The expressions ϕ(n)+σ(n)−3nϕ(n)+σ(n)−3n and ϕ(n)+σ(n)−4nϕ(n)+σ(n)−4n are unusual among linear co... more The expressions ϕ(n)+σ(n)−3nϕ(n)+σ(n)−3n and ϕ(n)+σ(n)−4nϕ(n)+σ(n)−4n are unusual among linear combinations of arithmetic functions in that they each vanish on a nonempty set of composite numbers. In 1966, Nicol proved that the set A:={n|(ϕ(n)+σ(n))/n∈N⩾3}A:={n|(ϕ(n)+σ(n))/n∈N⩾3} contains a2⋅3⋅(2a−2⋅7−1)2a⋅3⋅(2a−2⋅7−1) if and only if 2a−2⋅7−12a−2⋅7−1 is prime and conjectured that AA contains no odd integers. A 2008 paper by Luca and Sandor completely classifies the elements of AA that have three distinct prime factors and observes that Nicol's conjecture holds for numbers with fewer than six distinct prime factors. In this paper we let AKAK denote the set of n∈An∈A with exactly K distinct prime factors and present a computer-implementable algorithm that decides whether Nicol's conjecture holds for a given AKAK. Using this algorithm, we verify Nicol's conjecture for A6A6 and completely classify the elements of A4A4. We prove that all but finitely many n∈A4n∈A4 have the form a2⋅3⋅p3⋅p42a⋅3⋅p3⋅p4, and that all but finitely many n∈A5n∈A5 are divisible by 6 and not 9. In addition, we prove that every AKAK is contained in a finite union of sequences that each have the form {p1a1i⋯pkaki⋅u⋅wi}i=1∞, where k⩾1k⩾1, p1,…,pkp1,…,pk are distinct primes, and each ajiaji as well as the least prime factor of wiwi go to infinity as i does.
Proceedings of the National Academy of Sciences of the United States of America, Jan 17, 2015
As humans dispersed out of Africa they adapted to new environmental challenges, including changes... more As humans dispersed out of Africa they adapted to new environmental challenges, including changes in exposure to mutagenic solar radiation. Humans in temperate latitudes have acquired light skin that is relatively transparent to UV light, and some evidence suggests that their DNA damage response pathways have also experienced local adaptation. This raises the possibility that different populations have experienced different selective pressures affecting genome integrity. Here, I present evidence that the rate of a particular mutation type has recently increased in the European population, rising in frequency by 50% during the 40,000-80,000 y since Europeans began diverging from Asians. A comparison of SNPs private to Africa, Asia, and Europe in the 1000 Genomes data reveals that private European variation is enriched for the transition 5'-TCC-3' → 5'-TTC-3'. Although it is not clear whether UV played a causal role in changing the European mutational spectrum, 5'-...
About 2% of human genetic polymorphisms have been hypothesized to arise via multinucleotide mutat... more About 2% of human genetic polymorphisms have been hypothesized to arise via multinucleotide mutations (MNMs), complex events that generate SNPs at multiple sites in a single generation. MNMs have the potential to accelerate the pace at which single genes evolve and to confound studies of demography and selection that assume all SNPs arise independently. In this paper, we examine clustered mutations that are segregating in a set of 1092 human genomes, demonstrating that the signature of MNM becomes enriched as large numbers of individuals are sampled. We estimate the percentage of linked SNP pairs that were generated by simultaneous mutation as a function of the distance between affected sites and show that MNMs exhibit a high percentage of transversions relative to transitions, findings that are reproducible in data from multiple sequencing platforms and cannot be attributed to sequencing error. Among tandem mutations that occur simultaneously at adjacent sites, we find an especially skewed distribution of ancestral and derived alleles, with GC ! AA, GA ! TT, and their reverse complements making up 27% of the total. These mutations have been previously shown to dominate the spectrum of the error-prone polymerase Pol z, suggesting that low-fidelity DNA replication by Pol z is at least partly responsible for the MNMs that are segregating in the human population. We develop statistical estimates of MNM prevalence that can be used to correct phylogenetic and population genetic inferences for the presence of complex mutations.
