An evolutionary maximum principle for density-dependent population dynamics in a fluctuating environment

The evolution of population dynamics in a stochastic environment is analysed under a general form of density-dependence with genetic variation in r and K, the intrinsic rate of increase and carrying capacity in the average environment, and in σ_e², the environmental variance of population growth rate. The continuous-time model assumes a large population size and a stationary distribution of environments with no autocorrelation. For a given population density, N, and genotype frequency, p, the expected selection gradient is always towards an increased population growth rate, and the expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Long-term evolution maximizes the expected value of the density-dependence function, averaged over the stationary distribution of N. In the θ-logistic model, where density dependence of population growth is a function of N^θ, long-term evolution maximizes E[N^θ]=[1−σ_e²/(2r)]K^θ. While σ_e² is always selected to decrease, r and K are always selected to increase, implying a genetic trade-off among them. By contrast, given the other parameters, θ has an intermediate optimum between 1.781 and 2 corresponding to the limits of high or low stochasticity.     

Estimation of the Amount of DNA Polymorphism When the Neutral Mutation Rate Varies among Sites

Knowing the amount of DNA polymorphism is essential to understand the mechanism of maintaining DNA polymorphism in a natural population. The amount of DNA polymorphism can be measured by the average number of nucleotide differences per site (π), the proportion of segregating (polymorphic) site (s) and the minimum number of mutations per site (s*). Since the latter two quantities depend on the sample size, θ is often used as a measure of the amount of DNA polymorphism, where θ = 4Nμ, N is the effective population size and μ is the neutral mutation rate per site per generation. It is known that θ estimated from π, s and s* under the infinite site model can be biased when the mutation rate varies among sites. We have therefore developed new methods for estimating θ under the finite site model. Using computer simulations, it has been shown that the new methods give almost unbiased estimates even when the mutation rate varies among sites substantially. Furthermore, we have also developed new statistics for testing neutrality by modifying Tajima's D statistic. Computer simulations suggest that the new test statistics can be used even when the mutation rate varies among sites.     

The Amount of DNA Polymorphism Maintained in a Finite Population When the Neutral Mutation Rate Varies among Sites

The expectations of the average number of nucleotide differences per site (π), the proportion of segregating site (s), the minimum number of mutations per site (s*) and some other quantities were derived under the finite site models with and without rate variation among sites, where the finite site models include Jukes and Cantor's model, the equal-input model and Kimura's model. As a model of rate variation, the gamma distribution was used. The results indicate that if distribution parameter α is small, the effect of rate variation on these quantities are substantial, so that the estimates of θ based on the infinite site model are substantially underestimated, where θ = 4Nv, N is the effective population size and v is the mutation rate per site per generation. New methods for estimating θ are also presented, which are based on the finite site models with and without rate variation. Using these methods, underestimation can be corrected.     

Modelling mitochondrial site polymorphisms to infer the number of segregating units and mutation rate

We present a mathematical model of mitochondrial inheritance evolving under neutral evolution to interpret the heteroplasmies observed at some sites. A comparison of the levels of heteroplasmies transmitted from mother to her offspring allows us to estimate the number N_x of inherited mitochondrial genomes (segregating units). The model demonstrates the necessity of accounting for both the multiplicity of an unknown number N_x, and the threshold θ, below which heteroplasmy cannot be detected reliably, in order to estimate the mitochondrial mutation rate μ_m in the maternal line of descent. Our model is applicable to pedigree studies of any eukaryotic species where site heteroplasmies are observed in regions of the mitochondria, provided neutrality can be assumed. The model is illustrated with an analysis of site heteroplasmies in the first hypervariable region of mitochondrial sequence data sampled from Adélie penguin families, providing an estimate N_x and μ_m. This estimate of μ_m was found to be consistent with earlier estimates from ancient DNA analysis.     

