Rosenberg lab at Stanford University

We are a mathematical, theoretical, and computational lab in genetics and evolution. Research in the lab addresses problems in evolutionary biology and human genetics through a combination of mathematical modeling, computer simulations, development of statistical methods, and inference from population-genetic data. Read more...


  • 2-14-2023"All galls are divided into three or more parts" — so reports a study from Shaili Mathur, describing a recursive decomposition used to enumerate labeled histories for galled trees. The study is the first to enumerate labeled histories for a class of phylogenetic network.

  • 12-13-2022 — A new study by Filippo Disanto et al. obtains asymptotic distributions for the number of root ancestral configurations of matching gene trees and species trees, under the Yule and uniform models describing the labeled tree topology. The results build on Filippo's earlier work on ancestral configurations [152] [161].

  • 11-15-2022Egor Lappo has been recognized with honorable mention for the 2023 AMS-MAA-SIAM Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student! Congrats to Egor.

  • 11-15-2022Egor Lappo extends his analysis of coalescent trees by producing new approximate results for expectations and variances of ratios of tree properties under the coalescent model. The results extend Egor's earlier analysis of covariances and correlations of tree properties [198].

  • 9-12-2022 — In the 200th year since Gregor Mendel's birth, a historical commentary discusses Mendel as an icon not only of genetics, but also of the intersection of mathematics and biology.

  • 9-6-2022 — PhD student Maike Morrison, working with former postdoc Nicolas Alcala, introduces a new method for measuring the variability in membership assignments observed in genetic cluster analysis. The method relies on a new and surprising use of the population-genetic statistic FST.

  • 8-29-2022 — PhD student Danny Cotter advances the study of X-chromosomal and autosomal coalescence times in consanguineous populations. Danny shows that coalescence in X-chromosomal first-cousin mating models behaves like the standard coalescent, except with a reduction in coalescence time that depends on the features of consanguinity. The study builds on three recent studies from the lab on coalescence in consanguineous populations [166] [194] [195].

  • 7-11-2022Rohan Mehta and collaborator Mike Steel introduce a general algorithm for computing the probability of reciprocal monophyly of arbitrarily many groups in an arbitrary species tree. The study generalizes earlier computations involving species trees with three and four monophyletic groups [172], and with two monophyletic groups in arbitrary species trees [141].

  • 5-26-2022Xiran Liu and Gili Greenbaum apply the Netstruct hierarchical clustering program to study cultural variation. The analysis, which adapts a method from population genetics for cultural data, reveals new features of variation in regional pronunciation in the eastern United States, folklore motifs and phonemic content of languages worldwide, and US first names.

  • 5-18-2022 — A team including Julia Palacios, Anand Bhaskar, and Filippo Disanto describes an enumeration of binary trees in each of several categories (ranked labeled, ranked unlabeled, unranked labeled, unranked unlabeled) that are compatible with a perfect phylogeny. The enumeration is a contribution to the study of the combinatorics of evolutionary trees.

  • 5-5-2022 — A special issue of Philosophical Transactions of the Royal Society B Biological Sciences with editors Doc Edge, Sohini Ramachandran, and Noah Rosenberg celebrates 50 years since Lewontin's apportionment of human diversity." The special issue covers the background and legacy of this important milestone in the understanding of human genetic variation as well as recent technical advances that connect to it. In the special issue, Nicolas Alcala contributes a study of FST in relation to the frequency of the most frequent allele for multiallelic loci in multiple populations, generalizing earlier results for multiallelic loci in two populations [102] and biallelic loci in multiple populations [149].

  • 3-21-2022Alissa Severson and a collaborative team report a genetic study of ancient burial sites and their continuity with modern members of the Muwekma Ohlone Tribe. The project, a collaboration with the tribal leadership, finds a component of genetic ancestry that connects two burial sites separated by hundreds of years with each other and with the modern tribal members. [Illinois News Bureau] [Stanford Report]

  • 11-30-2021 — Under the coalescent model, a genealogical tree possess a series of features: its height, length, sum of external branches, sum of internal branches, and mean basal branch length. Egor Alimpiev has calculated the covariance and correlation coefficients of all these pairs of random variables, providing a compendium of existing and new fundamental results for the coalescent model. The calculation builds on a previous calculation for one of the pairs considered [154].

  • 11-11-2021 — The Sackin index is one of the most commonly used measures of tree balance. Undergraduate Matt King reports a simple new proof of a result that finds the mean value of the Sackin index across all labeled topologies on n leaves. The proof makes use of an identity that has been called by Graham, Knuth & Patashnik a "remarkable property of the 'middle' elements of Pascal's triangle."

