. (2014). Evidence for increased levels of positive and negative selection on the X chromosome versus autosomes in humans. Molecular biology and evolution, 31(9), 226782.More infoPartially recessive variants under positive selection are expected to go to fixation more quickly on the X chromosome as a result of hemizygosity, an effect known as fasterX. Conversely, purifying selection is expected to reduce substitution rates more effectively on the X chromosome. Previous work in humans contrasted divergence on the autosomes and X chromosome, with results tending to support the fasterX effect. However, no study has yet incorporated both divergence and polymorphism to quantify the effects of both purifying and positive selection, which are opposing forces with respect to divergence. In this study, we develop a framework that integrates previously developed theory addressing differential rates of X and autosomal evolution with methods that jointly estimate the level of purifying and positive selection via modeling of the distribution of fitness effects (DFE). We then utilize this framework to estimate the proportion of nonsynonymous substitutions fixed by positive selection (α) using exome sequence data from a West African population. We find that varying the female to male breeding ratio (β) has minimal impact on the DFE for the X chromosome, especially when compared with the effect of varying the dominance coefficient of deleterious alleles (h). Estimates of α range from 46% to 51% and from 4% to 24% for the X chromosome and autosomes, respectively. While dependent on h, the magnitude of the difference between α values estimated for these two systems is highly statistically significant over a range of biologically realistic parameter values, suggesting fasterX has been operating in humans.
 Watkins, J., Visscher, K., Bailey, B. L., Watkins, J., Visscher, K., & Bailey, B. L. (2014). A stochastic model of translation with 1 programmed ribosomal frameshifting. Physical Biology, 11(1), 116.More infoAbstract: Many viruses produce multiple proteins from a single mRNA sequence by encoding overlapping genes. One mechanism to decode both genes, which reside in alternate reading frames, is 1 programmed ribosomal frameshifting. Although recognized for over 25 years, the molecular and physical mechanism of 1 frameshifting remains poorly understood. We have developed a mathematical model that treats mRNA translation and associated 1 frameshifting as a stochastic process in which the transition probabilities are based on the energetics of local molecular interactions. The model predicts both the location and efficiency of 1 frameshift events in HIV1. Moreover, we compute 1 frameshift efficiencies upon mutations in the viral mRNA sequence and variations in relative tRNA abundances, predictions that are directly testable in experiment. © 2014 IOP Publishing Ltd.
 Mendez, F. L., Watkins, J. C., & Hammer, M. F. (2013). Neandertal origin of genetic variation at the cluster of OAS immunity genes. Molecular biology and evolution, 30(4), 798801.More infoAnalyses of ancient DNA from extinct humans reveal signals of at least two independent hybridization events in the history of nonAfrican populations. To date, there are very few examples of specific genetic variants that have been rigorously identified as introgressive. Here, we survey DNA sequence variation in the OAS gene cluster on chromosome 12 and provide strong evidence that a haplotype extending for ~185 kb introgressed from Neandertals. This haplotype is nearly restricted to Eurasians and is estimated to have diverged from the Neandertal sequence ~125 kya. Despite the potential for novel functional variation, the observed frequency of this haplotype is consistent with neutral introgression. This is the second locus in the human genome, after STAT2, carrying distinct haplotypes that appear to have introgressed separately from both Neandertals and Denisova.
 Mendez, F. L., Watkins, J. C., & Hammer, M. F. (2012). A haplotype at STAT2 Introgressed from neanderthals and serves as a candidate of positive selection in Papua New Guinea. American journal of human genetics, 91(2), 26574.More infoSignals of archaic admixture have been identified through comparisons of the draft Neanderthal and Denisova genomes with those of living humans. Studies of individual loci contributing to these genomewide average signals are required for characterization of the introgression process and investigation of whether archaic variants conferred an adaptive advantage to the ancestors of contemporary human populations. However, no definitive case of adaptive introgression has yet been described. Here we provide a DNA sequence analysis of the innate immune gene STAT2 and show that a haplotype carried by many Eurasians (but not subSaharan Africans) has a sequence that closely matches that of the Neanderthal STAT2. This haplotype, referred to as N, was discovered through a resequencing survey of the entire coding region of STAT2 in a global sample of 90 individuals. Analyses of publicly available complete genome sequence data show that haplotype N shares a recent common ancestor with the Neanderthal sequence (~80 thousand years ago) and is found throughout Eurasia at an average frequency of ~5%. Interestingly, N is found in Melanesian populations at ~10fold higher frequency (~54%) than in Eurasian populations. A neutrality test that controls for demography rejects the hypothesis that a variant of N rose to high frequency in Melanesia by genetic drift alone. Although we are not able to pinpoint the precise target of positive selection, we identify nonsynonymous mutations in ERBB3, ESYT1, and STAT2all of which are part of the same 250 kb introgressive haplotypeas good candidates.