Multiple coping strategies maintain stability of a small mammal population in a resource‐restricted environment

In semi‐arid environments, aperiodic rainfall pulses determine plant production and resource availability for higher trophic levels, creating strong bottom‐up regulation. The influence of climatic factors on population vital rates often shapes the dynamics of small mammal populations in such resource‐restricted environments. Using a 21‐year biannual capture–recapture dataset (1993 to 2014), we examined the impacts of climatic factors on the population dynamics of the brush mouse (Peromyscus boylii) in semi‐arid oak woodland of coastal‐central California. We applied Pradel''s temporal symmetry model to estimate capture probability (p), apparent survival (φ), recruitment (f), and realized population growth rate (λ) of the brush mouse and examined the effects of temperature, rainfall, and El Niño on these demographic parameters. The population was stable during the study period with a monthly realized population growth rate of 0.993 ± SE 0.032, but growth varied over time from 0.680 ± 0.054 to 1.450 ± 0.083. Monthly survival estimates averaged 0.789 ± 0.005 and monthly recruitment estimates averaged 0.175 ± 0.038. Survival probability and realized population growth rate were positively correlated with rainfall and negatively correlated with temperature. In contrast, recruitment was negatively correlated with rainfall and positively correlated with temperature. Brush mice maintained their population through multiple coping strategies, with high recruitment during warmer and drier periods and higher survival during cooler and wetter conditions. Although climatic change in coastal‐central California will likely favor recruitment over survival, varying strategies may serve as a mechanism by which brush mice maintain resilience in the face of climate change. Our results indicate that rainfall and temperature are both important drivers of brush mouse population dynamics and will play a significant role in predicting the future viability of brush mice under a changing climate.     

Maximum Likelihood Estimation of Population Parameters

One of the most important parameters in population genetics is θ = 4N(e)μ where N(e) is the effective population size and μ is the rate of mutation per gene per generation. We study two related problems, using the maximum likelihood method and the theory of coalescence. One problem is the potential improvement of accuracy in estimating the parameter θ over existing methods and the other is the estimation of parameter λ which is the ratio of two θ's. The minimum variances of estimates of the parameter θ are derived under two idealized situations. These minimum variances serve as the lower bounds of the variances of all possible estimates of θ in practice. We then show that Watterson's estimate of θ based on the number of segregating sites is asymptotically an optimal estimate of θ. However, for a finite sample of sequences, substantial improvement over Watterson's estimate is possible when θ is large. The maximum likelihood estimate of λ = θ(1)/θ(2) is obtained and the properties of the estimate are discussed.     

DNA Polymorphism in a Subdivided Population: The Expected Number of Segregating Sites in the Two-Subpopulation Model

Using the two subpopulation model, the expected numbers of segregating sites in a number of DNA sequences randomly sampled from a subdivided population were examined for several types of population subdivisions. It is shown that, in the case where the pattern of migration is symmetrical such as the finite island model, the expected number of segregating sites is independent of the migration rate when two or three DNA sequences are randomly sampled from the same subpopulation, but depends on the migration rate when more than three DNA sequences are sampled. It is also shown that the population subdivision can increase the amount of DNA polymorphism even in a subpopulation in some cases.     

A Phylogenetic Estimator of Effective Population Size or Mutation Rate

A new estimator of the essential parameter θ = 4N(e)μ from DNA polymorphism data is developed under the neutral Wright-Fisher model without recombination and population subdivision, where N(e) is the effective population size and μ is the mutation rate per locus per generation. The new estimator has a variance only slightly larger than the minimum variance of all possible unbiased estimators of the parameter and is substantially smaller than that of any existing estimator. The high efficiency of the new estimator is achieved by making full use of phylogenetic information in a sample of DNA sequences from a population. An example of estimating θ by the new method is presented using the mitochondrial sequences from an American Indian population.     

Effects of Photo and Genotype-Based Misidentification Error on Estimates of Survival,Detection and State Transition using Multistate Survival Models

We simulated multistate capture histories (CHs) by varying state survival (ϕ), detection (p) and transition (ψ), number of total capture occasions and releases per capture occasion and then modified these scenarios to mimic false rejection error (FRE), a common misidentification error, resulting from the failure to match samples of the same individual. We then fit a multistate model and estimated accuracy, bias and precision of state-specific ϕ, p and ψ to better understand the effects of FRE on different simulation scenarios. As expected, ϕ, and p, decreased in accuracy with FRE, with lower accuracy when CHs were simulated under a shorter-term study and a lower number of releases per capture occasion (lower sample size). Accuracy of ψ estimates were robust to FRE except in those CH scenarios simulated using low sample size. The effect of FRE on bias was not consistent among parameters and differed by CH scenario. As expected, ϕ was negatively biased with increased FRE (except for the low ϕ low p CH scenario simulated with a low sample size), but we found that the magnitude of bias differed by scenario (high p CH scenarios were more negatively biased). State transition was relatively unbiased, except for the low p CH scenarios simulated with a low sample size, which were positively biased with FRE, and high p CH scenarios simulated with a low sample size. The effect of FRE on precision was not consistent among parameters and differed by scenario and sample size. Precision of ϕ decreased with FRE and was lowest with the low ϕ low p CH scenarios. Precision of p estimates also decreased with FRE under all scenarios, except the low ϕ high p CH scenarios. However, precision of ψ increased with FRE, except for those CH scenarios simulated with a low sample size. Our results demonstrate how FRE leads to loss of accuracy in parameter estimates in a multistate model with the exception of ψ when estimated using an adequate sample size.     