  • 8-24-2021 — For a caterpillar species tree, undergraduate Egor Alimpiev studies coalescent histories in a family of gene trees, the p-pseudocaterpillar gene trees. For this family, his study investigates a claim that the number of coalescent histories is affected by a tradeoff between the number of possible sequences of coalescences and the number of species tree branches on which those sequences can take place. He finds a very nice symmetry. The work extends a study by a previous undergraduate in the lab, Zoe Himwich [176]

  • 8-24-2021 — PhD student Danny Cotter continues the investigation of coalescence times in consanguineous populations, considering the mean time to coalescence for a pair of lineages on the X chromosome in each of four first-cousin mating models. He finds that matrilateral first-cousin mating reduces X-chromosomal coalescence times to a greater extent than patrilateral first-cousin mating. The work builds on two studies led by co-author Alissa Severson [166] [194].

  • 5-28-2021 — In a new article led by PhD student Alissa Severson, the distribution of coalescence times is computed in a diploid model of a consanguineous population. Using a separation-of-time-scales approach, the study shows that the time to the most recent common ancestor for pairs of lineages in separate mating pairs follows a coalescent model with a reduced effective popualtion size. The study builds on a previous theoretical study that examined the mean pairwise coalescence time [166].

  • 5-25-2021 — Congratulations! Alissa Severson has successfully defended her PhD, "The effect of relatedness and population structure on patterns of genomic sharing."

  • 5-21-2021Jaehee Kim, Doc Edge, and Amy Goldberg report a study of the decoupling of a phenotype from admixture levels in an admixed population whose source populations differed in phenotype. As time proceeds, the phenotype of an individual comes to reveal less and less information about the individual's admixture level, particularly if mating occurs randomly in the admixed population. [Stanford Report]

  • 3-11-2021Gili Greenbaum and Jaehee Kim report a population-genetic model of gene drives and their potential to "spill over" from one population to another. In the model, an engineered gene drive is introduced into a target population with the goal of overtaking the extant population. Under what circumstances can the introduced gene drive be prevented from overtaking genotypes in non-target populations? The study finds a narrow set of circumstances.

  • 2-8-2021 — Last year we celebrated the 50th anniversary of the journal Theoretical Population Biology. The anniversary came just as the role for mathematical epidemiology models of COVID-19 began receiving intense attention. A recent editorial discusses the connections between decades of population biology modeling and the COVID-19 pandemic.

  • 2-2-2021 — In a genome scan of rats in New York City, former rotation student Arbel Harpak identifies genes associated with metabolism, diet, the nervous system, and locomotion as possible targets of natural selection. The results add to a growing understanding of adaptation in human-commensal species.

  • 12-18-2020 — Colijn & Plazzotta (2018) introduced a clever new way to associate the unlabeled binary rooted trees with the positive integers. A new paper explores the mathematical properties of the Colijn-Plazzotta enumeration. In particular, the study obtains an upper bound on the sequence providing the smallest Colijn-Plazzotta rank assigned to some tree with n leaves, and an asymptotic equivalence for the sequence providing the largest Colijn-Plazzotta rank assigned to some tree with n leaves.

  • 12-2-2020 — Admixture inflates the genetic diversity of the admixed population above that of the source populations — or does it? Simina Boca and Lucy Huang explore the effect of admixture on heterozygosity, examining when an admixed population has heterozygosity greater than that of source populations. The study also characterizes the level of admixture that gives rise to the greatest heterozygosity for a given set of source population allele frequencies.

  • 11-17-2020 — Studies of phylogenetic tree spaces have often focused on unranked labeled trees (panel C below), unranked unlabeled trees (panel D), or sometimes, ranked labeled trees (panel A). In a new study, Jaehee Kim introduces metrics for calculating distances between ranked unlabeled trees, an understudied type of tree that is useful in tracking pathogen lineages (panel B). The finds shows that the metrics can be used to cluster trees arising from a shared generative model, and to distinguish between those that have arisen by different models.

  • 8-4-2020Alyssa Fortier and Jaehee Kim examine the use of ancestry inference as a step to improve relatedness profiling in forensic genetics. By reducing the potential for misspecification of allele frequencies in likelihood calculations, inference of the genetic ancestry of the forensic sample can avoid a false positive inference of relatedness.

  • 7-29-2020Amy Goldberg and Ananya Rastogi report a study of "Assortative mating by population of origin in a mechanistic model of admixture." This work analyzes a model in which individuals mate assortatively in a setting with two ancestral populations and an admixed populaton. The study builds on several previous models from the lab. [82] [122] [133]

  • 6-11-2020Rohan Mehta reports an article entitled "Modelling anti-vaccine sentiment as a cultural pathogen." The paper describes a coupled contagion: the spread of an anti-vaccine sentiment, and the spread of the disease against which the vaccine protects. The dynamics illustrate how spread of sentiment against a vaccine generates and magnifies outbreaks of the associated disease. [Stanford Report]

  • 5-29-2020 — The long-awaited 50th anniversary special issue of Theoretical Population Biology has been published. The special issue contains commentaries on major research areas developed in TPB, commentaries on historic papers, biograpical commentaries, and research articles — including a study by Ilana Arbisser on FST and the triangle inequality. [Stanford Report]

  • 4-24-2020 — Using a combination of coalescent theory and simulation, Kim et al. study the probability under a birth-death process that species trees lie in the "anomaly zone," the region of the parameter space in which species trees can disagree with the gene tree they are most likely to produce. The work buils on earlier studies of the anomaly zone [30] [47], ranked gene trees [85] [97], and joint simulation of species trees and gene trees [140].

