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

Many possible inferences -Effective population size -Inbreeding/selfing -Mating success -Bottlenecks -Time of isolation -Migration/dispersal

Many possible inferences -Effective population size -Inbreeding/selfing -Mating success -Bottlenecks -Time of isolation -Migration/dispersal

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

Avg Dispersal = 10 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100 Population structure

Measuring population structure -F statistics – standardized variance in allele frequencies among different population components (e.g., individual-to-subpopulation; subpopulation-to-total) -Other measures (assignment tests, AMOVA, Hierarchical F, IBD, Genetic Distances, Moran’s I, etc etc etc) -For more 

Heterozygosity -Hardy-Weinberg Equilibrium (well-mixed): 1 locus, 2 alleles, freq(1)=p, freq(2)=q HWE => p 2 + 2pq +q 2 -Deviations from HWE Deviations of observed frequency of heterozygotes (Hobs) from those expected under HWE (Hexp) can occur due to non-random mating and sub-population structure

F F = fixation index and is a measure of how much the observed heterozygosity deviates from HWE F = (He - Ho)/He HI HI = observed heterozygosity over ALL subpopulations. HI HI = (  Hi)/k where Hi is the observed H of the ith supopulation and k = number of subpopulations sampled. F statistics

H S H S = Average expected heterozygosity within each subpopulation. H S = (  H Is )/k Where H Is is the expected H within the ith subpopulation and is equal to 1 -  p i 2 where p i 2 is the frequency of each allele. F statistics

H T H T = Expected heterozygosity within the total population. H T = 1 -  x i 2 where x i 2 is the frequency of each allele averaged over ALL subpopulations. F IT F IT measures the overall deviations from HWE taking into account factors acting within subpopulations and population subdivision. F statistics

F IT F IT = (H T - H I )/H T and ranges from - 1 to +1 because factors acting within subpopulations can either increase or decrease Ho relative to HWE. Large negative values indicate overdominance selection or outbreeding (H o > H e ). Large positive values indicate inbreeding or genetic differentiation among subpopulations (H o < H e ). F statistics

F IS F IS measures deviations from HWE within subpopulations taking into account only those factors acting within subpopulations F IS F IS = (H S - H I )/H S and ranges from -1 to +1 Positive F IS values indicate inbreeding or mating occurring among closely related individuals more often than expected under random mating. Individuals will possess a large proportion of the same alleles due to common ancestry.

F ST F ST measures the degree of differentiation among subpopulations -- possibly due to population subdivision. F ST = (HT - H S )/H T and ranges from 0 to 1. F ST estimates this differentiation by comparing He within subpopulations to He in the total population. F ST will always be positive because He in subpopulations can never be greater than He in the total population.

F IS = 1 - (H I /H S ) F IT = 1 - (H I /H T ) F ST = 1 - (H S /H T ) SUMMARY- F statistics

Fst and Migration (Wright’s 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?” Other models include 1D and 2D “stepping stones”, but these too have limitations, such as a highly restrictive definition of dispersal and assumption of an infinite number of demes or a circular/toroidal arrangement.

Limitations II. Many factors besides migration can affect Fst at any given point in space and time -Bottlenecks -Inbreeding/asexual reproduction -Non-equilibrium -Patchiness/geometry of gene flow -Definition of subpopulations -Dispersal barriers -Cryptic speciation

Lag Distance Standardized Variance Among Populations -Differentiation among populations increases with geographic distance (Wright 1943) Isolation-by-Distance (IBD)

A dynamic equilibrium between drift and migration

-Differentiation among populations increases with geographic distance (Wright 1943) Isolation-by-Distance (IBD) Data from Rocha-Olivares and Vetter, 1999, Can. J. Fish. Aquat. Sci. Isolation by distance in a sedentary marine fish

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)

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

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

Siegel et al (MEPS 260:83-96) r 2 = 0.802, p<0.001 n=32 Planktonic Larval Duration (days) Genetic Dispersal Scale (km)

Dispersal Scale vs. Developmental Mode INVERTEBRATES Planktotrophic Lecithotrophic Non-planktonic n=29 n=6 n=13

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?

Goal: a more realistic and flexible population genetic model -Explicit modeling of population dynamics & dispersal

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) )

Genotypic structure (tracked somewhat analagously to age structure)

Initial Questions -What does the approach to equilibrium look like? What is effect of non-equilibrium on dispersal estimates? -Effects of range edges/range size -Effects of temporal and spatial variation in demography (disturbance; spatial heterogeneity) -Effects of flow -How does the IBD signal “average” dispersal when the scale/pattern of dispersal is variable across the range?

Model Features

-Timescales -Population dynamics -Dispersal -Initial distribution/genetic structure -Spatial domain (barriers, etc) -Temporal variation -Different genetic markers

Avg Dispersal = 12 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100 An example run

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

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

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

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