Brian Kinlan UC Santa Barbara Integral-difference model simulations of marine population genetics

Population genetic structure -Analytical models date back to Fisher, Wright, Malecot 1930’s -1950’s -Neutral theory -Can give insight into population history and demography -Many simplifying assumptions -One of the most troublesome – Equilibrium -Simulations to understand real data?

Glossary Allele Locus Heterozygosity Polymorphism Deme Marker (e.g., Allozyme, Microsatellite, mtDNA) Hardy-Weinberg Equilibrium Genetic Drift

Measuring population structure -F statistics – standardized variance in allele frequencies among different population components (e.g., individual-to-subpopulation; subpopulation-to-total) F ST = 1 - (H S /H T )

Population structure vs. t=0; no structuret=500; structure

Inferring Migration from Genetic Structure: Island Model Fst = 1/(1+4Nm) Nm = ¼ (1-Fst)/Fst

Limitations I. Assumptions must be used to estimate Nm from Fst For strict Island Model these include: 1. An infinite number of populations 2. m is equal among all pairs of populations 3. There is no selection or mutation 4. There is an equilibrium between drift and migration “Fantasy Island?”

Lag Distance Standardized Variance Among Populations -Differentiation among populations increases with geographic distance (Wright 1943) -Dynamic equilibrium between drift and migration Inferring Migration from Genetic Structure: Isolation-by-Distance (IBD)

Palumbi Simulation Assumptions Palumbi, 2003, Ecol. App. 1. Kernel 3. Effective population size 2. Gene flow model N e = 1000 per deme Linear array of subpopulations Probability of dispersal Distance from source Laplacian

Calibrating the IBD Slope to Measure Dispersal Palumbi 2003 (Ecol. App.) -Simulations can predict the isolation-by-distance slope expected for a given average dispersal distance (Palumbi 2003 Ecol. Appl., Kinlan and Gaines 2003 Ecology)

Kinlan & Gaines (2003) Ecology 84(8): Genetic Estimates of Dispersal from IBD

Genetic Dispersion Scale (km) Modeled Dispersion Scale, D d (km) From Siegel, Kinlan, Gaylord & Gaines 2003 (MEPS 260:83-96)

But how well do these results hold up to the variability and complexity of the real-world marine environment?

Basic Integro-difference model of population dynamics A1MAAFKLdx x t1 x t x t xxxx       ()' ''' AAdultabundance[#/km] M Natural mortality FFecundity KDispersionkernel x t x' x [spawners/adult] [(settler/km)/total settled larvae]  LPost-settlementrecruitment x [adult/settler] (Ricker form L(x)  e -CA(x) )

Avg Dispersal = 10 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100 …Add genetic structure

Model Features -Coupled population dynamics & genetics -Temporal variation – mortality, fecundity, dispersal, settlement -Spatial heterogeneity – barriers, variable mortality, fecundity, dispersal, settlement -Timescales of adult & juv. movement & reproduction flexible (larval pool, discrete or overlap generations) -Initial distribution flexible; can study range expansions or stable pops, founder effects -Different genetic markers – effect of mutation rate, mutation model, number of loci, selection (future)

Avg Dispersal = 12 km Domain = 1000 km Spacing = 5 km 1000 generations Ne~100 Q1: How fast does IBD slope approach equilibrium?

Avg Dispersal = 10 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100 t=20 t=200 t=1000

Avg Dispersal = 12 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100 Palumbi model prediction Dd= 12.6 km t=20 t=200 t=1000

Avg Dispersal = 12 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100 Palumbi model prediction Dd= 38 km t=20 t=200 t=1000

Avg Dispersal = 2 km; Domain = 100 km; Spacing = 5 km; 800 generations; Ne=100 t=20 t=400 t=800

Avg Dispersal = 2 km; Domain = 100 km; Spacing = 5 km; 800 generations; Ne=100 t=20 t=400 t=800 Palumbi model prediction Dd= 1.6 km

Diverging Currents

Spatial Pattern of Abundance T=300 years; Mortality = 0.5 (mean lifespan = 2 years); Reproduction every year; 75 populations spaced 20 km apart over 1500 km of coast U= 10 cm/s; σ u =12 cm/s Dispersal: approximates an organism with 30 day PLD in a mean flow of 10 cm/s with a velocity variance of 12 cm/s (based on Siegel et al. 2003) Currents diverge at the midpoint (but there is some exchange across this point due to eddies and flow reversals represented by the velocity variance).

Pairwise Fst vs. Distance After T=10 (green), 50 (red), and 300 (black) years Mean ± 1 SE of Fst across all possible pairs at each distance lag T=300 T=50 T=10

Spatial Pattern of Allelic Richness After T=300 years 1 Microsatellite Locus (mutation rate = 1e-03; initial number of alleles = 10; symmetric stepwise mutation model)

Spatial Pattern of Genotype Presence/Absence After T=300 years 1 Microsatellite Locus (mutation rate = 1e-03; initial number of alleles = 10; symmetric stepwise mutation model)

Converging Currents

Strong Convergent Flow

LOCUS 1 (3 alleles)LOCUS 2 (2 alleles)

Unidirectional Currents

Current Mean drift = +15km Std = 5 km Strong Unidirectional Mean Flow 2 km spacing on 300 km domain t=20,60,100 Ne~1000

LOCUS 1 (2 alleles) LOCUS 2 (2 alleles)

Dispersal Barriers

X (km) Lag distance (km) Number of individuals Abundance vs. SpaceIBD

Dispersal Barriers Figure 1: Using a numerical gene-tracking integro-different model with a step-wise stochastic mutation rate at 3 loci, pairwise genetic distance (GST, (Nei 1973) patterns after stability for a 500-km coastline allowing for panmixia (A), and asymmetrical dispersal across a “border” placed in the center of the coastline (B). As expected (Rousset 1997b), pairwise genetic distance plateaus over large distances. The decline at greater distance lags is likely attributable to the asymmetrical barrier.

-Next steps Spiky kernels? Fishing effects? MPA’s?