 Mendez, F. L., Watkins, J. C., & Hammer, M. F. (2012). Global genetic variation at OAS1 provides evidence of archaic admixture in Melanesian populations. Molecular biology and evolution, 29(6), 151320.More infoRecent analysis of DNA extracted from two Eurasian forms of archaic human shows that more genetic variants are shared with humans currently living in Eurasia than with anatomically modern humans in subSaharan Africa. Although these genomewide average measures of genetic similarity are consistent with the hypothesis of archaic admixture in Eurasia, analyses of individual loci exhibiting the signal of archaic introgression are needed to test alternative hypotheses and investigate the admixture process. Here, we provide a detailed sequence analysis of the innate immune gene OAS1, a locus with a divergent Melanesian haplotype that is very similar to the Denisova sequence from the Altai region of Siberia. We resequenced a 7kb region encompassing the OAS1 gene in 88 individuals from six Old World populations (San, Biaka, Mandenka, French Basque, Han Chinese, and Papua New Guineans) and discovered previously unknown and ancient genetic variation. The 5' region of this gene has unusual patterns of diversity, including 1) higher levels of nucleotide diversity in Papuans than in subSaharan Africans, 2) very deep ancestry with an estimated time to the most recent common ancestor of >3 myr, and 3) a basal branching pattern with Papuan individuals on either side of the rooted network. A global geographic survey of >1,500 individuals showed that the divergent Papuan haplotype is nearly restricted to populations from eastern Indonesia and Melanesia. Polymorphic sites within this haplotype are shared with the draft Denisova genome over a span of ∼90 kb and are associated with an extended block of linkage disequilibrium, supporting the hypothesis that this haplotype introgressed from an archaic source that likely lived in Eurasia.
 Veeramah, K. R., Wegmann, D., Woerner, A., Mendez, F. L., Watkins, J. C., DestroBisol, G., Soodyall, H., Louie, L., & Hammer, M. F. (2012). An early divergence of KhoeSan ancestors from those of other modern humans is supported by an ABCbased analysis of autosomal resequencing data. Molecular biology and evolution, 29(2), 61730.More infoSubSaharan Africa has consistently been shown to be the most genetically diverse region in the world. Despite the fact that a substantial portion of this variation is partitioned between groups practicing a variety of subsistence strategies and speaking diverse languages, there is currently no consensus on the genetic relationships of subSaharan African populations. San (a subgroup of KhoeSan) and many Pygmy groups maintain huntergatherer lifestyles and cluster together in autosomalbased analysis, whereas nonPygmy NigerKordofanian speakers (nonPygmy NKs) predominantly practice agriculture and show substantial genetic homogeneity despite their wide geographic range throughout subSaharan Africa. However, KhoeSan, who speak a set of relatively unique clickbased languages, have long been thought to be an early branch of anatomically modern humans based on phylogenetic analysis. To formally test models of divergence among the ancestors of modern African populations, we resequenced a sample of San, Eastern, and Western Pygmies and nonPygmy NKs individuals at 40 nongenic (∼2 kb) regions and then analyzed these data within an Approximate Bayesian Computation (ABC) framework. We find substantial support for a model of an early divergence of KhoeSan ancestors from a protoPygmynonPygmy NKs group ∼110 thousand years ago over a model incorporating a protoKhoeSanPygmy huntergatherer divergence from the ancestors of nonPygmy NKs. The results of our analyses are consistent with previously identified signals of a strong bottleneck in Mbuti Pygmies and a relatively recent expansion of nonPygmy NKs. We also develop a number of methodologies that utilize "pseudoobserved" data sets to optimize our ABCbased inference. This approach is likely to prove to be an invaluable tool for demographic inference using genomewide resequencing data.