High genetic diversity at the extreme range edge: nucleotide variation at nuclear loci in Scots pine (Pinus sylvestris L.) in Scotland

Nucleotide polymorphism at 12 nuclear loci was studied in Scots pine populations across an environmental gradient in Scotland, to evaluate the impacts of demographic history and selection on genetic diversity. At eight loci, diversity patterns were compared between Scottish and continental European populations. At these loci, a similar level of diversity (θ_sil=∼0.01) was found in Scottish vs mainland European populations, contrary to expectations for recent colonization, however, less rapid decay of linkage disequilibrium was observed in the former (ρ=0.0086±0.0009, ρ=0.0245±0.0022, respectively). Scottish populations also showed a deficit of rare nucleotide variants (multi-locus Tajima''s D=0.316 vs D=−0.379) and differed significantly from mainland populations in allelic frequency and/or haplotype structure at several loci. Within Scotland, western populations showed slightly reduced nucleotide diversity (π_tot=0.0068) compared with those from the south and east (0.0079 and 0.0083, respectively) and about three times higher recombination to diversity ratio (ρ/θ=0.71 vs 0.15 and 0.18, respectively). By comparison with results from coalescent simulations, the observed allelic frequency spectrum in the western populations was compatible with a relatively recent bottleneck (0.00175 × 4N_e generations) that reduced the population to about 2% of the present size. However, heterogeneity in the allelic frequency distribution among geographical regions in Scotland suggests that subsequent admixture of populations with different demographic histories may also have played a role.     

How Many Processed Pseudogenes Are Accumulated in a Gene Family?

A simple kinetic model is developed that describes the accumulation of processed pseudogenes in a functional gene family. Insertion of new pseudogenes occurs at rate ν per gene and is countered by spontaneous deletion (at rate δ per DNA segment) of segments containing processed pseudogenes. If there are k functional genes in a gene family, the equilibrium number of processed pseudogenes is k(ν/δ), and the percentage of functional genes in the gene family at equilibrium is 1/[1 + (ν/δ)]. ν/δ values estimated for five gene families ranged from 1.7 to 15. This fairly narrow range suggests that the rates of formation and deletion of processed pseudogenes may be positively correlated for these families. If δ is sufficiently large relative to the per nucleotide mutation rate µ (δ > 20µ), processed pseudogenes will show high homology with each other, even in the absence of gene conversion between pseudogenes. We argue that formation of processed pseudogenes may share common pathways with transposable elements and retroviruses, creating the potential for correlated responses in the evolution of processed pseudogenes due to direct selection for control of transposable elements and/or retroviruses. Finally, we discuss the nature of the selective forces that may act directly or indirectly to influence the evolution of processed pseudogenes.

Anything produced by evolution is bound to be a bit of a mess—S. Brenner

     

The Dynamics of Finite Haploid Populations with Overlapping Generations. II. the Diffusion Approximation

The dynamics of a gene in a haploid population can be explained approximately by considering the average reproductive value of the gene. The dynamics of the average reproductive value are similar to those of a gene in a population with nonoverlapping generations with the following modifications: The effective population size, N_e, replaces N; the average mutation rates µ* and ν* replace µ and ν; the average overall selection r*+(T-1)s** replaces s; and time is measured in terms of generations, T. The implications of the average selection coefficient to adaptive life histories are discussed.     

