One-dimensional Stepping-stone Model Research Articles

Where do we draw the line? A simulation approach for evaluating management of marine fish stocks with isolation-by-distance stock structure

There is no established management protocol for stocks subject to isolation-by-distance (IBD) stock structure. This study examines several management strategies for two marine fish species subject to IBD using simulation: Pacific cod (Gadus macrocephalus) in the Aleutian Islands (AI) and northern rockfish (Sebastes polyspinis) in the Eastern Bering Sea (EBS) and AI. A one-dimensional stepping stone model was used to model IBD and was intended to mimic regions where marine species are exploited along a continental shelf. The performance of spatial assessment and management methods depended on how the range was split. Splitting anywhere within the managed area led to fewer demes falling below target and threshold biomass levels and higher yield than managing the entire area as a single unit. Equilibrium yield was maximized when each deme was assessed and managed separately and under catch cascading, in which harvest quotas within a management unit are spatially allocated based upon the distribution of survey biomass. The longer-lived rockfish declined more slowly than Pacific cod and experienced greater depletion in biomass under disproportionate fishing effort because of lower productivity. Overall, splitting a management area of the size simulated in the model improved performance measures, and the optimal management strategy grouped management units by demes with similar relative fishing effort.

Gene flow within and between catchments in the threatened riparian plant Myricaria germanica.

One of the major distinctions of riparian habitats is their linearity. In linear habitats, gene flow is predicted to follow a one-dimensional stepping stone model, characterized by bidirectional gene flow between neighboring populations. Here, we studied the genetic structure of Myricaria germanica, a threatened riparian shrub which is capable of both wind and water dispersal. Our data led us to reject the ‘one catchment – one gene pool’ hypothesis as we found support for two gene pools, rather than four as expected in a study area including four catchments. This result also implies that in the history of the studied populations, dispersal across catchments has occurred. Two contemporary catchment-crossing migration events were detected, albeit between spatially proximate catchments. Allelic richness and inbreeding coefficients differed substantially between gene pools. There was significant isolation by distance, and our data confirmed the one-dimensional stepping-stone model of gene flow. Contemporary migration was bidirectional within the studied catchments, implying that dispersal vectors other than water are important for M. germanica.

A Quantitative Test of Population Genetics Using Spatiogenetic Patterns in Bacterial Colonies

It is widely accepted that population-genetics theory is the cornerstone of evolutionary analyses. Empirical tests of the theory, however, are challenging because of the complex relationships between space, dispersal, and evolution. Critically, we lack quantitative validation of the spatial models of population genetics. Here we combine analytics, on- and off-lattice simulations, and experiments with bacteria to perform quantitative tests of the theory. We study two bacterial species, the gut microbe Escherichia coli and the opportunistic pathogen Pseudomonas aeruginosa, and show that spatiogenetic patterns in colony biofilms of both species are accurately described by an extension of the one-dimensional stepping-stone model. We use one empirical measure, genetic diversity at the colony periphery, to parameterize our models and show that we can then accurately predict another key variable: the degree of short-range cell migration along an edge. Moreover, the model allows us to estimate other key parameters, including effective population size (density) at the expansion frontier. While our experimental system is a simplification of natural microbial community, we argue that it constitutes proof of principle that the spatial models of population genetics can quantitatively capture organismal evolution.

WHEN GAPS REALLY ARE GAPS: STATISTICAL PHYLOGEOGRAPHY OF HYDROTHERMAL VENT INVERTEBRATES

The invertebrate animals endemic to deep-sea hydrothermal vents are distributed intermittently along relatively linear oceanic ridge axes. A one-dimensional stepping-stone model, therefore, provides a reasonable starting hypothesis of population structure for these species. Nevertheless, population genetic studies of many species from eastern Pacific vents did not detect the expected signatures of isolation-by-distance (IBD). Instead, distinct patterns of geographical subdivision have been attributed to the unique dispersal modes of individual species, topographical discontinuities of the ridge axes, nonequilibrium metapopulation scenarios and cryptic species. Here, we reexamined these inferences in light of expectations generated by computer simulations of a one-dimensional stepping-stone model. We evaluated whether the previously inferred subdivisions are statistically robust to an alternative explanation that continuous stepping-stone migration has occurred along the ridge axes but discontinuities in the sampling design (gaps) have generated the apparent disjunctions. We found that previous inferences about barriers to gene flow (vicariance) were supported in many cases, but that failures to detect evidence for IBD could be explained by low statistical power associated with the sampling effort. The simulation approaches presented here might be useful for testing the significance of inferred phylogeographic gaps in other species.