 Hammer, M. F., Woerner, A. E., Mendez, F. L., Watkins, J. C., & Wall, J. D. (2011). Genetic evidence for archaic admixture in Africa. Proceedings of the National Academy of Sciences of the United States of America, 108(37), 151238.More infoA longdebated question concerns the fate of archaic forms of the genus Homo: did they go extinct without interbreeding with anatomically modern humans, or are their genes present in contemporary populations? This question is typically focused on the genetic contribution of archaic forms outside of Africa. Here we use DNA sequence data gathered from 61 noncoding autosomal regions in a sample of three subSaharan African populations (Mandenka, Biaka, and San) to test models of African archaic admixture. We use two complementary approximatelikelihood approaches and a model of human evolution that involves recent population structure, with and without gene flow from an archaic population. Extensive simulation results reject the null model of no admixture and allow us to infer that contemporary African populations contain a small proportion of genetic material (≈ 2%) that introgressed ≈ 35 kya from an archaic population that split from the ancestors of anatomically modern humans ≈ 700 kya. Three candidate regions showing deep haplotype divergence, unusual patterns of linkage disequilibrium, and small basal clade size are identified and the distributions of introgressive haplotypes surveyed in a sample of populations from across subSaharan Africa. One candidate locus with an unusual segment of DNA that extends for >31 kb on chromosome 4 seems to have introgressed into modern Africans from a nowextinct taxon that may have lived in central Africa. Taken together our results suggest that polymorphisms present in extant populations introgressed via relatively recent interbreeding with hominin forms that diverged from the ancestors of modern humans in the LowerMiddle Pleistocene.
 Hammer, M. F., Woerner, A. E., Mendez, F. L., Watkins, J. C., Cox, M. P., & Wall, J. D. (2010). The ratio of human X chromosome to autosome diversity is positively correlated with genetic distance from genes. Nature genetics, 42(10), 8301.More infoThe ratio of Xlinked to autosomal diversity was estimated from an analysis of six human genome sequences and found to deviate from the expected value of 0.75. However, the direction of this deviation depends on whether a particular sequence is close to or far from the nearest gene. This pattern may be explained by stronger locally acting selection on Xlinked genes compared with autosomal genes, combined with larger effective population sizes for females than for males.
 Marsteller, P., de Pillis, L., Findley, A., Joplin, K., Pelesko, J., Nelson, K., Thompson, K., Usher, D., & Watkins, J. (2010). Toward integration: from quantitative biology to mathbiobiomath?. CBE life sciences education, 9(3), 16571.More infoIn response to the call of BIO2010 for integrating quantitative skills into undergraduate biology education, 30 Howard Hughes Medical Institute (HHMI) Program Directors at the 2006 HHMI Program Directors Meeting established a consortium to investigate, implement, develop, and disseminate best practices resulting from the integration of math and biology. With the assistance of an HHMIfunded minigrant, led by Karl Joplin of East Tennessee State University, and support in institutional HHMI grants at Emory and University of Delaware, these institutions held a series of summer institutes and workshops to document progress toward and address the challenges of implementing a more quantitative approach to undergraduate biology education. This report summarizes the results of the four summer institutes (20072010). The group developed four draft white papers, a wiki site, and a listserv. One major outcome of these meetings is this issue of CBELife Sciences Education, which resulted from proposals at our 2008 meeting and a January 2009 planning session. Many of the papers in this issue emerged from or were influenced by these meetings.
 Watkins, J. C. (2010). Convergence time to the Ewens sampling formula in the infinite alleles Moran model. Journal of Mathematical Biology, 60(2), 189206.More infoPMID: 19288263;Abstract: In this paper, we establish an upper bound for time to convergence to stationarity for the discrete time infinite alleles Moran model. If M is the population size and μ is the mutation rate, this bound gives a cutoff time of log(M μ)/μ generations. The stationary distribution for this process in the case of sampling without replacement is the Ewens sampling formula. We show that the bound for the total variation distance from the generation t distribution to the Ewens sampling formula is well approximated by one of the extreme value distributions, namely, a standard Gumbel distribution. Beginning with the card shuffling examples of Aldous and Diaconis and extending the ideas of Donnelly and Rodrigues for the two allele model, this model adds to the list of Markov chains that show evidence for the cutoff phenomenon. Because of the broad use of infinite alleles models, this cutoff sets the time scale of applicability for statistical tests based on the Ewens sampling formula and other tests of neutrality in a number of population genetic studies. © SpringerVerlag 2009.
 Watkins, J. C. (2010). On a calculusbased statistics course for life science students. CBE Life Sciences Education, 9(3), 298310.More infoPMID: 20810962;PMCID: PMC2931677;Abstract: The choice of pedagogy in statistics should take advantage of the quantitative capabilities and scientific background of the students. In this article, we propose a model for a statistics course that assumes student competency in calculus and a broadening knowledge in biology. We illustrate our methods and practices through examples from the curriculum. © 2010 The American Society for Cell Biology.