Correcting Estimators of θ and Tajima's D for Ascertainment Biases Caused by the Single-Nucleotide Polymorphism Discovery Process

Most single-nucleotide polymorphism (SNP) data suffer from an ascertainment bias caused by the process of SNP discovery followed by SNP genotyping. The final genotyped data are biased toward an excess of common alleles compared to directly sequenced data, making standard genetic methods of analysis inapplicable to this type of data. We here derive corrected estimators of the fundamental population genetic parameter θ = 4N_eμ (N_e, effective population size; μ, mutation rate) on the basis of the average number of pairwise differences and on the basis of the number of segregating sites. We also derive the variances and covariances of these estimators and provide a corrected version of Tajima's D statistic. We reanalyze a human genomewide SNP data set and find substantial differences in the results with or without ascertainment bias correction.     

Phylogenetic Analysis of Bacterial Communities in Mesophilic and Thermophilic Bioreactors Treating Pharmaceutical Wastewater

The phylogenetic diversity of the bacterial communities supported by a seven-stage, full-scale biological wastewater treatment plant was studied. These reactors were operated at both mesophilic (28 to 32°C) and thermophilic (50 to 58°C) temperatures. Community fingerprint analysis by denaturing gradient gel electrophoresis (DGGE) of the PCR-amplified V3 region of the 16S rRNA gene from the domain Bacteria revealed that these seven reactors supported three distinct microbial communities. A band-counting analysis of the PCR-DGGE results suggested that elevated reactor temperatures corresponded with reduced species richness. Cloning of nearly complete 16S rRNA genes also suggested a reduced species richness in the thermophilic reactors by comparing the number of clones with different nucleotide inserts versus the total number of clones screened. While these results imply that elevated temperature can reduce species richness, other factors also could have impacted the number of populations that were detected. Nearly complete 16S rDNA sequence analysis showed that the thermophilic reactors were dominated by members from the β subdivision of the division Proteobacteria (β-proteobacteria) in addition to anaerobic phylotypes from the low-G+C gram-positive and Synergistes divisions. The mesophilic reactors, however, included at least six bacterial divisions, including Cytophaga-Flavobacterium-Bacteroides, Synergistes, Planctomycetes, low-G+C gram-positives, Holophaga-Acidobacterium, and Proteobacteria (α-proteobacteria, β-proteobacteria, γ-proteobacteria and δ-proteobacteria subdivisions). The two PCR-based techniques detected the presence of similar bacterial populations but failed to coincide on the relative distribution of these phylotypes. This suggested that at least one of these methods is insufficiently quantitative to determine total community biodiversity—a function of both the total number of species present (richness) and their relative distribution (evenness).     

Mitochondrial Genome Analysis of Primary Open Angle Glaucoma Patients

Primary open angle glaucoma (POAG) is a multi-factorial optic disc neuropathy characterized by accelerating damage of the retinal ganglion cells and atrophy of the optic nerve head. The vulnerability of the optic nerve damage leading to POAG has been postulated to result from oxidative stress and mitochondrial dysfunction. In this study, we investigated the possible involvement of the mitochondrial genomic variants in 101 patients and 71 controls by direct sequencing of the entire mitochondrial genome. The number of variable positions in the mtDNA with respect to the revised Cambridge Reference Sequence (rCRS), have been designated “Segregating Sites”. The segregating sites present only in the patients or controls have been designated “Unique Segregating Sites (USS)”. The population mutation rate (θ = 4N_eμ) as estimated by Watterson’s θ (θ_w), considering only the USS, was significantly higher among the patients (p = 9.8×10⁻¹⁵) compared to controls. The difference in θ_w and the number of USS were more pronounced when restricted to the coding region (p<1.31×10⁻²¹ and p = 0.006607, respectively). Further analysis of the region revealed non-synonymous variations were significantly higher in Complex I among the patients (p = 0.0053). Similar trends were retained when USS was considered only within complex I (frequency 0.49 vs 0.31 with p<0.0001 and mutation rate p-value <1.49×10⁻⁴³) and ND5 within its gene cluster (frequency 0.47 vs 0.23 with p<0.0001 and mutation rate p-value <4.42×10⁻⁴⁷). ND5 is involved in the proton pumping mechanism. Incidentally, glaucomatous trabecular meshwork cells have been reported to be more sensitive to inhibition of complex I activity. Thus mutations in ND5, expected to inhibit complex I activity, could lead to generation of oxidative stress and favor glaucomatous condition.     