One-dimensional stepping stone models, sardine genetics and Brownian local time

Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability ν, or a voter model on ℤ in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that if Mν/L and L/K2 converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.

Multiscale analysis of Hymenocallis coronaria (Amaryllidaceae) genetic diversity, genetic structure, and gene movement under the influence of unidirectional stream flow

Understanding gene movement patterns in unidirectional flow environments and their effect on patterns of genetic diversity and genetic structure is necessary to manage these systems. Hypotheses and models to explain genetic patterns in streams are rare, and the results of macrophyte studies are inconsistent. This study addresses Ritland's (Canadian Journal of Botany 67: 2017-2024) unidirectional diversity hypothesis, the one-dimensional stepping stone model, and the metapopulation model within and among populations. Hymenocallis coronaria, an aquatic macrophyte of rocky river shoals of the SE USA, was sampled in four river basins. Within populations and among populations <16.2 km apart had significant isolation by distance. However, the rate of gene flow decay was not consistent with a one-dimensional stepping stone model, nor was evidence strong or consistent for Ritland's hypothesis. Some evidence indicates that localized metapopulation processes may be affecting genetic diversity and structure; however, gene flow patterns inconsistent with the assumptions of the linear and unidirectional models are also a possible influence. We discuss three variants on the one-dimensional stepping stone model. Future research in linear environments should examine the expectations of these models. This study is also one of the first efforts to calculate population genetic parameters using a new program, TETRASAT.

Equilibrium between Genetic Drift and Migration at Various Mutation Rates: Simulation Analysis

Rates of approach to equilibrium values of F(ST)/R(ST) at various mutation rates and using different mutation models (K-allele model KAM and stepwise model SMM) were analyzed numerically for the finite island model and the one-dimensional stepping stone models of migration, using simulation. In the island model of migration and the KAM mutation model, the rate of approach to the equilibrium F(ST) value was appreciably higher and the equilibrium value was almost twofold lower at micro (mutation rate) = m (migration rate) than at micro << m. In the one-dimensional stepping stone model of migration and the KAM model of mutation, the mutation rate significantly affected both the rate of approaching F(ST) equilibrium and the equilibrium value. In both island and one-dimensional stepping stone models and SMM, R(ST) was not influenced by various mutation rates. The rate of approach to the equilibrium values of both F(ST) and R(ST) was lower for the stepping stone model than to the island model. RST was rather resistant to deviations from the SMM mutation model.

The Influence of Spatial Inhomogeneities on Neutral Models of Geographical Variation IV. Discontinuities in the Population Density and Migration Rate

The equilibrium structure of the infinite, one-dimensional stepping-stone model with coincident discontinuities in the population density and migration rate is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into an infinite linear array of equally large, panmictic colonies that exchange gametes isotropically. The population density and the migration rate have a discontinuity at the origin, but are elsewhere uniform. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus without selection; every allele mutates to new alleles at the same rate. The three dimensionless parameters in the theory are α=(ρ+/ρ−)2 (V+/V−)3/2, and β±=4ρ±2uV± , where ρ+ (ρ−) and V+ (V−) designate the population density and variance of gametic dispersion per generation to the right (left) of the discontinuity, respectively, and u denotes the mutation rate. The characteristic length on the right (left) is V+/(2u) (V−/(2u)). The probability of identity is continuous at the origin, but its partial derivatives have a discontinuity unless migration is conservative (ρ−V−=ρ+V+). At least for nonconservative migration, the probability of identity (including the expected homozygosity) can be nonmonotonic even if the migration rate is uniform and the population density is monotonic. Thus, there can be a nonmonotonic genetic response in a neutral model to a monotonic environment.