The expressions φ(n) + σ (n) − 3n and φ(n) + σ (n) − 4n are unusual among linear combinations of ... more The expressions φ(n) + σ (n) − 3n and φ(n) + σ (n) − 4n are unusual among linear combinations of arithmetic functions in that they each vanish on a nonempty set of composite numbers. In 1966, Nicol proved that the set A := {n | (φ(n) + σ (n))/n ∈ N 3 } contains 2 a · 3 · (2 a−2 · 7 − 1) if and only if 2 a−2 · 7 − 1 is prime and conjectured that A contains no odd integers. A 2008 paper by Luca and Sandor completely classifies the elements of A that have three distinct prime factors and observes that Nicol's conjecture holds for numbers with fewer than six distinct prime factors. In this paper we let A K denote the set of n ∈ A with exactly K distinct prime factors and present a computer-implementable algorithm that decides whether Nicol's conjecture holds for a given A K . Using this algorithm, we verify Nicol's conjecture for A 6 and completely classify the elements of A 4 . We prove that all but finitely many n ∈ A 4 have the form 2 a · 3 · p 3 · p 4 , and that all but finitely many n ∈ A 5 are divisible by 6 and not 9. In addition, we prove that every A K is contained in a finite union of sequences that each have the form {p
There has been much recent excitement about the use of genetics to elucidate ancestral history an... more There has been much recent excitement about the use of genetics to elucidate ancestral history and demography. Whole genome data from humans and other species are revealing complex stories of divergence and admixture that were left undiscovered by previous smaller data sets. A central challenge is to estimate the timing of past admixture and divergence events, for example the time at which Neanderthals exchanged genetic material with humans and the time at which modern humans left Africa. Here, we present a method for using sequence data to jointly estimate the timing and magnitude of past admixture events, along with population divergence times and changes in effective population size. We infer demography from a collection of pairwise sequence alignments by summarizing their length distribution of tracts of identity by state (IBS) and maximizing an analytic composite likelihood derived from a Markovian coalescent approximation. Recent gene flow between populations leaves behind long tracts of identity by descent (IBD), and these tracts give our method power by influencing the distribution of shared IBS tracts. In simulated data, we accurately infer the timing and strength of admixture events, population size changes, and divergence times over a variety of ancient and recent time scales. Using the same technique, we analyze deeply sequenced trio parents from the 1000 Genomes project. The data show evidence of extensive gene flow between Africa and Europe after the time of divergence as well as substructure and gene flow among ancestral hominids. In particular, we infer that recent African-European gene flow and ancient ghost admixture into Europe are both necessary to explain the spectrum of IBS sharing in the trios, rejecting simpler models that contain less population structure.
We introduce a method for comparing a test genome with numerous genomes from a reference populati... more We introduce a method for comparing a test genome with numerous genomes from a reference population. Sites in the test genome are given a weight, w, that depends on the allele frequency, x, in the reference population. The projection of the test genome onto the reference population is the average weight for each x, [Formula: see text]. The weight is assigned in such a way that, if the test genome is a random sample from the reference population, then [Formula: see text]. Using analytic theory, numerical analysis, and simulations, we show how the projection depends on the time of population splitting, the history of admixture, and changes in past population size. The projection is sensitive to small amounts of past admixture, the direction of admixture, and admixture from a population not sampled (a ghost population). We compute the projections of several human and two archaic genomes onto three reference populations from the 1000 Genomes project-Europeans, Han Chinese, and Yoruba-an...
