Transcript Slide 1

Integral-difference model simulations of
marine population genetics
Brian Kinlan
UC Santa Barbara
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
t=500; structure
t=0; no structure
vs.
Population structure
Avg Dispersal = 10 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100
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)
http://genetics.nbii.gov/population.html
-For more 
http://dorakmt.tripod.com/genetics/popgen.html
Heterozygosity
-Hardy-Weinberg Equilibrium (well-mixed):
1 locus, 2 alleles, freq(1)=p, freq(2)=q
HWE => p2 + 2pq +q2
-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 statistics
F = fixation index and is a measure of how much the
observed heterozygosity deviates from HWE
F = (He - Ho)/He
HI = observed heterozygosity over ALL
subpopulations.
HI = (Hi)/k where Hi is the observed H of
the ith supopulation and k = number of
subpopulations sampled.
F statistics
HS = Average expected heterozygosity within
each subpopulation.
HS = (HIs)/k
Where HIs is the expected H within the ith
subpopulation and is equal to 1 - pi2 where
pi2 is the frequency of each allele.
F statistics
HT = Expected heterozygosity within the total
population.
HT = 1 - xi2
where xi2 is the frequency of each allele averaged
over ALL subpopulations.
FIT measures the overall deviations from HWE
taking into account factors acting within
subpopulations and population subdivision.
F statistics
FIT = (HT - HI)/HT 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 (Ho > He).
Large positive values indicate inbreeding or
genetic differentiation among subpopulations
(Ho < He).
FIS measures deviations from HWE within
subpopulations taking into account only those
factors acting within subpopulations
FIS = (HS - HI)/HS and ranges from -1 to +1
Positive FIS 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.
FST measures the degree of differentiation
among subpopulations -- possibly due to population
subdivision.
FST = (HT - HS)/HT and ranges from 0 to 1.
FST estimates this differentiation by comparing
He within subpopulations to He in the total
population.
FST will always be positive because He in
subpopulations can never be greater than He in
the total population.
SUMMARY- F statistics
FIS = 1 - (HI/HS)
FIT = 1 - (HI/HT)
FST = 1 - (HS/HT)
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
Isolation-by-Distance (IBD)
Standardized Variance
Among Populations
-Differentiation among populations increases with geographic
distance (Wright 1943)
Lag Distance
A dynamic equilibrium
between drift and migration
Isolation-by-Distance (IBD)
-Differentiation among populations increases with geographic
distance (Wright 1943)
Isolation by distance in a sedentary marine fish
0.08
Genetic differentiation (FST)
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
500
1000
1500
2000
2500
3000
-0.01
Geographic distance between populations (km)
Data from Rocha-Olivares and Vetter, 1999, Can. J. Fish. Aquat. Sci.
Calibrating the IBD Slope
to Measure Dispersal
-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 2003 (Ecol. App.)
Palumbi 2003 - Simulation Assumptions
Probability of dispersal
1. Kernel
Laplacian
Distance from source
2. Gene flow model
Linear array of subpopulations
3. Effective population size
Ne = 1000 per deme
Palumbi, 2003, Ecol. App.
Genetic Estimates of
Dispersal from IBD
Kinlan & Gaines (2003) Ecology 84(8):2007-2020
Genetic Dispersal Scale
(km)
r2 = 0.802, p<0.001
n=32
Planktonic Larval Duration (days)
Siegel et al. 2003 (MEPS 260:83-96)
Dispersal Scale vs. Developmental Mode
INVERTEBRATES
Genetic Disp. Scale vs. Larval Mode -- Inverts
Dispersal Scale (km)
140
n=29
n=6
P
Planktotrophic
L
Lecithotrophic
n=13
120
100
80
60
40
20
0
Developmental Mode
N
Non-planktonic
Modeled Dispersion Scale, Dd (km)
Genetic Dispersion Scale (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

A
t +1
x

= (1  M ) A tx + A tx ' Fx ' K x  x ' L x dx '

A
t
x
Adult abundance [#/km]
M
Natural mortality
Fx'
Fecundity [spawners / adult]
A tx
Adult abundance [#/km]
M
Hx
Natural mortality
Harvest mortality
Fx'
Fecundity [spawners / adult]
P
Larval mortality [larvae / spawner]
Lx
Post - settlement recruitment [adult / settler]
K x  x' Dispersion kernel [(settler / km) / total settled larvae]
K x  x' Dispersion kernel [(settler / km) / total settled larvae]
Lx
Post - settlement recruitment [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
Model Features
-Timescales
-Population dynamics
-Dispersal
-Initial distribution/genetic structure
-Spatial domain (barriers, etc)
-Temporal variation
-Different genetic markers
An example run
Avg Dispersal = 12 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100
t=1000
t=200
t=20
Avg Dispersal = 10 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 = 12 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100
t=800
t=400
t=20
Avg Dispersal = 2 km; Domain = 100 km; Spacing = 5 km; 800 generations; Ne=100
Palumbi model
prediction
t=800
Dd= 1.6 km
t=400
t=20
Avg Dispersal = 2 km; Domain = 100 km; Spacing = 5 km; 800 generations; Ne=100
1.8
total settlers= 13 total part = 100
-Next steps
Spiky kernels?
Fishing effects?
MPA’s?
U = 5 Ustd = 15 To = 14 Tf = 21
2
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-100
-50
0
50
100
150
alongcoast (km)
200
250
300
350