class: center, middle, inverse, title-slide # Empirical results and interpretation ### Jinliang Yang ### Nov. 9th, 2018 --- # Theoretical maximum limit ### Without considering mutation `\begin{align*} & R_{max} = 2N_eih^2\sigma_P \\ \end{align*}` Set an upper limit to what can be expected. -- ---- Dr. __W. G. Hill__ in 1982, developed theory to include mutations as a wellspring of continuing new variation upon which selection could produce response. -- ### Considering mutation `\begin{align*} & R_{max} = \frac{2N_eih^2\sigma_M^2}{\sigma_P} \\ \end{align*}` Where `\(\sigma_M^2\)` is the __additive genetic variance__ arising from new mutations each generation. --- # Considering mutation `\begin{align*} & R_{max} = \frac{2N_eih^2\sigma_M^2}{\sigma_P} \\ \end{align*}` Where `\(\sigma_M^2\)` is the __additive genetic variance__ arising from new mutations each generation. -- ### Important notes: 1. It takes at least 20 generations for mutations to be __high enough in frequency__. 2. Additive variance is a function of allele frequency. -- ### Take home message: Increasing the mutation rate through __mutagenesis__ is not expected to enhance response in the short term. --- # Number of effective factors `\begin{align*} & \frac{R_T^2}{\sigma_A^2} = 8n \\ \end{align*}` -- `\begin{align*} & n = \frac{R_T^2}{8\sigma_A^2} \\ \end{align*}` Where `\(n\)` is the number of loci with many assumptions 1) unrelated loci, 2) same effect size, ... -- If takes LD into consideration, the above equation actually defines __the number of effective factors__. -- The numer of loci affecting a trait is larger than the number of effective factor, resulting from - __LD between loci__ - large __differences in effect size__ between loci --- # Evolvability ### General definition: The ability for a species or breeding population - to adapt to its environment through natural selection - be improved through artificial selection is known as __evolvability__. -- ### Other possible definition: - The ability of a population to respond to selection - The ability of a genomic architecture to facilitate change - The ability of a genetic system to produce and maintain potentially adaptive genetic variants --- # E. Coli Long-term selection experiment (LTEE) .pull-left[ - Richard Lenski and co-workers, at Michigan State University since 1988 - Selection for fitness under glucose-limited conditions for __> 50,000 generations__ with __12 independent populations__ - Transferring 0.1 ml of culture into 9.9 ml of fresh medium each day. (about 6.6 life cycles) ] .pull-right[ <div align="center"> <img src="ecoli.png" height=350> </div> Rich makes the 10,000th transfer. ] --- # E. Coli Long-term selection experiment __Response__: - They store a sample every 75 days, or about 500 generations. - The relative fitness can be compared to ancestral populations by competing two populations one another and counting the number of cells from each population. -- <div align="center"> <img src="response.png" height=250> </div> The figure is from Dawkins 2009. - Left: the response for 1/12 populations. - Right: responses of all 12 populations. --- # Mutations <div align="center"> <img src="mut.png" height=250> </div> The figure is from Tenaillon 2016. Mutations accumulated over time. - Left: total mutations over time in the 12 LTEE. - Right: total mutations rescaled to reveal the trajectories for the six populations that did not become __hypermutable__ for point mutations, and for the other six before they evolved hypermutability. --- # Mutations <div align="center"> <img src="mutmap.png" height=350> </div> The figure is from Barrick 2009. Mutations accumulated over time. - Inside ring represents genome of clone from 2000 generations. - Outside ring represents clone from 20K generations. - In between, intermediate clones. ??? Note that, mutations in the form of SNPs, deletions, insertions, and inversions have accumulated across the generations. --- # Large effect mutation <div align="center"> <img src="largeeffect.png" height=150> </div> The