class: middle background-image: url("background.png") background-position: right background-size: auto # .orange[Population Genetics] ### .fancy[Neutral & Non-neutral Processes <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:brickred;overflow:visible;position:relative;"><path d="M.1 494.1c-1.1 9.5 6.3 17.8 15.9 17.8l32.3.1c8.1 0 14.9-5.9 16-13.9.7-4.9 1.8-11.1 3.4-18.1H380c1.6 6.9 2.9 13.2 3.5 18.1 1.1 8 7.9 14 16 13.9l32.3-.1c9.6 0 17.1-8.3 15.9-17.8-4.6-37.9-25.6-129-118.9-207.7-17.6 12.4-37.1 24.2-58.5 35.4 6.2 4.6 11.4 9.4 17 14.2H159.7c21.3-18.1 47-35.6 78.7-51.4C410.5 199.1 442.1 65.8 447.9 17.9 449 8.4 441.6.1 432 .1L399.6 0c-8.1 0-14.9 5.9-16 13.9-.7 4.9-1.8 11.1-3.4 18.1H67.8c-1.6-7-2.7-13.1-3.4-18.1-1.1-8-7.9-14-16-13.9L16.1.1C6.5.1-1 8.4.1 17.9 5.3 60.8 31.4 171.8 160 256 31.5 340.2 5.3 451.2.1 494.1zM224 219.6c-25.1-13.7-46.4-28.4-64.3-43.6h128.5c-17.8 15.2-39.1 30-64.2 43.6zM355.1 96c-5.8 10.4-12.8 21.1-21 32H114c-8.3-10.9-15.3-21.6-21-32h262.1zM92.9 416c5.8-10.4 12.8-21.1 21-32h219.4c8.3 10.9 15.4 21.6 21.2 32H92.9z"/></svg>] --- class: center, middle background-image: url("https://live.staticflickr.com/65535/51198451479_5ce952b659_c_d.jpg") background-position: center --- class: inverse, middle background-image: url("background.png") background-position: right background-size: auto # .blue[Foundations] ## .fancy[The Algebra of Alleles & Genotypes] --- # Allele Frequencies .pull-left[ Consider the alleles `\(A\)` and `\(B\)` which occur at: `\begin{equation} f(A) = p \end{equation}` and `\begin{equation} f(B) = 1 - p = q \end{equation}` which implies `\begin{equation} p + q = 1.0 \end{equation}` ] -- .pull-right[ Alleles coalesce into **genotypes** which occur at the following frequency expectations. **Homozygotes** `\begin{equation} f(AA) = p^2 \\ f(BB) = q^2 \end{equation}` **Heterozygotes** `\begin{equation} f(AB) + f(BA) = 2pq = 1- (p^2 + q^2) \end{equation}` ] --- # Mating Frequencies The frequency of genotypes through time, can be model using the following assumptions. .pull-left[ For simplicity, let's consider the genotypes at this locus is defined by the following .redinline[**genotype frequencies**]. `\(f(AA) = p^2 = P\)` `\(f(AB) = 2pq = Q\)` `\(f(BB) = q^2 = R\)` ] -- .pull-right[ Then, the expections for the next generation suggest that the genotypes coalesce in random and the probabilities of each mating pair could be parameterized as follows: | AA | AB | BB ---|----|----|---- AA | `\(P^2\)` | `\(PQ\)` | `\(PR\)` AB | `\(PQ\)` | `\(Q^2\)` | `\(QR\)` AB | `\(PR\)` | `\(QR\)` | `\(R^2\)` ] --- # Samples of Frequencies From samples of `\(N\)` individuals with observerations of `\(N_{AA}\)`, `\(N_{AB}\)`, and `\(N_{BB}\)` for each genoptype. .pull-left[ ### Genotype Frequencies `\(P = \frac{N_{AA}}{N}\)`, `\(Q = \frac{N_{AB}}{N}\)`, and `\(R = \frac{N_{BB}}{N}\)` ] -- .pull-right[ ### Allele Frequencies `\(f(A) = P + \frac{Q}{2} = p\)` and `\(f(B) = R + \frac{Q}{2} = q\)` ] --- # Independent Assortment of Alleles And when we have any of these mating events occur, we can predict the frequency of alleles in the **next generation**. .pull-left[ <center> <b>Homozygotes</b> </center> | A | A -------|----|---- **A** | AA | AA **A** | AA | AA | A | A -------|----|---- **B** | AB | AB **B** | AB | AB ] .pull-right[ <center> <b>Mixed</b> </center> | A | A -------|----|---- **A** | AA | AA **B** | AB | AB | A | B -------|----|---- **A** | A1 | AB **B** | AB | BB ] The old "Punnet Square" --- class: middle background-image: url("https://live.staticflickr.com/65535/51202021951_40cb6f88ae_c_d.jpg") background-position: center background-size: cover --- class: middle background-image: url("https://live.staticflickr.com/65535/51201303937_6a8c37be22_c_d.jpg") background-position: center background-size: cover --- # Mating Frequencies for `\(N_{t+1}\)` Given the genotype frequencies of `\(P\)`, `\(Q\)`, and `\(R\)`, random mating of individuals are expected to create the Parents | `\(\mathbf{P(Parents)}\)` | `\(\mathbf{P(AA)}\)` | `\(\mathbf{P(AB)}\)` | `\(\mathbf{P(BB)}\)` ------------|:----------:|:-------:|:-------:|:-------: `\(AA\;x\;AA\)` | `\(P^2\)` | `\(P^2\)` | | `\(AA\;x\;AB\)` | `\(PQ\)` | `\(PQ/2\)` | `\(PQ/2\)` | `\(AA\;x\;BB\)` | `\(PR\)` | | `\(PR\)` | `\(AB\;x\;AB\)` | `\(Q^2\)` | `\(Q^2/4\)` | `\(Q^2/2\)` | `\(Q^2/4\)` `\(AB\;x\;BB\)` | `\(QR\)` | | `\(QR/2\)` | `\(QR/2\)` `\(BB\;x\;BB\)` | `\(R^2\)` | | | `\(R^2\)` --- # Genotype Frequencies for `\(N_{t+1}\)` .pull-left[ ### Homozygotes $$ `\begin{aligned} P_{t+1} &= P_{t}^2 + \frac{P_tQ_t}{2} + \frac{Q^2_t}{4} \\\\ &= \left( P_t + \frac{Q_t}{2} \right)^2 \\\\ &= p^2 \end{aligned}` $$ ] -- .pull-right[ ### Heterozygotes $$ `\begin{aligned} Q_{t+1} &= \frac{P_tQ_t}{2} + P_tR_t + \frac{Q^2_t}{2} + \frac{Q_tR_t}{2}\\ &= 2\left(P_t + \frac{Q_t}{2} \right)\left(R_t + \frac{Q_t}{2}\right)\\ &= 2pq \end{aligned}` $$ ] -- #### Consequences So expectation .redinline[through time] is that with **random mating**, gentoype frequencies maintain `\(P = p^2\)`, `\(Q = 2pq\)`, and `\(R = q^2\)` --- # Hardy-Weinberg Equilibrium  --- class: inverse, middle background-image: url("background.png") background-position: right background-size: auto # .red[Popgen in R] ## .fancy[The `gstudio` library] --- # The `gstudio` Package For this course, we will be using the `gstudio` package. It is provided in a [Github Repository](https://github.com/dyerlab/gstudio) for simplicity. It can be installed as: ```r remotes::install_github("dyerlab/popgraph") remotes::install_github("dyerlab/gstudio") ``` I created this package because I found the current offerings to be at odds between how my data workflow is constructed and how R packages dealt with Genetic Data. --- # S2 vs S3 Configurations There are two common ways (and several additional ones) that you can program objects in `R` such as a genotype. Using the spatial packages `sp` and `sf` I'll show the differences. .pull-left[ ### S2 Configuration `sp` Objects such as the `SpatialPointsDataFrame` are special entities, which have the following components: - attributes: `crs`, `bbox`, `centroid` - data: `data.frame` *Consequences* The *object* is what we work with and stuff *inside of it* is where the data live making spatial data somehow different than other data we work with. ] -- .pull-right[ ### S3 Configuration `sf` Objects are fundamental data types: - Contained within native `data.frame`, `list` containers. *Consequences* The data integrate directly into `%>%` workflows just like all the rest of your data putting spatial components on an equal footing with all the other data we work with. ] --- # The `locus` Object ```r library( gstudio ) ``` A genotype is represented as a `locus` object in `R`, which is just like a `numeric`, `logical`, `character` or `factor` variable. .pull-left[ #### Missing Locus ```r loc_empty <- locus() loc_empty ``` ``` ## [1] NA ``` #### Haploid Locus ```r loc_hap <- locus("A") loc_hap ``` ``` ## [1] "A" ``` ] -- .pull-right[ #### Diploid Locus ```r loc_hom <- locus( c("A","A") ) loc_het <- locus( "A:B" ) c(loc_hom, loc_het) ``` ``` ## [1] "A:A" "A:B" ``` ```r class( loc_hom ) ``` ``` ## [1] "locus" ``` ] --- # The Beetle Data The Bark Beetle, *Araptus attenuatus*, is a Sonoran Desert endemic parasite that lives on within the plant *Euphorbia lomelii*. .pull-left[  ] .pull-right[ .center[] ] --- .pull-left[ # Host Plant Vicariance - Obligate - Desert Endemic - Geologic Sources of Vicariance - Alternative Scenarios - Coalescent Theory Tests [ref](https://rodneydyer.com/publication/garrick-et-al-2009-mol-ecol/) ] .pull-right[  ] --- .pull-left[ # Araptus attenuatus - mtDNA Sequence - Spatial Locations - Main clade diverged ~4.6MYA ] .pull-right[  ] --- # Genotypes These data were genotyped and are provided in the `gstudio` package as example data. As in most workflows, they are provided within a `data.frame` object along with any other data types. ```r data( arapat ) class( arapat ) ``` ``` ## [1] "data.frame" ``` ```r names( arapat ) ``` ``` ## [1] "Species" "Cluster" "Population" "ID" "Latitude" ## [6] "Longitude" "LTRS" "WNT" "EN" "EF" ## [11] "ZMP" "AML" "ATPS" "MP20" ``` ```r column_class( arapat, "locus") ``` ``` ## [1] "LTRS" "WNT" "EN" "EF" "ZMP" "AML" "ATPS" "MP20" ``` --- ```r summary( arapat ) ``` ``` ## Species Cluster Population ID Latitude ## Cape : 75 CBP-C :150 32 : 19 101_10A: 1 Min. :23.08 ## Mainland : 36 NBP-C : 84 75 : 11 101_1A : 1 1st Qu.:24.59 ## Peninsula:252 SBP-C : 18 Const : 11 101_2A : 1 Median :26.25 ## SCBP-A: 75 12 : 10 101_3A : 1 Mean :26.25 ## SON-B : 36 153 : 10 101_4A : 1 3rd Qu.:27.53 ## 157 : 10 101_5A : 1 Max. :29.33 ## (Other):292 (Other):357 ## Longitude LTRS WNT EN EF ## Min. :-114.3 01:01 :147 03:03 :108 01:01 :225 01:01 :219 ## 1st Qu.:-113.0 01:02 : 86 01:01 : 82 01:02 : 52 01:02 : 52 ## Median :-111.5 02:02 :130 01:03 : 77 02:02 : 38 02:02 : 90 ## Mean :-111.7 02:02 : 62 03:03 : 22 NA's : 2 ## 3rd Qu.:-110.5 03:04 : 8 01:03 : 7 ## Max. :-109.1 (Other): 15 (Other): 16 ## NA's : 11 NA's : 3 ## ZMP AML ATPS MP20 ## 01:01 : 46 08:08 : 51 05:05 :155 05:07 : 64 ## 01:02 : 51 07:07 : 42 03:03 : 69 07:07 : 53 ## 02:02 :233 07:08 : 42 09:09 : 66 18:18 : 52 ## NA's : 33 04:04 : 41 02:02 : 30 05:05 : 48 ## 07:09 : 22 07:09 : 14 05:06 : 22 ## (Other):142 08:08 : 9 (Other):119 ## NA's : 23 (Other): 20 NA's : 5 ``` --- # Genotype Frequencies Because the columns of data are recognized as `locus` object, we can get frequencies for: .pull-left[ #### Genotypes ```r genotype_frequencies( arapat$LTRS ) ``` ``` ## Genotype Observed Expected ## 1 01:01 147 99.44904 ## 2 01:02 86 181.10193 ## 3 02:02 130 82.44904 ``` ] .pull-right[ #### Alleles ```r frequencies( arapat$LTRS ) ``` ``` ## Allele Frequency ## 1 01 0.523416 ## 2 02 0.476584 ``` ] --- # Allele Frequencies We can also get frequencies for multiple loci. ```r freqs <- frequencies( arapat ) head( freqs ) ``` ``` ## Locus Allele Frequency ## 1 LTRS 01 0.52341598 ## 2 LTRS 02 0.47658402 ## 3 WNT 01 0.35795455 ## 4 WNT 02 0.18181818 ## 5 WNT 03 0.43039773 ## 6 WNT 04 0.02698864 ``` --- # Allele Frequencies Or for subdivisions of the population. ```r freqs <- frequencies( arapat, stratum = "Cluster" ) head( freqs ) ``` ``` ## Stratum