Brian KinlanUC 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)

FST = 1 - (HS/HT)

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

Sta

nd

ard

ized

Var

ian

ce

Am

on

g P

op

ula

tio

ns

-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 2003 - Simulation Assumptions

Palumbi, 2003, Ecol. App.

1. Kernel

3. Effective population size

2. Gene flow model

Ne = 1000 per deme

Linear array of subpopulations

Pro

bab

ility

of

dis

per

sal

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):2007-2020

Genetic Estimates of Dispersal from IBD

Genetic Dispersion Scale (km)

Mod

ele

d D

isp

ers

ion

Sca

le,

Dd

(km

)

From Siegel, Kinlan, Gaylord & Gaines 2003 (MEPS 260:83-96)

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

marine environment?marine environment?

Basic Integro-difference model of population dynamics

A Adult abundance [#/km]

M Natural mortality

H Harvest mortality

F Fecundity

P Larval mortality

L Post - settlement recruitment

K Dispersion kernel

xt

x

x'

x

x x'

[spawners / adult]

[larvae / spawner]

[adult / settler]

[(settler / km) / total settled larvae]

A 1 M A A F K L dxxt 1

xt

xt

x x x x

( ) '' ' '

A Adult abundance [#/km]

M Natural mortality

F Fecundity

K Dispersion kernel

xt

x'

x x'

[spawners / adult]

[(settler / km) / total settled larvae]

L Post - settlement recruitmentx [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=800Palumbi model prediction

Dd= 1.6 km

Diverging Currents

Spatial Pattern of AbundanceT=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; σu=12 cm/s

U= 10 cm/s; σu=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. DistanceAfter T=10 (green), 50 (red), and 300 (black) yearsMean ± 1 SE of Fst across all possible pairs at each distance lag

T=300

T=50

T=10

Spatial Pattern of Allelic RichnessAfter T=300 years1 Microsatellite Locus (mutation rate = 1e-03; initial number of alleles = 10; symmetric stepwise mutation model)

Spatial Pattern of Genotype Presence/AbsenceAfter T=300 years1 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

Dispersal Barriers

X (km) Lag distance (km)

Nu

mb

er o

f in

div

idu

als

Abundance vs. Space IBD

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.

-100 -50 0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

U = 5 Ustd = 15 To = 14 T

f = 21

tota

l set

tlers

= 1

3 t

otal

par

t =

100

alongcoast (km)

-Next stepsNext steps Spiky kernels?Spiky kernels?Fishing effects?Fishing effects?

MPA’s?MPA’s?