figure is from Dawkins 2009. -- - Increase in bacterial population density after 33,000 generations of one of the twelve populations. - This population accumulated the mutations necessary to metabolize citrate, greatly increasing the food source availability. - and thus greatly increase in the bacterial population before resources were exhausted each day. --- # Mutations affecting mutation rate Recently, Wielgoss et al., 2013 reported an interesting interaction between mutations affecting mutation rate, which involves mutations falling in genes controlling the __cellular repair machinery__. <div align="center"> <img src="mm.png" height=350> </div> The figure is from Wielgoss 2013. --- # Mutations affecting mutation rate <div align="center"> <img src="mm.png" height=150> </div> The figure is from Wielgoss 2013. -- - A mutation in __mutT__ between 20,000 and 30,000 generations dramatically increased the mutation rate. - Then, two mutations in __mutY__ (__mutY-E__ and __mutY-L__) between generations 35,000 and 40,000, which decreased the mutation rate -- The interesting part is that because __mutY__ is involved in DNA repair, mutations in __mutY__ are expected to be hypermutators themselves. - But in the background of the __mutT__, it actually decrease the mutation rate, because these mutations coincidentally repair the mutations made by __mutT__. --- # Review P = G + E G = A + D ### Breeding value and dominance deviation | Genotype | Genotypic Value | Value as deviated from mean | Breeding Value | Dominance Deviation | | :-------: | :-------: | :-----------: | :-----------: | :-------: | :-------: | | `\(A_1A_1\)` | `\(a\)` | `\(2q(\alpha - qd)\)` | `\(2q\alpha\)` | `\(-2q^2d\)` | | `\(A_1A_2\)` | `\(d\)` | `\((q-p)\alpha + 2pqd\)` | `\((q-p)\alpha\)` | `\(2pqd\)` | | `\(A_2A_2\)` | `\(-a\)` | `\(-2p(\alpha + pd)\)` | `\(-2p\alpha\)` | `\(-2p^2d\)` | --- # Review ### Genetic variance partitioning `\begin{align*} & \sigma_{G}^2 = \sigma_A^2 + \sigma_D^2 \end{align*}` -- The additive genetic variance __in a HWE population__ is: `\begin{align*} \sigma_A^2 & = 2pq\alpha^2 \\ & = 2pq(a + d(q-p))^2 \\ \sigma_D^2 & = (2pqd)^2 \\ \end{align*}` --- # Repeatability One way to increase the accuracy of measurements is to use the average of multiple measurements instead of just one. `\begin{align*} & r_n = \frac{\sigma_G^2 + \sigma_{Eg}^2}{\sigma_{P(n)}^2} \\ & \sigma_{P(n)}^2 = \sigma_G^2 + \sigma_{Eg}^2 + \frac{1}{n}\sigma_{Es}^2 \\ \end{align*}` --- # Intraclass correlation - The proportion of between-family variance to the total variance - __t = Var(B)/Var(Total)__ --- # Genetic covariances for general relatives `\begin{align*} Cov_G(X, Y) = 2f_{XY}\sigma_A^2 + \Delta_{XY}\sigma_D^2 \\ \end{align*}` -- | Relationship | | Coancestry | r (of `\(\sigma^2_A\)`) | u (of `\(\sigma^2_D\)`) | | :-------: | :-------: | :-----------: | :-----------: | :-------: | :-------: | | First degree | Parent:offspring | 1/4 | 1/2 | 0 | | Second degree | Half sibs | 1/8 | 1/4 | 0 | | | Full sibs | 1/4 | 1/2 | __1/4__ | | | Grantparent:offspring | 1/8 | 1/4 | 0 | | Third degree | great-grantparent:offspring | 1/16 | 1/8 | 0 | --- # Interpretation of variance components for sib design | Observational | | Covariance and causal components estimated | | :------------: | :-------: | | | Sires | `\(\sigma_s^2 = Cov(HS)\)` | `\(=\frac{1}{4}\sigma_A^2\)` | | Dams | `\(\sigma_d^2 = Cov(FS) - Cov(HS)\)` | `\(=\frac{1}{4}\sigma_A^2 + \frac{1}{4}\sigma_D^2\)` | | Progeny | `\(\sigma_w^2 = V_P - Cov(FS)\)` | `\(= \frac{1}{2}\sigma_A^2 +\frac{3}{4}\sigma_D^2\)` | | Total | `\(\sigma_T^2 = V_P = \sigma_s^2 + \sigma_d^2 + \sigma_w^2\)` | `\(=\sigma_A^2 + \sigma_D^2 + \sigma_E^2\)` | | Sires + Dams | `\(\sigma_s^2 + \sigma_d^2 = Cov(FS)\)` | `\(=\frac{1}{2}\sigma_A^2 + \frac{1}{4}\sigma_D^2\)` | --- # Ways to calculate heritability ### Regression - `\(h^2 = b_{AP}\)` is the __regression coefficient__ of breeding value on phenotypic value: `\(A = h^2P\)`. ### Correlation Coefficient - `\(r_{AP} = h\)`