REINFORCING SELECTION IS EFFECTIVE UNDER A RELATIVELY BROAD SET OF CONDITIONS IN A MOSAIC HYBRID ZONE.

Many hybrid zones have a mosaic structure, yet we know of no theoretical work that examines the impact of mosaicism on the outcome of evolution. We developed a computer simulation model designed to test whether the outcome of reinforcing selection differs in a mosaic and a clinal hybrid zone. Our model was a one-dimensional stepping-stone model. The mosaic and clinal hybrid zones that we modeled were, respectively, a mosaic maintained by differential fitness of the interacting taxa in patchy habitats and a tension zone. We modeled changes in gene frequency at two biallelic loci, A and B. Hybrids at the A locus were selected against. An allele at the B locus caused assortative mating at the A locus, which promoted reinforcement; there was a selective cost to this allele. In a mosaic hybrid zone, spatial variation in the fitness of A-locus homozygotes in different patches caused gene and genotype frequencies at the A and B loci to differ greatly from those in a tension zone. Compared to a tension zone, a mosaic hybrid zone had a broader region in which hybrids could be formed and, thus, a broader region in which the assortative-mating allele provided a net selective advantage (via decreased production of the less fit A-locus hybrids). This caused the assortative-mating allele to be favored under a broader set of conditions in a mosaic hybrid zone than in a tension zone. In mosaic and tension hybrid zones, both low and high levels of migration could prevent the establishment of the allele that promoted reinforcement, but the allele could establish under a wider range of migration rates in a mosaic than in a tension zone. In a tension zone, both low and high levels of selection against A-locus hybrids could prevent the establishment of the assortative-mating allele. In a mosaic hybrid zone, the assortative-mating allele established under lower levels of selection against hybrids than in a tension zone, and high levels of selection did not impede the establishment of this allele. Overall, our work illustrates how the structure of a hybrid zone can alter the outcome of an important evolutionary process, in this case, reinforcement.

Population Genetic Structure of the Northern Idaho Ground Squirrel (Spermophilus brunneus brunneus)

Spermophilus brunneus is one of the rarest species of North American ground squirrels. It occurs in just five counties in western Idaho. We sampled blood from 14–64 individuals in 11 populations of the northern subspecies S. b. brunneus . Analyses of allozymes at five polymorphic loci revealed significant population structure ( F st = 0.167). Four populations located within a 2-km radius in Bear Meadow (Adams Co.) were indistinguishable with respect to allelic frequencies (Nei's D = 0.002), but the remaining seven, more distant populations (also in Adams Co.) were differentiated from this group and from each other. F ST values ranged from 0.034 to 0.124 when the four populations in Bear Meadow were pooled and compared with each of the seven isolated populations. Regression of Slatkin's M against geographic distance among the 55 pairwise comparisons of populations suggested a strong effect of isolation-by-distance, consistent with a one-dimensional stepping-stone model of gene flow. Significant genetic structure in S. b. brunneus populations apparently is due to genetic drift in populations with small Ne (i.e., 20–50), reinforced by lack of gene flow following recent habitat fragmentation. Fragmentation results from the shrinking of meadows due to invasion by conifers, itself the consequence of fire suppression in the past 100 years.

Does the Mediterranean killifish Aphanius fasciatus (Teleostei: Cyprinodontidae) fit the one-dimensional stepping-stone model of gene flow?

Five samples (30 individuals each) of the Mediterranean cyprinodontid fish, Aphanius fasciatus, were analysed by allozyme electrophoresis for a study concerning genetic structure and gene flow among populations. Forty-three loci, twelve of which were polymorphic at P0.99 level, were analysed. A. fasciatus showed low levels of genetic polymorphism, with expected heterozygosity values ranging from 0.027 (SE = 0.013) to 0.064 (SE = 0.023). The modified Rogers' genetic distances ranged from 0.053 to 0.202. The observed pattern of genetic differentiation among populations, obtained by multidimensional scaling of the modified Rogers' genetic distances, was consistent with geographic distribution. Coancestry coefficient (θ = 0.302, SE = 0.045) indicated a very high degree of genetic subdivision among populations and the estimate of gene flow (the effective number of migrants per generation, Nm = 0.445), computed by Wright's method, showed very restricted gene flow among populations. The regression analysis carried out using log-transformed values of geographic distance against M^ (pairwise values of Nm) gave a regression coefficient, b = −1.03, indicating that a restricted migration rate among populations can occur according to the one-dimensional stepping-stone model. The amount and modality of gene flow in A. fasciatus inferred from the results of the present study are consistent with the absence of dispersal stages and with the fragmented nature of brackish-water habitats along the coasts.