In many areas of computational biology, hidden Markov models (HMMs) have been used to model local... more In many areas of computational biology, hidden Markov models (HMMs) have been used to model local genomic features. In particular, coalescent HMMs have been used to infer ancient population sizes, migration rates, divergence times, and other parameters such as mutation and recombination rates. As more loci, sequences, and hidden states are added to the model, however, the runtime of coalescent HMMs can quickly become prohibitive. Here we present a new algorithm for reducing the runtime of coalescent HMMs from quadratic in the number of hidden time states to linear, without making any additional approximations. Our algorithm can be incorporated into various coalescent HMMs, including the popular method PSMC for inferring variable effective population sizes. Here we implement this algorithm to speed up our demographic inference method diCal, which is equivalent to PSMC when applied to a sample of two haplotypes. We demonstrate that the linear-time method can reconstruct a population size change history more accurately than the quadratic-time method, given similar computation resources. We also apply the method to data from the 1000 Genomes project, inferring a high-resolution history of size changes in the European population.
Polar bears are uniquely adapted to life in the High Arctic and have undergone drastic physiologi... more Polar bears are uniquely adapted to life in the High Arctic and have undergone drastic physiological changes in response to Arctic climates and a hyper-lipid diet of primarily marine mammal prey. We analyzed 89 complete genomes of polar bear and brown bear using population genomic modeling and show that the species diverged only 479-343 thousand years BP. We find that genes on the polar bear lineage have been under stronger positive selection than in brown bears; nine of the top 16 genes under strong positive selection are associated with cardiomyopathy and vascular disease, implying important reorganization of the cardiovascular system. One of the genes showing the strongest evidence of selection, APOB, encodes the primary lipoprotein component of low-density lipoprotein (LDL); functional mutations in APOB may explain how polar bears are able to cope with life-long elevated LDL levels that are associated with high risk of heart disease in humans.
Throughout history, the population size of modern humans has varied considerably due to changes i... more Throughout history, the population size of modern humans has varied considerably due to changes in environment, culture, and technology. More accurate estimates of population size changes, and when they occurred, should provide a clearer picture of human colonization history and help remove confounding effects from natural selection inference. Demography influences the pattern of genetic variation in a population, and thus genomic data of multiple individuals sampled from one or more present-day populations contain valuable information about the past demographic history. Recently, Li and Durbin developed a coalescent-based hidden Markov model, called the pairwise sequentially Markovian coalescent (PSMC), for a pair of chromosomes (or one diploid individual) to estimate past population sizes. This is an efficient, useful approach, but its accuracy in the very recent past is hampered by the fact that, because of the small sample size, only few coalescence events occur in that period. Multiple genomes from the same population contain more information about the recent past, but are also more computationally challenging to study jointly in a coalescent framework. Here, we present a new coalescent-based method that can efficiently infer population size changes from multiple genomes, providing access to a new store of information about the recent past. Our work generalizes the recently developed sequentially Markov conditional sampling distribution framework, which provides an accurate approximation of the probability of observing a newly sampled haplotype given a set of previously sampled haplotypes. Simulation results demonstrate that we can accurately reconstruct the true population histories, with a significant improvement over the PSMC in the recent past. We apply our method, called diCal, to the genomes of multiple human individuals of European and African ancestry to obtain a detailed population size change history during recent times.