Locus Allele Frequency ## 1 CBP-C LTRS 01 0.630000000 ## 2 CBP-C LTRS 02 0.370000000 ## 3 CBP-C WNT 01 0.287162162 ## 4 CBP-C WNT 03 0.641891892 ## 5 CBP-C WNT 04 0.064189189 ## 6 CBP-C WNT 05 0.006756757 ``` ```r freqs <- frequencies( arapat, stratum = "Population" ) head( freqs ) ``` ``` ## Stratum Locus Allele Frequency ## 1 101 LTRS 01 0.2777778 ## 2 101 LTRS 02 0.7222222 ## 3 101 WNT 01 1.0000000 ## 4 101 EN 01 0.6111111 ## 5 101 EN 03 0.3888889 ## 6 101 EF 01 0.7142857 ``` --- # Allele Frequencies Or whatever combination you are interested in getting ```r frequencies( arapat, stratum = "Cluster", loci = c("EN","LTRS") ) %>% head() ``` ``` ## Stratum Locus Allele Frequency ## 1 CBP-C EN 01 0.983108108 ## 2 CBP-C EN 03 0.003378378 ## 3 CBP-C EN 04 0.013513514 ## 4 CBP-C LTRS 01 0.630000000 ## 5 CBP-C LTRS 02 0.370000000 ## 6 NBP-C EN 01 0.755952381 ``` --- # Visualizing .pull-left[ We can get allele frequencies from the data by using the function `frequencies` and it will return a `data.frame` object that we can pipe into subsequent analyses. ```r arapat %>% frequencies( loci = "LTRS") %>% ggplot( aes(Allele,Frequency) ) + geom_bar( stat="identity") ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-16-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Built-in `geom_`'s .pull-left[ OR, you can use some of the `geom_` functions provided in `gstudio` which can take care of the mapping for you (e.g., no need to do `aes()`) such as: ```r ggplot() + geom_frequencies( freqs ) ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-18-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Allele Frequencies .pull-left[ ```r freqs <- frequencies( arapat, loci="EN", stratum="Cluster") ggplot() + geom_frequencies( freqs ) + facet_grid(Stratum~.) ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-20-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Allele Frequencies .pull-left[ Or compare across substrata for an individual locus ```r ggplot() + geom_locus( aes( x=LTRS, fill=Cluster), data=arapat ) ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-22-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Loading In Data Data should be easy to get *into* `R` and `gstudio` supports the following text formats for loading data in.  --- # Gentoypes Separated by Symbols .pull-left[ #### Separated Genotypes <pre> Population,Latitude,Longitude,Loc1,Loc2 Olympia,47.15,-122.89,12:14,15:15 Bellingham,48.75,-122.49,12:12,15:17 St. Louis,38.81,-89.98,12:10,NA Ames,42.26,-93.47,10:14,NA:NA Richmond,37.74,-77.16,10:10,17:17 </pre> ] .pull-right[ #### The `R` Code ```r path <- system.file("extdata","data_separated.csv", package="gstudio") read_population( path, type = "separated", locus.columns = 4:5) ``` ``` ## Population Latitude Longitude Loc1 Loc2 ## 1 Olympia 47.15 -122.89 12:14 15:15 ## 2 Bellingham 48.75 -122.49 12:12 15:17 ## 3 St. Louis 38.81 -89.98 10:12 ## 4 Ames 42.26 -93.47 10:14 ## 5 Richmond 37.74 -77.16 10:10 17:17 ``` ] --- # Genotypes Contained in Separate Columns .pull-left[ #### Alleles In Columns <pre> Population,Latitude,Longitude,Loc1,Loc1,Loc2,Loc2 Olympia,47.15,-122.89,14,12,15,15 Bellingham,48.75,-122.49,12,12,17,15 St. Louis,38.81,-89.98,12,10,NA,NA Ames,42.26,-93.47,10,14,NA,NA Richmond,37.74,-77.16,10,10,17,17 </pre> ] -- .pull-right[ #### The `R` Code ```r path <- system.file("extdata","data_2_column.csv", package="gstudio") read_population( path, type = "column", locus.columns = 4:7) ``` ``` ## Population Latitude Longitude Loc1 Loc2 ## 1 Olympia 47.15 -122.89 12:14 15:15 ## 2 Bellingham 48.75 -122.49 12:12 15:17 ## 3 St. Louis 38.81 -89.98 10:12 ## 4 Ames 42.26 -93.47 10:14 ## 5 Richmond 37.74 -77.16 10:10 17:17 ``` ] --- # SNP Genotypes As Minor Allele .pull-left[ #### Alleles In Columns <pre> Population,Latitude,Longitude,snp1,snp2,snp3,snp4 Olympia,47.15,-122.89,1,2,1,1 Bellingham,48.75,-122.49,1,0,1,2 St. Louis,38.81,-89.98,1,0,NA,NA Ames,42.26,-93.47,0,1,NA,NA Richmond,37.74,-77.16,0,2,1,2 </pre> ] -- .pull-right[ #### The `R` Code ```r path <- system.file("extdata","data_snp.csv", package="gstudio") read_population( path, type = "snp", locus.columns = 4:7) ``` ``` ## Population Latitude Longitude snp1 snp2 snp3 snp4 ## 1 Olympia 47.15 -122.89 A:B B:B A:B A:B ## 2 Bellingham 48.75 -122.49 A:B A:A A:B B:B ## 3 St. Louis 38.81 -89.98 A:B A:A ## 4 Ames 42.26 -93.47 A:A A:B ## 5 Richmond 37.74 -77.16 A:A B:B A:B B:B ``` ] --- class: inverse, middle background-image: url("background.png") background-position: right background-size: auto # .green[Evoluationary Forces] ## .fancy[Deviants from Hardy-Weinberg] **OR** what we are really trying to measure... --- # Genetic Drift .pull-left[ Genetic drift is the *stochastic change in allele frequencies through time* due to the population size. When `\(N = \infty\)` then `\(P_{t+1} = P_{t}\)`, etc. However, when `\(N << \infty\)`, allele frequencies are expected to change. ```r Alleles <- c("A","B") sample_sizes <- c(5,10,20,50,100) data.frame( N = rep( sample_sizes, each=50), FreqA = NA) -> df for( i in 1:nrow(df)){ a <- sample( Alleles, size=df$N[i], replace=TRUE) df$FreqA[i] <- sum( a == "A") / length(a) } ``` ] -- .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-27-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Genetic Drift - The Binomial .pull-left[ The amount of drift is inversely proportional to the population size `\(N\)`. More specifically, we can treat a simple locus like this one as a *binomial random variable* $$ P(N_A|N,p_A) = \frac{2N!}{N_A!(2N-N_A)!}p_A^{N_A}(1-p_A)^{2N-N_A} $$ For simplicity, let's assume that a population of `\(N=20\)` individuals is randomly mating and has a locus where `\(p=q\)`. At the next generation the probability distribution for the number of `\(A\)` alleles is given by the expansion. This means that it is possible that even if everyone starts as a heterozygote at this generation, you may have a non-zero probability of .redinline[losing one of the alleles due to drift!] ] -- .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-28-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Skewness in Allele Frequencies This probability increases the further allele frequencies diverge from `\(p=q\)`. Here is an example where `\(p=0.1\)` <img src="slides_files/figure-html/unnamed-chunk-29-1.png" width="504" style="display: block; margin: auto;" /> --- class: center, middle background-image: url("https://live.staticflickr.com/65535/51198451189_1c19fd70d7_c_d.jpg") background-position: center --- # Drifting Away - Simulations .pull-left[ We can simulate mating in `R` for ```r freqs <- frequencies( arapat, loci="LTRS") freqs ``` ``` ## Locus Allele Frequency ## 1 LTRS 01 0.523416 ## 2 LTRS 02 0.476584 ``` ] .pull-right[ ```r pop <- make_population(freqs) summary( pop ) ``` ``` ## ID