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Brian Kinlan UC Santa Barbara

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

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Brian Kinlan UC Santa Barbara

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

  2. 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?

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

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

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

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

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

  8. 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  http://genetics.nbii.gov/population.html http://dorakmt.tripod.com/genetics/popgen.html

  9. 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

  10. 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.

  11. 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.

  12. 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.

  13. 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).

  14. 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.

  15. FSTmeasures 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.

  16. SUMMARY-F statistics FIS = 1 - (HI/HS) FIT = 1 - (HI/HT) FST = 1 - (HS/HT)

  17. Fst and Migration(Wright’s Island Model) Fst = 1/(1+4Nm) Nm = ¼ (1-Fst)/Fst

  18. 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.

  19. 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

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

  21. A dynamic equilibrium between drift and migration

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

  23. 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.)

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

  25. Genetic Estimates of Dispersal from IBD Kinlan & Gaines (2003) Ecology 84(8):2007-2020

  26. r2 = 0.802, p<0.001 n=32 Genetic Dispersal Scale (km) Planktonic Larval Duration (days) Siegel et al. 2003 (MEPS 260:83-96)

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

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

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

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

  31. Integro-difference model of population dynamics ¥  + = - + t 1 t t A ( 1 M ) A A F K L dx ' - ' ' ' x x x x x x x -¥ t A Adult abundance [#/km] x M Nat ural morta lity F Fecundity [spawners / adult] x' K Dispersion kernel [(settler / km) / total sett led larvae ] - x x' L Post - settlement recruitmen t [adult / settler] x (Ricker form L(x)  e-CA(x))

  32. Genotypic structure (tracked somewhat analagously to age structure)

  33. 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?

  34. Model Features

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

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

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

  38. Palumbi model prediction Dd= 12.6 km Avg Dispersal = 12 km; Domain = 1000 km; Spacing = 5 km; 1000 generations; Ne=100

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

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

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

  42. Next stepsSpiky kernels?Fishing effects?MPA’s?

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