Genetic diversity in partially selfing populations with the stepping-stone structure

A method to compute identity coefficients of two genes in the stepping-stone model with partial selfing is developed. The identity coefficients in partially selfing populations are computed from those in populations without selfing as functions of s (selfing rate), m (migration rate), N (subpopulation size), n (number of subpopulations) and u (mutation rate). For small m, 1/N and u, it is shown that approximate formulae for the identity coefficients of two genes from different individuals are the same as those in random mating populations if we replace N in the latter with N(1−s/2). Thus, the effects of selfing on genetic variability are summarized as reducing variation within subpopulations and increasing differentiation among subpopulations by reducing the subpopulation size. The extent of biparental inbreeding as measured by the genotypic correlation between truly outcrossed mates was computed in the one-dimensional stepping-stone model. The correlation was shown to be independent of the selfing rate and starts to fall off as the migration rate increases when mN is larger than 0.1.

Recent evolution of population structure in Australian barramundi, Lates calcarifer (Bloch): An example of isolation by distance in one dimension

New and previously published genetic data from 6000 barramundi comprising 50 collections across tropical Australia were analysed for evidence of population subdivision. Sixteen discrete populations were identified, including four populations that were identified from new collections. Duplicate collections from two localities were statistically homogeneous after seven years between collections. Environmental and genetic factors that yielded the observed genetic pattern were investigated. Geological evidence of sea-level changes, when compared with bathymetry data for the region, reveals that barramundi must have recently recolonized many of the coastal estuaries of tropical Australia. This recolonization resulted from the inundation of the Gulf of Carpentaria and Torres Strait by a rapid rise in sea level of at least 130 m between 18 000 and 6000 years ago. The genetic data clearly indicate that, as the population spread into new habitats, there was a corresponding decrease in genetic diversity. This observed decrease has been maintained despite continued migration between populations. The 'onedimensional stepping stone' migration model, which most closely fits the observed population structure, predicts that the observed level of population subdivision (FST = 0.064) is maintained against substantial gene flow between adjacent populations. This contrasts with the predictions of the often-used 'island model' which gives estimates of Nem at least two orders of magnitude lower than those from the onedimensional stepping-stone model.

The Influence of Spatial Inhomogeneities on Neutral Models of Geographical Variation III. Migration across a Geographical Barrier

The equilibrium structure of the infinite, one-dimensional stepping-stone model with a geographical barrier is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into an infinite linear array of equally large, panmictic colonies that exchange gametes symmetrically. Migration is reduced across the geographical barrier, but is otherwise uniform. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. The two dimensionless parameters in the theory are β = 4ρ0[formula] and κ where ρ0, u, V0, and κ represent the population density, mutation rate, variance of gametic dispersion per generation, and penetrability of the barrier, respectively. The characteristic length is [formula]. Relative to a homogeneous infinite habitat, the barrier raises the probability of identity if the two points of observation are on the same side and lowers it if they are on opposite sides. The former effect is moderate or small, but the latter is large unless transmission is high (κ ⪢ 1); genetic differentiation across the barrier is very strong for low transmission (κ ⪡ 1). For points of observation on the same side, the influence of the barrier is significant only if the proximal point is within a few characteristic lengths. Upper and lower bounds on the probability of identity are established, and approximations are derived for four cases: (i) low expected homozygosity (β ⪢1), (ii) high transmission, (iii) low transmission, and (iv) at least one point distant.

The coalescent and the genealogical process in geographically structured population

We shall extend Kingman's coalescent to the geographically structured population model with migration among colonies. It is described by a continuous-time Markov chain, which is proved to be a dual process of the diffusion process of stepping-stone model. We shall derive a system of equations for the spatial distribution of a common ancestor of sampled genes from colonies and the mean time to getting to one common ancestor. These equations are solved in three particular models; a two-population model, the island model and the one-dimensional stepping-stone model with symmetric nearest-neighbour migration.