 Watkins, J., & Watkins, J. C. (2010). Convergence time to the Ewens sampling formula in the infinite alleles Moran model. Journal of mathematical biology, 60(2).More infoIn this paper, we establish an upper bound for time to convergence to stationarity for the discrete time infinite alleles Moran model. If M is the population size and mu is the mutation rate, this bound gives a cutoff time of log(Mmu)/mu generations. The stationary distribution for this process in the case of sampling without replacement is the Ewens sampling formula. We show that the bound for the total variation distance from the generation t distribution to the Ewens sampling formula is well approximated by one of the extreme value distributions, namely, a standard Gumbel distribution. Beginning with the card shuffling examples of Aldous and Diaconis and extending the ideas of Donnelly and Rodrigues for the two allele model, this model adds to the list of Markov chains that show evidence for the cutoff phenomenon. Because of the broad use of infinite alleles models, this cutoff sets the time scale of applicability for statistical tests based on the Ewens sampling formula and other tests of neutrality in a number of population genetic studies.
 Watkins, J., & Watkins, J. C. (2010). On a calculusbased statistics course for life science students. CBE life sciences education, 9(3).More infoThe choice of pedagogy in statistics should take advantage of the quantitative capabilities and scientific background of the students. In this article, we propose a model for a statistics course that assumes student competency in calculus and a broadening knowledge in biology. We illustrate our methods and practices through examples from the curriculum.
 Lansing, J. S., Watkins, J. C., Hallmark, B., Cox, M. P., Karafet, T. M., Sudoyo, H., & Hammer, M. F. (2008). Male dominance rarely skews the frequency distribution of Y chromosome haplotypes in human populations. Proceedings of the National Academy of Sciences of the United States of America, 105(33), 1164550.More infoA central tenet of evolutionary social science holds that behaviors, such as those associated with social dominance, produce fitness effects that are subject to cultural selection. However, evidence for such selection is inconclusive because it is based on shortterm statistical associations between behavior and fertility. Here, we show that the evolutionary effects of dominance at the population level can be detected using noncoding regions of DNA. Highly variable polymorphisms on the nonrecombining portion of the Y chromosome can be used to trace lines of descent from a common male ancestor. Thus, it is possible to test for the persistence of differential fertility among patrilines. We examine haplotype distributions defined by 12 short tandem repeats in a sample of 1269 men from 41 Indonesian communities and test for departures from neutral mutationdrift equilibrium based on the Ewens sampling formula. Our tests reject the neutral model in only 5 communities. Analysis and simulations show that we have sufficient power to detect such departures under varying demographic conditions, including founder effects, bottlenecks, and migration, and at varying levels of social dominance. We conclude that patrilines seldom are dominant for more than a few generations, and thus traits or behaviors that are strictly paternally inherited are unlikely to be under strong cultural selection.
 Lansing, J. S., Cox, M. P., Downey, S. S., Gabler, B. M., Hallmark, B., Karafet, T. M., Norquest, P., Schoenfelder, J. W., Sudoyo, H., Watkins, J. C., & Hammer, M. F. (2007). Coevolution of languages and genes on the island of Sumba, eastern Indonesia. Proceedings of the National Academy of Sciences of the United States of America, 104(41), 160226.More infoNumerous studies indicate strong associations between languages and genes among human populations at the global scale, but all broader scale genetic and linguistic patterns must arise from processes originating at the community level. We examine linguistic and genetic variation in a contact zone on the eastern Indonesian island of Sumba, where Neolithic Austronesian farming communities settled and began interacting with aboriginal foraging societies approximately 3,500 years ago. Phylogenetic reconstruction based on a 200word Swadesh list sampled from 29 localities supports the hypothesis that Sumbanese languages derive from a single ancestral Austronesian language. However, the proportion of cognates (words with a common origin) traceable to ProtoAustronesian (PAn) varies among language subgroups distributed across the island. Interestingly, a positive correlation was found between the percentage of Y chromosome lineages that derive from Austronesian (as opposed to aboriginal) ancestors and the retention of PAn cognates. We also find a striking correlation between the percentage of PAn cognates and geographic distance from the site where many Sumbanese believe their ancestors arrived on the island. These languagegenegeography correlations, unprecedented at such a fine scale, imply that historical patterns of social interaction between expanding farmers and resident huntergatherers largely explain communitylevel language evolution on Sumba. We propose a model to explain linguistic and demographic coevolution at fine spatial and temporal scales.