Glycogen synthase kinase 3β inhibition enhanced proliferation,migration and functional re-endothelialization of endothelial progenitor cells in hypercholesterolemia microenvironment

Hypercholesterolemia impairs the quantity and function of endothelial progenitor cell. We hypothesized that glycogen synthase kinase 3β activity is involved in regulating biological function of endothelial progenitor cells in hypercholesterolemia microenvironment. For study, endothelial progenitor cells derived from apolipoprotein E-deficient mice fed with high-fat diet were used. Glycogen synthase kinase 3β activity was interfered with glycogen synthase kinase 3β inhibitor lithium chloride or transduced with replication defective adenovirus vector expressing catalytically inactive glycogen synthase kinase 3β (GSK3β-KM). Functions of endothelial progenitor cells, proliferation, migration, secretion and network formation of endothelial progenitor cells were assessed in vitro. The expression of phospho-glycogen synthase kinase 3β, β-catenin and cyclinD1 in endothelial progenitor cells was detected by Western blot. The in vivo function re-endothelialization and vasodilation were also analyzed by artery injury model transplanted with glycogen synthase kinase 3β-inhibited endothelial progenitor cells. We demonstrated that while the proliferation, migration, network formation as well as VEGF and NO secretion were impaired in apolipoprotein E-deficient endothelial progenitor cells, glycogen synthase kinase 3β inhibition significantly improved all these functions. Apolipoprotein E-deficient endothelial progenitor cells showed decreased phospho-glycogen synthase kinase 3β, β-catenin and cyclinD1 expression, whereas these signals were enhanced by glycogen synthase kinase 3β inhibition and accompanied with β-catenin nuclear translocation. Our in vivo model showed that glycogen synthase kinase 3β inhibition remarkably increased re-endothelial and vasodilation. Taken together, our data suggest that inhibition of glycogen synthase kinase 3β is associated with endothelial progenitor cell biological functions both in vitro and in vivo. It might be an important interference target in hypercholesterolemia microenvironment.     

MICB Allele Genotyping on Microarrays by Improving the Specificity of Extension Primers

Major histocompatibility complex (MHC) class I chain-related gene B (MICB) encodes a ligand for activating NKG2D that expressed in natural killer cells, γδ T cells, and αβ CD8⁺ T cells, which is associated with autoimmune diseases, cancer, and infectious diseases. Here, we have established a system for genotyping MICB alleles using allele-specific primer extension (ASPE) on microarrays. Thirty-six high quality, allele-specific extension primers were evaluated using strict and reliable cut-off values using mean fluorescence intensity (MFI), whereby an MFI >30,000 represented a positive signal and an MFI <10,000 represented a negative signal. Eight allele-specific extension primers were found to be false positives, five of which were improved by adjusting their length, and three of which were optimized by refractory modification. The MICB alleles (*002:01, *003, *005:02/*010, *005:03, *008, *009N, *018, and *024) present in the quality control panel could be exactly defined by 22 allele-specific extension primers. MICB genotypes that were identified by ASPE on microarrays were in full concordance with those identified by PCR-sequence-based typing. In conclusion, we have developed a method for genotyping MICB alleles using ASPE on microarrays; which can be applicable for large-scale single nucleotide polymorphism typing studies of population and disease associations.     

Translesion Synthesis across 1,N6-(2-Hydroxy-3-hydroxymethylpropan-1,3-diyl)-2′-deoxyadenosine (1,N6-γ-HMHP-dA) Adducts by Human and Archebacterial DNA Polymerases

The 1,N⁶-(2-Hydroxy-3-hydroxymethylpropan-1,3-diyl)-2′-deoxyadenosine (1,N⁶-γ-HMHP-dA) adducts are formed upon bifunctional alkylation of adenine nucleobases in DNA by 1,2,3,4-diepoxybutane, the putative ultimate carcinogenic metabolite of 1,3-butadiene. The presence of a substituted 1,N⁶-propano group on 1,N⁶-γ-HMHP-dA is expected to block the Watson-Crick base pairing of the adducted adenine with thymine, potentially contributing to mutagenesis. In this study, the enzymology of replication past site-specific 1,N⁶-γ-HMHP-dA lesions in the presence of human DNA polymerases (hpols) β, η, κ, and ι and archebacterial polymerase Dpo4 was investigated. Run-on gel analysis with all four dNTPs revealed that hpol η, κ, and Dpo4 were able to copy the modified template. In contrast, hpol ι inserted a single base opposite 1,N⁶-γ-HMHP-dA but was unable to extend beyond the damaged site, and a complete replication block was observed with hpol β. Single nucleotide incorporation experiments indicated that although hpol η, κ, and Dpo4 incorporated the correct nucleotide (dTMP) opposite the lesion, dGMP and dAMP were inserted with a comparable frequency. HPLC-ESI-MS/MS analysis of primer extension products confirmed the ability of bypass polymerases to insert dTMP, dAMP, or dGMP opposite 1,N⁶-γ-HMHP-dA and detected large amounts of −1 and −2 deletion products. Taken together, these results indicate that hpol η and κ enzymes bypass 1,N⁶-γ-HMHP-dA lesions in an error-prone fashion, potentially contributing to A→T and A→C transversions and frameshift mutations observed in cells following treatment with 1,2,3,4-diepoxybutane.     

