Article Figures & Data
Figures
- Figure 1
Decay of the error function between the equilibrium density of allele frequencies and its polynomial approximation and piecewise linear approximation. The horizontal axis denotes the number of polynomials for the lower curve and the number of grid points for the upper curve.
- Figure 2
Simulation of two Brownian paths on a two-population tree. The plot illustrates different types of initial conditions for the paths. The initial condition can be either a random allele frequency in the ancestral population at mutation–drift equilibrium (solid lines) or a de novo mutation that arises in one of the populations after the ancestral population leaves the state of equilibrium (shaded lines).
- Figure 3
Decay of the chi-square statistic in MultiPop (top) and ∂a∂i (bottom). Four different demographic scenarios with two simultaneous populations and 50 chromosomes sampled per population are considered. For simplicity, the average scaled migration rate is used to label each scenario. The observed AFS were constructed using Π = 50,000 independent loci produced with Monte Carlo simulations.
- Figure 4
Decay of the chi-square statistic in MultiPop (top) and ∂a∂i (bottom). Three different demographic scenarios with three simultaneous populations and 20 chromosomes sampled per population are considered. For simplicity, the average scaled migration rate is used to label each scenario. The observed AFS were constructed using Π = 50,000 independent loci produced with Monte Carlo simulations.
- Figure 5
A graphical representation of a four-population model for the human expansion out of Africa and peopling of the Americas. The nonconstancy of the population sizes of CEU, CHB, and MEX is modeled by means of an exponential growth model with growth rates rEU, rAS, and rMX.
Tables
- Table 1 Comparison of numerical approximations of the AFS and the simulated AFS
Model no. Intensity of migration (2Nm) MPop vs. Monte Carlo (chi-square statistic) ∂a∂i vs. Monte Carlo (chi-square statistic) 1 0 0.001265 0.002479 2 <0.5 0.00894 0.005955 3 1 0.01076 0.006457 4 >1 0.01140 0.006286 5 < 0.5 0.007558 0.04684 6 1 0.01511 0.02882 7 >1 0.02979 0.01791 Chi-square statistics associated with seven demographic scenarios are shown. The AFS computed by MultiPop (MPop) corresponds to the Λ = 35 AFS, and the AFS computed by ∂a∂i corresponds to a grid size of 40 grid points per population. The frequency spectra were normalized in all the cases such that the total number of SNPs was 1.
- Table 2 Comparison of maximum-likelihood estimates using different numerical approximations
Model Model parameter True value θ MPop θ ∂a∂i 1 4NAu 1 1 1 1 N1/NA 1 1.015 1.042 1 N2/NA 0.4538 0.4385 0.4421 1 T/2NA 0.01875 0.01788 0.01840 2 4NAu 1 1 1 2 N1/NA 1 1.08 1.042 2 N2/NA 0.4538 0.6467 0.6379 2 T/2NA 0.01875 0.02304 0.02720 2 2NAm1→2 0.3 0.5706 3.529 2 2NAm2→1 0.88 0.8735 3.228 3 4NAu 1 1 1 3 N1/NA 1 1.099 1.044 3 N2/NA 0.4538 0.6648 0.6497 3 T/2NA 0.01875 0.02287 0.02783 3 2NAm1→2 0.8 0.5873 4.022 3 2NAm2→1 1.76 0.9469 3.965 4 4NAu 1 1 1 4 N1/NA 1 1.086 1.068 4 N2/NA 0.4538 0.6712 0.6493 4 T/2NA 0.01875 0.02258 0.02592 4 2NAm1→2 1.12 0.6190 2.574 4 2NAm2→1 2.64 0.9910 3.798 5 4NAu 1 1 1 5 N1/NA 0.3630 0.3713 0.3522 5 N2/NA 0.1630 0.1540 0.1440 5 N3/NA 0.06302 0.05673 0.05569 5 Ta/2NA 0.022 0.02158 0.02041 5 Tb/2NA 0.013 0.01225 0.01152 6 4NAu 1 1 1 6 N1/NA 0.3630 0.3713 0.3344 6 N2/NA 0.1630 0.17855 0.1527 6 N3/NA 0.06302 0.05950 0.06112 6 Ta/2NA 0.022 0.02996 0.01749 6 Tb/2NA 0.013 0.01211 0.01249 6 2NAma1→2 0.05 0.04562 0.0001828 6 2NAma2→1 5 0.02345 0.0009895 6 2NAmb1→2 2 0.002115 0.003311 6 2NAmb1→3 0.3 0.3897 1.505 6 2NAmb2→1 0.005 