 Norquest, P. K., Lansing, J. S., Hammer, M. F., Watkins, J. C., Karafet, T. M., Cox, M. P., Downey, S. S., Gabler, B. M., Schoenfelder, J. W., & Sudoyo, H. (2007). Coevolution of languages and genes on the island of Sumba, eastern Indonesia. Proceedings of the National Academy of Sciences, 104(41), 16022–16026. doi:10.1073/pnas.0704451104More infoNumerous studies indicate strong associations between languages and genes among human populations at the global scale, but all broader scale genetic and linguistic patterns must arise from processes originating at the community level. We examine linguistic and genetic variation in a contact zone on the eastern Indonesian island of Sumba, where Neolithic Austronesian farming communities settled and began interacting with aboriginal foraging societies ≈3,500 years ago. Phylogenetic reconstruction based on a 200word Swadesh list sampled from 29 localities supports the hypothesis that Sumbanese languages derive from a single ancestral Austronesian language. However, the proportion of cognates (words with a common origin) traceable to ProtoAustronesian (PAn) varies among language subgroups distributed across the island. Interestingly, a positive correlation was found between the percentage of Y chromosome lineages that derive from Austronesian (as opposed to aboriginal) ancestors and the retention of PAn cognates. We also find a striking correlation between the percentage of PAn cognates and geographic distance from the site where many Sumbanese believe their ancestors arrived on the island. These language–gene–geography correlations, unprecedented at such a fine scale, imply that historical patterns of social interaction between expanding farmers and resident huntergatherers largely explain communitylevel language evolution on Sumba. We propose a model to explain linguistic and demographic coevolution at fine spatial and temporal scales.
 Watkins, J. C. (2007). Microsatellite evolution: Markov transition functions for a suite of models. Theoretical Population Biology, 71(2), 147159.More infoPMID: 17123560;Abstract: This paper takes from the collection of models considered by Whittaker et al. [2003. Likelihoodbased estimation of microsatellite mutation rates. Genetics 164, 781787] derived from direct observation of microsatellite mutation in parentchild pairs and provides analytical expressions for the probability distributions for the change in number of repeats over any given number of generations. The mathematical framework for this analysis is the theory of Markov processes. We find these expressions using two approaches, approximating by circulant matrices and solving a partial differential equation satisfied by the generating function. The impact of the differing choice of models is examined using likelihood estimates for time to most recent common ancestor. The analysis presented here may play a role in elucidating the connections between these two approaches and shows promise in reconciling differences between estimates for mutation rates based on Whittaker's approach and methods based on phylogenetic analyses. © 2006 Elsevier Inc. All rights reserved.
 Watkins, J., & Watkins, J. C. (2007). Microsatellite evolution: Markov transition functions for a suite of models. Theoretical population biology, 71(2).More infoThis paper takes from the collection of models considered by Whittaker et al. [2003. Likelihoodbased estimation of microsatellite mutation rates. Genetics 164, 781787] derived from direct observation of microsatellite mutation in parentchild pairs and provides analytical expressions for the probability distributions for the change in number of repeats over any given number of generations. The mathematical framework for this analysis is the theory of Markov processes. We find these expressions using two approaches, approximating by circulant matrices and solving a partial differential equation satisfied by the generating function. The impact of the differing choice of models is examined using likelihood estimates for time to most recent common ancestor. The analysis presented here may play a role in elucidating the connections between these two approaches and shows promise in reconciling differences between estimates for mutation rates based on Whittaker's approach and methods based on phylogenetic analyses.