How and when the Americas were populated remains contentious. Using ancient and modern genome-wid... more How and when the Americas were populated remains contentious. Using ancient and modern genome-wide data, we find that the ancestors of all present-day Native Americans, including Athabascans and Amerindians, entered the Americas as a single migration wave from Siberia no earlier than 23 thousand years ago (KYA), and after no more than 8,000-year isolation period in Beringia. Following their arrival to the Americas, ancestral Native Americans diversified into two basal genetic branches around 13 KYA, one that is now dispersed across North and South America and the other is restricted to North America. Subsequent gene flow resulted in some Native Americans sharing ancestry with present-day East Asians (including Siberians) and, more distantly, Australo-Melanesians. Putative…
The expressions ϕ(n)+σ(n)−3nϕ(n)+σ(n)−3n and ϕ(n)+σ(n)−4nϕ(n)+σ(n)−4n are unusual among linear co... more The expressions ϕ(n)+σ(n)−3nϕ(n)+σ(n)−3n and ϕ(n)+σ(n)−4nϕ(n)+σ(n)−4n are unusual among linear combinations of arithmetic functions in that they each vanish on a nonempty set of composite numbers. In 1966, Nicol proved that the set A:={n|(ϕ(n)+σ(n))/n∈N⩾3}A:={n|(ϕ(n)+σ(n))/n∈N⩾3} contains a2⋅3⋅(2a−2⋅7−1)2a⋅3⋅(2a−2⋅7−1) if and only if 2a−2⋅7−12a−2⋅7−1 is prime and conjectured that AA contains no odd integers. A 2008 paper by Luca and Sandor completely classifies the elements of AA that have three distinct prime factors and observes that Nicol's conjecture holds for numbers with fewer than six distinct prime factors. In this paper we let AKAK denote the set of n∈An∈A with exactly K distinct prime factors and present a computer-implementable algorithm that decides whether Nicol's conjecture holds for a given AKAK. Using this algorithm, we verify Nicol's conjecture for A6A6 and completely classify the elements of A4A4. We prove that all but finitely many n∈A4n∈A4 have the form a2⋅3⋅p3⋅p42a⋅3⋅p3⋅p4, and that all but finitely many n∈A5n∈A5 are divisible by 6 and not 9. In addition, we prove that every AKAK is contained in a finite union of sequences that each have the form {p1a1i⋯pkaki⋅u⋅wi}i=1∞, where k⩾1k⩾1, p1,…,pkp1,…,pk are distinct primes, and each ajiaji as well as the least prime factor of wiwi go to infinity as i does.
Proceedings of the National Academy of Sciences of the United States of America, Jan 17, 2015
As humans dispersed out of Africa they adapted to new environmental challenges, including changes... more As humans dispersed out of Africa they adapted to new environmental challenges, including changes in exposure to mutagenic solar radiation. Humans in temperate latitudes have acquired light skin that is relatively transparent to UV light, and some evidence suggests that their DNA damage response pathways have also experienced local adaptation. This raises the possibility that different populations have experienced different selective pressures affecting genome integrity. Here, I present evidence that the rate of a particular mutation type has recently increased in the European population, rising in frequency by 50% during the 40,000-80,000 y since Europeans began diverging from Asians. A comparison of SNPs private to Africa, Asia, and Europe in the 1000 Genomes data reveals that private European variation is enriched for the transition 5'-TCC-3' → 5'-TTC-3'. Although it is not clear whether UV played a causal role in changing the European mutational spectrum, 5'-...
About 2% of human genetic polymorphisms have been hypothesized to arise via multinucleotide mutat... more About 2% of human genetic polymorphisms have been hypothesized to arise via multinucleotide mutations (MNMs), complex events that generate SNPs at multiple sites in a single generation. MNMs have the potential to accelerate the pace at which single genes evolve and to confound studies of demography and selection that assume all SNPs arise independently. In this paper, we examine clustered mutations that are segregating in a set of 1092 human genomes, demonstrating that the signature of MNM becomes enriched as large numbers of individuals are sampled. We estimate the percentage of linked SNP pairs that were generated by simultaneous mutation as a function of the distance between affected sites and show that MNMs exhibit a high percentage of transversions relative to transitions, findings that are reproducible in data from multiple sequencing platforms and cannot be attributed to sequencing error. Among tandem mutations that occur simultaneously at adjacent sites, we find an especially skewed distribution of ancestral and derived alleles, with GC ! AA, GA ! TT, and their reverse complements making up 27% of the total. These mutations have been previously shown to dominate the spectrum of the error-prone polymerase Pol z, suggesting that low-fidelity DNA replication by Pol z is at least partly responsible for the MNMs that are segregating in the human population. We develop statistical estimates of MNM prevalence that can be used to correct phylogenetic and population genetic inferences for the presence of complex mutations.