  • 3-20-2020 — A new study examines the mathematical connections between homozygosity and heterozygosity statistics and measures of health care fragmentation in health services research. The study relies on results from related studies in the lab [87] [158].

  • 3-10-2020 — PhD graduate Jonathan Kang reports a new study of five measures of linkage disequilibrium. Jonathan computes mathematical bounds on linkage disequilibrium measures in relation to the allele frequencies at a pair of loci, analyzing the implications of these bounds in human genetic data. The study builds on an earlier analysis of the r2 measure [51].

  • 1-9-2020 — A paper by Zoe Himwich, recent Stanford graduate in mathematics, studies coalescent histories for non-matching caterpillar gene trees and species trees. This study in enumerative combinatorics identifies new connections to the Catalan numbers, Dyck paths, and roadblocked monotonic paths not crossing the diagonal of a square lattice. The paper builds on two earlier studies of coalescent histories for caterpillar-like tree families [111] [142].

  • 12-9-2019Gili Greenbaum introduces a new network-based approach to inference of population structure. The method relies on detection of "communities" in genetic distance matrices and can be used to produce a new way of displaying population structure — a "population structure tree."

  • 12-8-2019 — The work of lab alumnus Brian Donovan is featured on the front page of the New York Times.

  • 11-1-2019Gili Greenbaum reports a study of dynamics of the spatial boundary between Neanderthals and Modern Humans before Modern Humans spread rapidly out of Africa. The question is not "why did Modern Humans replace Neanderthals so quickly?" Rather, Gili asks "why did Modern Humans not replace Neanderthals for so long?" The proposed answer lies in the dynamics of infectious disease. [Haaretz] [Stanford Report]

  • 10-1-2019 — A new study by Rohan Mehta computes probabilities under the coalescent model of reciprocal monophyly for sets of gene lineages from three and four species. The computation extends an earlier computation that permitted only two sets of lineages [141]. The study appears in a special issue of Theoretical Population Biology celebrating Marc Feldman's 75th birthday.

  • 9-23-2019Nicolas Alcala studies the coalescent theory of all possible symmetric migration models involving at most four demes. His paper examines coalescent quantities such as the time to the most recent common ancestor under the models, determining how these quantities relate to network properties such as the mean number of edges per vertex and the density of edges. The study introduces a network perspective for coalescent models — applying it to empirical examples on tigers and birds of genus Sholicola in India. PhD graduate Amy Goldberg also contributed to the project.

  • 9-9-2019 — A new paper led by Rohan Mehta examines the behavior of the FST measure of genetic differentiation on haplotypic data. The study illustrates how incrementing the length of the haplotype window tends to decrease FST — but sometimes increases it. The work is closely related to several of the lab's papers on FST [102] [121] [149] [165]. Check out the video abstract drawn and narrated by co-author Alison Feder.

  • 5-8-2019 — In a collaboration with the Stanford Conservation Program, we have developed a stochastic population occupancy model to examine two decades of occupancy data from the campus populations of the California red-legged frog (Rana draytonii). The model seeks to explain population declines of R. draytonii in campus creeks and suggests conservation management approaches for reversing these declines. The study was led by Nicolas Alcala.

  • 5-2-2019 — A new study led by Alissa Severson examines the relationship between runs of homozygosity and identity-by-descent tracts. The paper determines for a diploid coalescent model the time to the most recent common ancestor, both for two haplotypes in the same individual and for two haplotypes in different individuals. The work provides theory that builds on empirical observations in an earlier study [144].

  • 4-29-2019Nicolas Alcala has a new study of mathematical bounds on three population-genetic statistics: GST', Jost's D, and FST. He shows that for biallelic markers whose mean frequency across a set of populations is fixed, these three statistics achieve their maximal values at the same configuration of allele frequencies across populations. The results extend Nicolas's earlier work on FST bounds as well as that of two other studies from the lab concerning bounds on FST [102] [121].

  • 3-26-2019Filippo Disanto reports a study of the enumeration of compact coalescent histories for matching gene trees and species trees. Compact coalescent histories represent a combinatorial structure that collapses standard coalescent histories into a smaller number of equivalence classes. The study extends the lab's work on enumeration of coalescent histories to a new structure.