 Karafet, T. M., Lansing, J. S., Redd, A. J., Reznikova, S., Watkins, J. C., Surata, S. P., Arthawiguna, W. A., Mayer, L., Bamshad, M., Jorde, L. B., & Hammer, M. F. (2005). Balinese Ychromosome perspective on the peopling of Indonesia: genetic contributions from preneolithic huntergatherers, Austronesian farmers, and Indian traders. Human biology, 77(1), 93114.More infoThe island of Bali lies near the center of the southern chain of islands in the Indonesian archipelago, which served as a steppingstone for early migrations of huntergatherers to Melanesia and Australia and for more recent migrations of Austronesian farmers from mainland Southeast Asia to the Pacific. Bali is the only Indonesian island with a population that currently practices the Hindu religion and preserves various other Indian cultural, linguistic, and artistic traditions (Lansing 1983). Here, we examine genetic variation on the Y chromosomes of 551 Balinese men to investigate the relative contributions of Austronesian farmers and preNeolithic huntergatherers to the contemporary Balinese paternal gene pool and to test the hypothesis of recent paternal gene flow from the Indian subcontinent. Seventyone Ychromosome binary polymorphisms (single nucleotide polymorphisms, SNPs) and 10 Ychromosomelinked short tandem repeats (STRs) were genotyped on a sample of 1,989 Y chromosomes from 20 populations representing Indonesia (including Bali), southern China, Southeast Asia, South Asia, the Near East, and Oceania. SNP genotyping revealed 22 Balinese lineages, 3 of which (OM95, OM119, and OM122) account for nearly 83.7% of Balinese Y chromosomes. Phylogeographic analyses suggest that all three major Ychromosome haplogroups migrated to Bali with the arrival of Austronesian speakers; however, STR diversity patterns associated with these haplogroups are complex and may be explained by multiple waves of Austronesian expansion to Indonesia by different routes. Approximately 2.2% of contemporary Balinese Y chromosomes (i.e., KM9*, KM230, and M lineages) may represent the preNeolithic component of the Indonesian paternal gene pool. In contrast, eight other haplogroups (e.g., within H, J, L, and R), making up approximately 12% of the Balinese paternal gene pool, appear to have migrated to Bali from India. These results indicate that the Austronesian expansion had a profound effect on the composition of the Balinese paternal gene pool and that cultural transmission from India to Bali was accompanied by substantial levels of gene flow.
 Lansing, J. S., Redd, A. J., Karafet, T. M., Watkins, J., Ardika, I. W., Surata, S. P., Schoenfelder, J. S., Campbell, M., Merriwether, A. M., & Hammer, M. F. (2004). An Indian trader in ancient Bali?. Antiquity, 78(300), 287293.More infoAbstract: DNA analysis of a tooth found with imported pottery in Bali offers a strong possibility of the presence of a trader of Indian extraction in the late first millennium BC.
 Watkins, J. C. (2004). The role of marriage rules in the structure of genetic relatedness. Theoretical Population Biology, 66(1), 1324.More infoPMID: 15225572;Abstract: In this work, we take a forward in time approach to compute the probabilities of nonidentity by descent for a population consisting of n sections obeying one of a class of marriage rules that is invariant under cyclical relabeling of sections. A perturbation method allows for exact asymptotics using the reciprocal of the section population as a small parameter. The analysis yields relatedness measures that generalize Wright's Fstatistics. © 2004 Elsevier Inc. All rights reserved.
 Watkins, J., & Watkins, J. C. (2004). The role of marriage rules in the structure of genetic relatedness. Theoretical population biology, 66(1).More infoIn this work, we take a forward in time approach to compute the probabilities of nonidentity by descent for a population consisting of n sections obeying one of a class of marriage rules that is invariant under cyclical relabeling of sections. A perturbation method allows for exact asymptotics using the reciprocal of the section population as a small parameter. The analysis yields relatedness measures that generalize Wright's Fstatistics.
 Anderson, K. R., Mendelson, N. H., & Watkins, J. C. (2000). A new mathematical approach predicts individual cell growth behavior using bacterial population information. Journal of Theoretical Biology, 202(1), 8794.More infoPMID: 10623502;Abstract: A theoretical methodology has been developed for studying the growth kinetics of bacterial cells. It utilizes the steadystate cell length distribution in a bacterial population to predict the dependency of growth and division rates on cell length and age. The mathematical model has been applied to the analysis of two bacterial populations, a wildtype strain of Bacillus subtilis, and a minicellproducing strain that carries the divIVB1 mutation. The results show that our model describes the wildtype population very well and that the assumptions typically used in traditional methods are unrealistic. In the case of the minicellproducing mutant we find evidence that the rate of cell division must be a function not only of cell size but also of cell age. (C) 2000 Academic Press.
 DeGrandiHoffman, G., & Watkins, J. C. (2000). The foraging activity of honey bees Apis mellifera and nonApis bees on hybrid sunflowers (Helianthus annuus) and its influence on crosspollination and seed set. Journal of Apicultural Research, 39(12), 3745.More infoAbstract: The repercussions of concurrent foraging by honey bee (Apis mellifera) and nonApis bee populations on crosspollination and seed set in hybrid sunflowers (Helianthus annuus) was investigated. The amount of sunflower pollen on the bodies of honey bees foraging in rows of malesterile (MS) sunflowers was positively correlated with the size of the nonApis bee population. The combined population of nonApis bees and honey bees foraging on malefertile (MF) and MS sunflowers also was positively correlated to seed set in MS rows. There were more honey bees than nonApis bees foraging in MF and MS rows, but there was no evidence of competition for resources between the two populations. The size of the honey bee population was positively correlated to the area of open flowers on sunflower capitula, while the nonApis population remained relatively constant throughout bloom. Results from this study indicate that a combined honey bee and nonApis bee population might result in better pollination of hybrid sunflowers than either population alone.