The expressions φ(n) + σ (n) − 3n and φ(n) + σ (n) − 4n are unusual among linear combinations of ... more The expressions φ(n) + σ (n) − 3n and φ(n) + σ (n) − 4n are unusual among linear combinations of arithmetic functions in that they each vanish on a nonempty set of composite numbers. In 1966, Nicol proved that the set A := {n | (φ(n) + σ (n))/n ∈ N 3 } contains 2 a · 3 · (2 a−2 · 7 − 1) if and only if 2 a−2 · 7 − 1 is prime and conjectured that A contains no odd integers. A 2008 paper by Luca and Sandor completely classifies the elements of A that have three distinct prime factors and observes that Nicol's conjecture holds for numbers with fewer than six distinct prime factors. In this paper we let A K denote the set of n ∈ A with exactly K distinct prime factors and present a computer-implementable algorithm that decides whether Nicol's conjecture holds for a given A K . Using this algorithm, we verify Nicol's conjecture for A 6 and completely classify the elements of A 4 . We prove that all but finitely many n ∈ A 4 have the form 2 a · 3 · p 3 · p 4 , and that all but finitely many n ∈ A 5 are divisible by 6 and not 9. In addition, we prove that every A K is contained in a finite union of sequences that each have the form {p
There has been much recent excitement about the use of genetics to elucidate ancestral history an... more There has been much recent excitement about the use of genetics to elucidate ancestral history and demography. Whole genome data from humans and other species are revealing complex stories of divergence and admixture that were left undiscovered by previous smaller data sets. A central challenge is to estimate the timing of past admixture and divergence events, for example the time at which Neanderthals exchanged genetic material with humans and the time at which modern humans left Africa. Here, we present a method for using sequence data to jointly estimate the timing and magnitude of past admixture events, along with population divergence times and changes in effective population size. We infer demography from a collection of pairwise sequence alignments by summarizing their length distribution of tracts of identity by state (IBS) and maximizing an analytic composite likelihood derived from a Markovian coalescent approximation. Recent gene flow between populations leaves behind long tracts of identity by descent (IBD), and these tracts give our method power by influencing the distribution of shared IBS tracts. In simulated data, we accurately infer the timing and strength of admixture events, population size changes, and divergence times over a variety of ancient and recent time scales. Using the same technique, we analyze deeply sequenced trio parents from the 1000 Genomes project. The data show evidence of extensive gene flow between Africa and Europe after the time of divergence as well as substructure and gene flow among ancestral hominids. In particular, we infer that recent African-European gene flow and ancient ghost admixture into Europe are both necessary to explain the spectrum of IBS sharing in the trios, rejecting simpler models that contain less population structure.
We introduce a method for comparing a test genome with numerous genomes from a reference populati... more We introduce a method for comparing a test genome with numerous genomes from a reference population. Sites in the test genome are given a weight, w, that depends on the allele frequency, x, in the reference population. The projection of the test genome onto the reference population is the average weight for each x, [Formula: see text]. The weight is assigned in such a way that, if the test genome is a random sample from the reference population, then [Formula: see text]. Using analytic theory, numerical analysis, and simulations, we show how the projection depends on the time of population splitting, the history of admixture, and changes in past population size. The projection is sensitive to small amounts of past admixture, the direction of admixture, and admixture from a population not sampled (a ghost population). We compute the projections of several human and two archaic genomes onto three reference populations from the 1000 Genomes project-Europeans, Han Chinese, and Yoruba-an...