  • 3-3-2019 — A new paper discusses challenges of interpreting differences in polygenic scores across populations. The paper builds from the models developed by Ph.D. graduate Doc Edge for analyzing the relationship between the magnitude of genetic and phenotypic differences among populations [129] [132].

  • 1-23-2019 — Two papers from the lab appear in a special issue of Bulletin of Mathematical Biology on Algebraic Methods in Phylogenetics.
    • Jaehee Kim, Filippo Disanto, and Naama Kopelman report a study of the properties of the neighbor-joining algorithm when applied to data from admixed populations. The study shows that tree properties conjectured by Kopelman et al. [99] do not necessarily hold for every distance matrix, but they do hold much more frequently than in a null model without an admixed taxon.

    • Filippo Disanto examines the number of nonequivalent ancestral configurations for matching gene trees and species trees. Nonequivalent ancestral configurations at first appear to be less numerous than ancestral configurations without applying the equivalence relation — studied previously by Filippo [152]. Here, Filippo shows that asymptotic growth for nonequivalent configurations is also exponential.
    This pair of studies extends the lab's work on theory of admixture and combinatorics of evolutionary trees.

  • Past news items


    E Alimpiev, NA Rosenberg (2022) A compendium of covariances and correlation coefficients of coalescent tree properties. Theoretical Population Biology 143: 1-13. [Abstract] [PDF]

    J Kim, MD Edge, A Goldberg, NA Rosenberg (2021) Skin deep: the decoupling of genetic admixture levels from phenotypes that differed between source populations. American Journal of Physical Anthropology 175: 406-421 (2021). [Abstract]

    NA Rosenberg (2021) On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees. Discrete Applied Mathematics 291: 88-98. [Abstract] [PDF]

    RS Mehta, NA Rosenberg (2020) Modelling anti-vaccine sentiment as a cultural pathogen. Evolutionary Human Sciences 2: e21. [Abstract] [PDF] [Supplement]

    IM Arbisser, NA Rosenberg (2020) FST and the triangle inequality for biallelic markers. Theoretical Population Biology 133: 117-129. [Abstract]

    NA Rosenberg (2020) Fifty years of Theoretical Population Biology. Theoretical Population Biology 133: 1-12. [Abstract]

    ZM Himwich, NA Rosenberg (2020) Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees. Advances in Applied Mathematics 113: 101939. [Abstract]

    AL Severson, S Carmi, NA Rosenberg (2019) The effect of consanguinity on between-individual identity-by-descent sharing. Genetics 212: 305-316. [Abstract] [PDF]

    NA Rosenberg, MD Edge, JK Pritchard, MW Feldman (2019) Interpreting polygenic scores, polygenic adaptation, and human phenotypic differences. Evolution, Medicine, and Public Health 2019: 26-34. [Abstract] [PDF]

    NA Rosenberg (2019) Enumeration of lonely pairs of gene trees and species trees by means of antipodal cherries. Advances in Applied Mathematics 102: 1-17. [Abstract] [PDF]

    J Kim, MD Edge, BFB Algee-Hewitt, JZ Li, NA Rosenberg (2018) Statistical detection of relatives typed with disjoint forensic and biomedical loci. Cell 175: 848-858. [Abstract] [PDF] [Supplement]

    AJ Aw, NA Rosenberg (2018) Bounding measures of genetic similarity and diversity using majorization. Journal of Mathematical Biology 77: 711-737. [Abstract] [PDF]

    IM Arbisser, EM Jewett, NA Rosenberg (2018) On the joint distribution of tree height and tree length under the coalescent. Theoretical Population Biology 122: 46-56. [Abstract] [PDF]

    N Alcala, NA Rosenberg (2017) Mathematical constraints on FST: biallelic markers in arbitrarily many populations. Genetics 206: 1581-1600. [Abstract] [PDF] [File S1] [File S2]

    JTL Kang, A Goldberg, MD Edge, DM Behar, NA Rosenberg (2016) Consanguinity rates predict long runs of homozygosity in Jewish populations. Human Heredity 82: 87-102. [Abstract] [PDF]

    RS Mehta, D Bryant, NA Rosenberg (2016) The probability of monophyly of a sample of gene lineages on a species tree. Proceedings of the National Academy of Sciences 113: 8002-8009. [Abstract] [PDF] [Supplement] [Software]

    A Goldberg, NA Rosenberg (2015) Beyond 2/3 and 1/3: the complex signatures of sex-biased admixture on the X chromosome. Genetics 201: 263-279. [Abstract] [PDF]

    NA Rosenberg, JTL Kang (2015) Genetic diversity and societally important disparities. Genetics 201: 1-12. [Abstract] [PDF] [Supplement]