A Fast Estimate for the Population Recombination Rate Based on Regression

Recombination is a fundamental evolutionary force. Therefore the population recombination rate ρ plays an important role in the analysis of population genetic data; however, it is notoriously difficult to estimate. This difficulty applies both to the accuracy of commonly used estimates and to the computational efforts required to obtain them. Some particularly popular methods are based on approximations to the likelihood. They require considerably less computational efforts than the full-likelihood method with not much less accuracy. Nevertheless, the computation of these approximate estimates can still be very time consuming, in particular when the sample size is large. Although auxiliary quantities for composite likelihood estimates can be computed in advance and stored in tables, these tables need to be recomputed if either the sample size or the mutation rate θ changes. Here we introduce a new method based on regression combined with boosting as a model selection technique. For large samples, it requires much less computational effort than other approximate methods, while providing similar levels of accuracy. Notably, for a sample of hundreds or thousands of individuals, the estimate of ρ using regression can be obtained on a single personal computer within a couple of minutes while other methods may need a couple of days or months (or even years). When the sample size is smaller (n ≤ 50), our new method remains computational efficient but produces biased estimates. We expect the new estimates to be helpful when analyzing large samples and/or many loci with possibly different mutation rates.     

Quantitative models characterizing seed germination responses to abscisic Acid and osmoticum

Mathematical models were developed to characterize the physiological bases of the responses of tomato (Lycopersicon esculentum Mill. cv T5) seed germination to water potential (ψ) and abscisic acid (ABA). Using probit analysis, three parameters were derived that can describe the germination time courses of a seed population at different ψ or ABA levels. For the response of seed germination to reduced ψ, these parameters are the mean base water potential (¯ψ_b, MPa), the standard deviation of the base water potential among seeds in the population (σ_ψb, MPa), and the “hydrotime constant” (θ_H, MPa·h). For the response to ABA, they are the log of the mean base ABA concentration ([unk]ABA_b, m), the standard deviation of the base ABA concentration among seeds in the population (σ_ABAb, log[m]), and the “ABA-time constant” (θ_ABA, log[m]·h). The values of ¯ψ_b and [unk]ABA_b provide quantitative estimates of the mean sensitivity of germination rate to ψ or ABA, whereas σ^ψ_b and σ_ABAb account for the variation in sensitivity among seeds in the population. The time constants, θ_H and θ_ABA, indicate the extent to which germination rate will be affected by a given change in ψ or ABA. Using only these parameters, germination time courses can be predicted with reasonable accuracy at any medium ψ according to the equation probit(g) = [ψ - (θ_H/t_g) - ¯ψ_b]/σ_ψb, or at any ABA concentration according to the equation probit(g) = [log[ABA] - (θ_ABA/t_g) - log[[unk]ABA_b]]/σ_ABAb, where t_g is the time to radicle emergence of percentage g, and ABA is the ABA concentration (m) in the incubation solution. In the presence of both ABA and reduced ψ, the same parameters can be used to predict seed germination time courses based upon strictly additive effects of ψ and ABA in delaying the time of radicle emergence. Further analysis indicates that ABA and ψ can act both independently and interactively to influence physiological processes preparatory for radicle growth, such as the accumulation of osmotic solutes in the embryo. The models provide quantitative values for the sensitivity of germination to ABA or ψ, allow evaluation of independent and interactive effects of the two factors, and have implications for understanding how ABA and ψ may regulate growth and development.