In many areas of computational biology, hidden Markov models (HMMs) have been used to model local... more In many areas of computational biology, hidden Markov models (HMMs) have been used to model local genomic features. In particular, coalescent HMMs have been used to infer ancient population sizes, migration rates, divergence times, and other parameters such as mutation and recombination rates. As more loci, sequences, and hidden states are added to the model, however, the runtime of coalescent HMMs can quickly become prohibitive. Here we present a new algorithm for reducing the runtime of coalescent HMMs from quadratic in the number of hidden time states to linear, without making any additional approximations. Our algorithm can be incorporated into various coalescent HMMs, including the popular method PSMC for inferring variable effective population sizes. Here we implement this algorithm to speed up our demographic inference method diCal, which is equivalent to PSMC when applied to a sample of two haplotypes. We demonstrate that the linear-time method can reconstruct a population size change history more accurately than the quadratic-time method, given similar computation resources. We also apply the method to data from the 1000 Genomes project, inferring a high-resolution history of size changes in the European population.
Polar bears are uniquely adapted to life in the High Arctic and have undergone drastic physiologi... more Polar bears are uniquely adapted to life in the High Arctic and have undergone drastic physiological changes in response to Arctic climates and a hyper-lipid diet of primarily marine mammal prey. We analyzed 89 complete genomes of polar bear and brown bear using population genomic modeling and show that the species diverged only 479-343 thousand years BP. We find that genes on the polar bear lineage have been under stronger positive selection than in brown bears; nine of the top 16 genes under strong positive selection are associated with cardiomyopathy and vascular disease, implying important reorganization of the cardiovascular system. One of the genes showing the strongest evidence of selection, APOB, encodes the primary lipoprotein component of low-density lipoprotein (LDL); functional mutations in APOB may explain how polar bears are able to cope with life-long elevated LDL levels that are associated with high risk of heart disease in humans.
Throughout history, the population size of modern humans has varied considerably due to changes i... more Throughout history, the population size of modern humans has varied considerably due to changes in environment, culture, and technology. More accurate estimates of population size changes, and when they occurred, should provide a clearer picture of human colonization history and help remove confounding effects from natural selection inference. Demography influences the pattern of genetic variation in a population, and thus genomic data of multiple individuals sampled from one or more present-day populations contain valuable information about the past demographic history. Recently, Li and Durbin developed a coalescent-based hidden Markov model, called the pairwise sequentially Markovian coalescent (PSMC), for a pair of chromosomes (or one diploid individual) to estimate past population sizes. This is an efficient, useful approach, but its accuracy in the very recent past is hampered by the fact that, because of the small sample size, only few coalescence events occur in that period. Multiple genomes from the same population contain more information about the recent past, but are also more computationally challenging to study jointly in a coalescent framework. Here, we present a new coalescent-based method that can efficiently infer population size changes from multiple genomes, providing access to a new store of information about the recent past. Our work generalizes the recently developed sequentially Markov conditional sampling distribution framework, which provides an accurate approximation of the probability of observing a newly sampled haplotype given a set of previously sampled haplotypes. Simulation results demonstrate that we can accurately reconstruct the true population histories, with a significant improvement over the PSMC in the recent past. We apply our method, called diCal, to the genomes of multiple human individuals of European and African ancestry to obtain a detailed population size change history during recent times.
How and when the Americas were populated remains contentious. Using ancient and modern genome-wid... more How and when the Americas were populated remains contentious. Using ancient and modern genome-wide data, we find that the ancestors of all present-day Native Americans, including Athabascans and Amerindians, entered the Americas as a single migration wave from Siberia no earlier than 23 thousand years ago (KYA), and after no more than 8,000-year isolation period in Beringia. Following their arrival to the Americas, ancestral Native Americans diversified into two basal genetic branches around 13 KYA, one that is now dispersed across North and South America and the other is restricted to North America. Subsequent gene flow resulted in some Native Americans sharing ancestry with present-day East Asians (including Siberians) and, more distantly, Australo-Melanesians. Putative…
Uploads
Papers by Kelley Harris