LTRS ## Min. : 1.00 01:01 : 5 ## 1st Qu.: 5.75 01:02 :10 ## Median :10.50 02:02 : 5 ## Mean :10.50 ## 3rd Qu.:15.25 ## Max. :20.00 ``` ] --- # Drifting Away - Simulations .pull-left[ Set up `data.frame` to receive frequencies ```r Lineage <- factor( rep( 1:10, each = 100 ) ) Generation <- rep( 1:100, times=10) df.drift <- data.frame( Lineage, Generation, p = NA) ``` Small function to estimate `\(f(A)\)`. ```r fA <- function( p ) { frequencies( p ) %>% filter( Locus == "LTRS", Allele == "01" ) %>% select( Frequency ) %>% as.numeric() } ``` ] -- .pull-right[ Simulate random mating across generations where we are creating a new generation equal in size to the previous one. ```r for( rep in levels( df.drift$Lineage) ) { p <- pop for( gen in 1:100 ) { f <- fA(p) df.drift[ df.drift$Generation == gen & df.drift$Lineage == rep, "p"] <- f p <- mate(p) } } ``` ] --- # Genetic Drift & Fixation .pull-left[ Even though all the replicate simulations started at the same location (e.g., `LTRS` allele frequencies), their eventual fate is stochastic. ```r ggplot( df.drift, aes(Generation,p,color=Lineage) ) + geom_line() ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-36-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Expectations of Drift As a result, there is an expectation on the number of random generations necessary for a population to go to .redinline[fixation] based upon the population size and allele frequencies. This can be approximated by: `\begin{equation} t_{fix} = \frac{-4N_e(1-p)\log{(1-p})}{p} \end{equation}` Where `\(N_e\)` is the *genetic effective number* of individuals. --- # Mutation .pull-left[ It it weren't for *mutation*, you would all look like this guy!  ] -- .pull-right[ Mutation changes allele frequencies very slowly relative to time-frames that most landscape genetics studies are conducted. Assuming the rate of mutation from `\(A \to B\)` is `\(\mu\)` and from `\(B \to A\)` is `\(\nu\)`, we expect to see a *per genertation* change equal to: $$ p_{t+1} = p_t(1-\mu) + (1-p_t)\nu $$ with an expected equilibrium at $$ \hat{p} = \frac{\nu}{\nu + \mu} $$ See also: - Stepwise mutation model (microsatellites) - Infinite allele model (Kimura & Crow, 1964) ] --- # So why do we care? In population genetics, the only things we have are *allele frequencies* and *genotype frequencies*. What we **must** do is deploy .redinline[sampling strategies] (e.g., experimental design) to answer specific questions that can be answered by looking at variation in *allele* and *genotype* frequencies. --- # Changes in Heterozygosity Microevolutionary forces that can impact heterozygosity. Decreased Heterozygosity | Increased Heterozygosity ---------------------------------|------------------------------- Inbreeding | Coding Errors Null Alleles | Gametic Gene Flow Positive Assortative Mating | Negative Assortative Mating Selection Against Heterozygotyes | Heterosis Wahlund Effects | Outbreeding Zygotic Gene Flow | --- class: middle background-image: url("https://live.staticflickr.com/65535/50367566131_85c1285e2f_o_d.png") background-position: right background-size: auto .pull-left[  ] .pull-right[ # 🙋🏻 Questions? If you have any questions for about<br/> the content presented herein<br/> now is your time. If you think of something later though, <br/>feel free to [ask me via email](mailto://rjdyer@vcu.edu) and I'll<br/> get back to you as soon as possible. ]