 DeGrandiHoffman, G., Watkins, J., Guerrero, P., & Erickson, E. (2000). Using honey bees to teach mathematics and science to high school students. American Bee Journal, 140(4), 293295.More infoAbstract: Honey bees have been an integral part of human civilization for centuries. They pollinate crops and supply us with honey and pollen. In fact, humans use almost everything that honey bees collect or produce from royal jelly in cosmetics, to wax for candies and propolis for finishing the wood of fine musical instruments. Honey bees also have another function. They make great tools for teaching the fundamentals of mathematics and biology to students of all ages.
 Watkins, J. C. (2000). Consistency and fluctuation theorems for discrete time structured population models having demographic stochasticity. Journal of Mathematical Biology, 41(3), 253271.More infoPMID: 11072758;Abstract: In this paper we prove a consistency theorem (law of large numbers) and a fluctuation theorem (central limit theorem) for structured population processes. The basic assumptions for these theorems are that the individuals have no statistically distinguishing features beyond their class and that the interaction between any two individuals is not too high. We apply these results to density dependent models of Leslie type and to a model for flour beetle dynamics.
 Mendelson, N. H., Bourque, A., Wilkening, K., Anderson, K. R., & Watkins, J. C. (1999). Organized cell swimming motions in Bacillus subtilis colonies: Patterns of shortlived whirls and jets. Journal of Bacteriology, 181(2), 600609.More infoPMID: 9882676;PMCID: PMC93416;Abstract: The swimming motions of cells within Bacillus subtilis colonies, as well as the associated fluid flows, were analyzed from video films produced during colony growth and expansion on wet agar surfaces. Individual cells in very wet dense populations moved at rates between 76 and 116 μm/s. Swimming cells were organized into patterns of whirls, each approximately 1,000 μm2, and jets of about 95 by 12 μm. Whirls and jets were shortlived, lasting only about 0.25 s. Patterns within given areas constantly repeated with a periodicity of approximately 1 s. Whirls of a given direction became disorganized and then reformed, usually into whirls moving in the opposite direction. Pattern elements were also organized with respect to one another in the colony. Neighboring whirls usually turned in opposite directions. This correlation decreased as a function of distance between whirls. Fluid flows associated with whirls and jets were measured by observing the movement of marker latex spheres added to colonies. The average velocity of markers traveling in whirls was 19 μm/s, whereas those traveling in jets moved at 27 μm/s. The paths followed by markers were aligned with the direction of cell motion, suggesting that cells create flows moving with them into whirls and along jets. When colonies became dry, swimming motions ceased except in regions close to the periphery and in isolated islands where cells traveled in slow whirls at about 4 μm/s. The addition of water resulted in immediate though transient rapid swimming (> 80 μm/s) in characteristic whirl and jet patterns. The rate of swimming decreased to 13 μm/s within 2 min, however, as the water diffused into the agar. Organized swimming patterns were nevertheless preserved throughout this period. These findings show that cell swimming in colonies is highly organized.
 DeGrandiHoffman, G., & Watkins, J. C. (1998). Queen development time and the Africanization of European honey bees. American Bee Journal, 138(6), 467469.More infoAbstract: Organisms that inhabit an area thrive because they have become well adapted to the environmental conditions that surround them. The general rule is that the distinguishing characteristics of a species change very slowly over time. Suppose an individual of a particular species migrates into a territory in which that same species is established. If the migrant individual is similar to the current inhabitants, then the small differences in traits that the new arrival might bring will mix into the population and the distinguishing traits will rarely, if ever, be seen. On the other hand, if the immigrant has distinctive genetic characteristics, it often cannot compete for survival and reproduction with the resident population. Thus, these distinguishing characteristics are rapidly removed from the population's gene pool.