The influence of spatial inhomogeneities on neutral models of geographical variation: II. The semi-infinite linear habitat

The equilibrium structure of the semi-infinite, one-dimensional stepping-stone model is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into a semi-infinite linear array of equally large, panmictic colonies that exchange gametes uniformly and symmetrically. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. Boundaries corresponding to an impenetrable geographical barrier and to contact with a region of extremely high population density or dispersal rate (a “density-mobility boundary”) are both treated. Relative to a homogeneous infinite habitat, a geographical barrier raises, and a density-mobility boundary lowers, the probability of identity. For fixed average position of the two points of observation, the probability of identity decreases with increasing separation. For fixed separation, the probability of identity decreases (increases) as the average position recedes from the barrier (density-mobility boundary). The sole dimensionless parameter in the theory is β = 4 ρo √2 uV o , where ρ o , u, and V o represent the population density, mutation rate, and variance of gametic dispersion per generation, respectively. The characteristic length is √ V o (2u) . Lower and upper bounds on the probability of identity are established; these yield a simple approximation for β ⪢ 1. The probability of identity is obtained as an integral; from this solution, approximations close to and far from the boundary are derived and the exact mean homozygosity at a geographical barrier is evaluated.

The influence of spatial inhomogeneities on neutral models of geographical variation: I. Formulation

The influence of spatial variation in the carrying capacity and migration rate of a geographical barrier on the one-dimensional stepping-stone model is studied. The monoecious, diploid population is subdivided into an infinite linear array of panmictic colonies that exchange gametes. In each deme, the rate of self-fertilization is equal to the reciprocal of the number of individuals in that deme. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. In the diffusion approximation, a partial differential equation that incorporates spatial (and temporal) variation in the carrying capacity and migration rate is derived for the probability of identity. Transition conditions that simultaneously take into account discontinuities in the carrying capacity and migration rate are established: the probability of identity is continuous, but its partial derivatives are not, their ratio being a simple function of the carrying capacities and migrational variance on the two sides of the inhomogeneity. The partial derivatives of the probability of identity are continuous across a geographical barrier, whereas the probability of identity itself has a discontinuity proportional to the partial derivative at the barrier, the constant of proportionality being a measure of the difficulty of crossing the barrier.

Population Structure of Sciaenops ocellatus and Cynoscion nebulosus (Pisces: Sciaenidae): Biochemical Variation, Genetic Subdivision and Dispersal

Genetic variation was examined in red drum (Sciaenops ocellatus) and spotted seatrout (Cynoscion nebulosus), sciaenid marine fishes that differ in life-history components such as age and size at maturity, longevity, and migratory behavior. Survey of 22 enzymes and seven structural proteins (40 loci) revealed that measures of genic variability and F-statistics were comparable to those for other sciaenids. Differentiation into subpopulations was only weakly evident for either red drum or spotted seatrout (F,, = 0.019 and 0.032). Spotted seatrout had lower heterozygosity (<1%), spatial clumping of rare alleles, and a regional cluster of localities in the eastern Gulf of Mexico. Red drum had higher heterozygosity (3%), a mosaic geographic pattern for protein variation and for a meristic polymorphism (number of tailspots), and one slightly divergent population on Florida's Atlantic Coast. As suggested from life-history components and supported by allelic dispersion, regional differentiation, and autocorrelation analysis, the less mobile and more rapidly-maturing species, spotted seatrout, displays isolation-by-distance. Less divergence of red drum in bays is contrary to expectation based on relative inshore abundance of the two species; adult reproduction by the larger and longer-lived species, red drum, produces an offshore effective population size which may equal or exceed colonies of spotted seatrout in bays. Annual exchange of spotted seatrout between bays, derived from a one-dimensional stepping stone model and estimated to be 5% of the breeding population, is compatible with previous movement data. Red drum may show Gulf-wide panmixia, but too little is known about the migratory adults to propose an appropriate population model.