0.007982 0.09358 6 2NAmb2→3 0.3 0.0007637 4.4e-09 6 2NAmb3→1 3 0.6831 3.398 6 2NAmb3→2 3 1.788 2.640 7 4NAu 1 1 1 7 N1/NA 0.3630 0.3874 0.3138 7 N2/NA 0.1630 0.1724 0.1445 7 N3/NA 0.06302 0.05843 0.05589 7 Ta/2NA 0.022 0.02415 0.01296 7 Tb/2NA 0.013 0.01155 0.01115 7 2NAma1→2 0.05 0.02982 0.7007 7 2NAma2→1 12 0.007439 0.004225 7 2NAmb1→2 6 0.0007166 0.0003699 7 2NAmb1→3 0.3 0.00090192 3.1354 7 2NAmb2→1 0.005 0.0003413 0.0020024 7 2NAmb2→3 0.3 0.0007713 0.0001095 7 2NAmb3→1 3 0.007362 3.232 7 2NAmb3→2 3 0.4090 0.01191 Maximum-likelihood estimates of seven different demographic scenarios are shown. Many results are biased due to numerical errors in the calculation of the frequency spectra (see subsection Numerical errors and bias-corrected confidence intervals for a detailed discussion on how numerical errors affect the location of the maximum-likelihood peak). In the case of MultiPop, 40 polynomials were used in the two-population models and 35 polynomials in the three-population models. We used 100 grid points in all models approximated with finite-difference schemes. The observed AFS were constructed using Π = 50,000 independent loci produced with Monte Carlo simulations. The maximum-likelihood peak was found by means of the BFGS method, using the coordinates of the true peak as the initial point in this local optimization algorithm.
- Table 3 Comparison of computing time
Model No. polynomials (MPop) Computing time of MPop (sec) Grid size (∂a∂i) Computing time of ∂a∂i (sec) 2 15 0.27 15 0.07 2 20 0.61 50 0.10 2 25 1.10 100 0.13 2 30 1.86 500 1.43 2 35 2.92 1000 5.68 6 15 3.1 15 0.16 6 20 7.95 50 0.86 6 25 17.31 100 6.33 6 30 33.3 200 68.41 6 35 57.41 300 253.85 Computing times required to evaluate an allele-frequency spectrum using MultiPop and ∂a∂i are shown. The demographic scenarios involved two populations and 50 chromosomes sampled per population or three populations and 20 chromosomes sampled per population. The CPU used to measure the computing times was an Intel Core(TM)2 Duo P8600 with speed 2.40 GHz.
- Table 4 Inference of a four-population model for the human expansion out of Africa and peopling of the Americas
Model parameters θ MPop 95% C.I. θ ∂a∂i, 1 95% C.I. θ ∂a∂i, 2 95% C.I. NA 10,400 8,670–12,200 7,300 4,400–10,100 NAF 17,300 10,900–29,300 12,300 11,500–13,900 NB 2,060 346–5,070 2,100 1,400–2,900 NEU0 1,710 1,030–3,020 1,000 500–1,900 1,500 700–2,100 rEU (generation−1) 0.0055 0.00351–0.0103 0.004 0.0015–0.0066 0.0023 0.0008–0.0045 NAS0 453 210–800 510 310–910 590 320–800 rAS (generation−1) 0.016 0.0102–0.0301 0.0055 0.0023–0.0088 0.0037 0.0016–0.006 mAF→B (×10−5) 6.06 0.0–13.6 25 15–34 mAF→B (×10−5) 1.63 0.0–3.47 3.0 2.0–6.0 mAF→B (×10−5) 0.487 0.0–1.07 25 15–34 mAF→B (×10−5) 1.54 0.0–2.96 9.6 2.3–17.4 TAF (yr) 125,400 54,300–250,000 220,000 100,000–510,000 TB (yr) 52,400 36,000–80,800 140,000 40,000–270,000 TEU→AS (yr) 29,500 23,500–38,000 21,200 17,200–26,500 26,400 18,100–43,100 NMX0 3,200 1,100–6,100 800 160–1,800 rMX (generation−1) 0.0071 0.0043–0.011 0.005 0.0014–0.0117 TMX (yr) 29,300 23,000–37,500 21,600 16,300–26,900 fMX (%) 20.4 3.2–41 48 42–60 Inference of parameters by means of maximum likelihood is shown. Confidence intervals were computed by means of nonparametric bootstrap. The estimated parameters θ with MultiPop correspond to the mean of the bootstrap distribution. The estimates by MultiPop are compared with the estimates by ∂a∂i with two three-population models. ∂a∂i 1 denotes the three-population model for the out-of-Africa event described in Gutenkunst et al. (2009). ∂a∂i 2 denotes the three-population model for the peopling of the Americas studied in Gutenkunst et al. (2009).