 DegrandiHoffman, G., Watkins, J. C., Collins, A. M., Loper, G. M., Martin, J. H., Arias, M. C., & Sheppard, W. S. (1998). Queen developmental time as a factor in the Africanization of European honey bee (Hymenoptera: Apidae) populations. Annals of the Entomological Society of America, 91(1), 5258.More infoAbstract: The development times of daughter queens from African and European matrilines mated to both African and European drones were recorded. Regardless of the matriline, African patriline queens completed their development and emerged 812 h before those with European paternity. A probability distribution function derived from the emergence time data indicated that because of differences in development times between patrilines, the probability that an African patriline queen will emerge 1st can be 23 times greater than the proportion of the African patrilines in the colony population. Because the 1st queen to emerge has the best chance of becoming the colony's new queen, differences in queen development times between African and European patrilines might be a factor contributing to the asymmetrical gene flow between African and European honey bee, Apis mellifera L., populations, and the eventual loss of European nuclear markers and behavioral attributes in European honey bee populations where African bees have migrated.
 Watkins, J. C. (1997). Mechanical models for cell movement  Locomotion, translocation, migration. Journal of Applied Probability, 34(4), 827846.More infoAbstract: This paper provides a detailed stochastic analysis of leucocyte cell movement based on the dynamics of a rigid body. The cell's behavior is studied in two relevant anisotropic environments displaying adhesion mediated movement (haptotaxis) and stimulus mediated movement (chemotaxis). This behavior is modeled by diffusion processes on three successively longer time scales, termed locomotion, translocation, and migration.
 Watkins, J. C., & Woessner, B. (1991). Diffusion models for chemotaxis: a statistical analysis of noninteractive unicellular movement. Mathematical Biosciences, 104(2), 271303.More infoPMID: 1804464;Abstract: A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a onedimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell phsyiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a threedimensional experimental setup of spherical symmetry. © 1991.
 Watkins, J. C. (1990). A remark on Kunita's decomposition theorem. Stochastic Processes and their Applications, 35(1), 8185.More infoAbstract: We use Michel Emery's stability theorem for stochastic differential equations to give a short proof for explicit solutions to linear stochastic differential equations over a solvable Lie group. © 1990.
 Watkins, J. C. (1987). Functional central limit theorems and their associated large deviation principles for products of random matrices. Probability Theory and Related Fields, 76(2), 133166.More infoAbstract: This paper establishes a functional central limit theorem for a product of random matrices. The sequence of matrices form a stationary process which is a φmixing. The individual matrices in the product become closer and closer to the identity matrix with longer and longer products. In addition, these perturbations from the identity matrix have mean zero. A large deviation principle for the limit process is proved. © 1987 SpringerVerlag.
 Watkins, J. C. (1985). Limit theorems for stationary random evolutions. Stochastic Processes and their Applications, 19(2), 189224.More infoAbstract: On a separable Banach space, let A(ξ1),A(ξ2),... be a strictly stationary sequence of infinitesimal operators, centered so that EA(ξi) = 0, i = 1,2,.... This paper characterizes the limit of the random evolutions Yn(t)=exp 1 nA(ξ[n2t])⋯exp 1 nA(ξ2)exp 1 nA(ξ1)Yn(0)as the solution to a martingale problem. This work is a direct extension of previous work on i.i.d. random evolutions. © 1985.
Presentations
 Watkins, J. C. (2018, January). Data and the Human Condition. AAAS. AAAS Washington DC: AAAS.
 Watkins, J. C. (2018, October). Multilevel Steiner Trees. TRIPODS Conference. Santa Clara, California: University of California, Santa Cruz.
 Watkins, J. C. (2017, September). On the Human Condition. Biostatistics Seminar. University of Arizona: College of Public Health.
 Watkins, J. C. (2014, June). BEEPOP: The Population Dynamics of the Honey Bee in the Hive and in the Wild. BioQuest/HHMI. University of Delaware: BioQuest/HHMI.More infoWorkshop on the Native American Summer Program at the University of Arizona
 Watkins, J. C. (2014, September). Curriculum & Classroom Strategies in a Statistics Course for Life Sciences Majors/Math Minors. High Performance Computing in Undergradraduate Quantitative Biologyu. Cold Spring Harbor, New York: Cold Spring Harbor.
Poster Presentations
 Watkins, J. C. (2018, October). A novel nonlinear dimension reduction approach to infer population structure for lowcoverage sequencing data. American Society of Human Genetics Annual Meeting. San Diego. California: American Society of Human Genetics.
 Bender, C., Watkins, J. C., Tolbert, L. P., Bender, C., Watkins, J. C., Tolbert, L. P., Bender, C., Watkins, J. C., Tolbert, L. P., Bender, C., Watkins, J. C., & Tolbert, L. P. (2012, October). Assessing Undergraduate Research and BioMath Efforts at the University of Arizona. Howard Hughes Medical Institute Program Directors' Meeting. Chevy Chase, Maryland: Howard Hughes Medical Institute.