diff --git a/Chapters/Genetic_drift_selection.tex b/Chapters/Genetic_drift_selection.tex index a459651..d5c69da 100644 --- a/Chapters/Genetic_drift_selection.tex +++ b/Chapters/Genetic_drift_selection.tex @@ -112,7 +112,7 @@ \section{Stochastic loss of strongly selected alleles} \end{equation} Solving this we find that \begin{equation} -p_F = 2s. +p_F = 2s. \label{eqn:prob_fix_strong} \end{equation} Thus even an allele with a $1\%$ selection coefficient has a $98\%$ probability of being lost when it is first introduced into the diff --git a/Chapters/Interaction_selection_mut_mig.tex b/Chapters/Interaction_selection_mut_mig.tex index 92e0e6a..4a8c192 100644 --- a/Chapters/Interaction_selection_mut_mig.tex +++ b/Chapters/Interaction_selection_mut_mig.tex @@ -98,7 +98,7 @@ \subsection{Mutation--Selection Balance} so eqn.\ \eqref{eqn:mut_sel_bal} is not valid. However, we can make an argument similar to the one above to show that, for truly recessive alleles, \begin{equation} - q_e = \sqrt{\frac{\mu}{s}}. + q_e = \sqrt{\frac{\mu}{s}}. \eqref{eqn:recess_mut_sel_bal} \end{equation} \graham{Add figure illustrating the freq as a function of h and s} \begin{marginfigure} @@ -370,7 +370,7 @@ \subsection{Migration--selection balance} selection model, resulting in a diploid migration--selection balance equilibrium frequency of \begin{equation} -q_{e,1} = \frac{m}{hs} +q_{e,1} = \frac{m}{hs} \label{eqn:mig_sel_eq} \end{equation} \begin{figure} @@ -477,7 +477,7 @@ \subsection{Migration--selection balance} tangent) of $q(x)$ at $x=0$. See Figure \ref{fig:cline_tangent}. Under this definition, the cline width is approximately \begin{equation} - 0.6 \sigma/\sqrt{s} ~\textrm{miles}, + 0.6 \sigma/\sqrt{s} ~\textrm{miles}, \label{eqn:cline_width} \end{equation} note that the units are miles here just because we defined the average dispersal distance ($\sigma$) in miles above. Thus the cline will be wider if individuals dispersal further, higher diff --git a/Chapters/Multi_trait_selection.tex b/Chapters/Multi_trait_selection.tex index eb09392..bdefe43 100644 --- a/Chapters/Multi_trait_selection.tex +++ b/Chapters/Multi_trait_selection.tex @@ -426,7 +426,7 @@ \subsection{Hamilton's Rule and the evolution of altruistic and % http://darwin-online.org.uk/content/frameset?pageseq=23&itemID=F8.2&viewtype=side -%% +%% Beaver https://twitter.com/EponymousBreeze/status/1226934486090145799/photo/1 \paragraph{Other forms of alturism} Kin-selection can favour altruism because individuals carrying altruistic alleles interact with other \emph{ related} individuals who tend to display altruistic phenotypes and so gain an advantage. However, there are other ways that altruistic behaviours can spread than just through the interactions with kin. diff --git a/Chapters/One_locus_selection.tex b/Chapters/One_locus_selection.tex index 0f13c88..8dbbe5d 100644 --- a/Chapters/One_locus_selection.tex +++ b/Chapters/One_locus_selection.tex @@ -111,11 +111,11 @@ \subsection{Haploid selection model} In practice, it is often helpful to parametrize the relative fitnesses $w_i$ in a specific way. For example, we may set $w_1 = 1$ and $w_2 = 1 - s$, where $s$ is called the selection coefficient. Using this parametrization, $s$ is simply the difference in relative fitnesses between the two alleles. Equation \eqref{eq:haploid_tau_gen} becomes \begin{equation} \label{eq:haploid_tau_gen_expl} - p_{t+\tau} = \frac{p_{t}}{p_{t} + q_{t} (1 - s)^{\tau}}, + p_{\tau} = \frac{p_{0}}{p_0 + q_0 (1 - s)^{\tau}}, \end{equation} as $w_2 / w_1 = 1 - s$. Then, if $s \ll 1$, we can approximate $(1-s)^{\tau}$ in the denominator by $\exp(-s\tau)$ to obtain \begin{equation} \label{eq:haploid_logistic growth} - p_{t+\tau} \approx \frac{p_t}{p_t + q_t e^{-s\tau}}. + p_{\tau} \approx \frac{p_0}{p_0 + q_0 e^{-s\tau}}. \end{equation} This equation takes the form of a logistic function. That is because we are looking at the relative frequencies of two `populations' (of @@ -849,7 +849,7 @@ \subsection{Heterozygote advantage} Using our $s_1$ and $s_2$ parametrization above, we see that the marginal fitnesses of the two alleles are equal when \begin{equation} - p_e = \frac{s_2}{s_1+s_2} + p_e = \frac{s_2}{s_1+s_2} \label{eqn:het_ad_eq} \end{equation} \begin{marginfigure} \begin{center} diff --git a/math_background/Equation_sheet.tex b/math_background/Equation_sheet.tex new file mode 100644 index 0000000..b889412 --- /dev/null +++ b/math_background/Equation_sheet.tex @@ -0,0 +1,44 @@ +% \usepackage{multicol} %%For eqn sheet https://tex.stackexchange.com/questions/152093/troubles-with-a-two-column-document-using-tufte-handout + + +%\d% ocumentclass[12pt,twocolumn]{article} +% % \usepackage{nicefrac} +% % \usepackage{amsmath} +% % \usepackage{amsfonts} +% % \usepackage{amssymb} +% % \usepackage[dvipsnames]{xcolor} +% % \newcounter{question}[section] %%modified from https://www.sharelatex.com/learn/Counters +% % %\newenvironment{question}[1][]{\refstepcounter{question}\par \begin{tcolorbox} +% % % \medskip \textbf{Question~\thequestion. #1}\rmfamily}{\medskip} \end{tcolorbox} +% % \newcommand{\E}{\mathbb{E}} +% % \renewcommand{\P}{\mathbb{P}} +% % \newcommand{\half}{\tfrac{1}{2}} + +%\newcommand{\wbar}{\overline{w}} +% New commands added by Simon: +%\newcommand{\fis}{F_{\mathrm{IS}}} +%\newcommand{\fit}{F_{\mathrm{IT}}} +%\newcommand{\fst}{F_{\mathrm{ST}}} +%\newcommand{\Wbar}{\overline{W}} + +%\newenvironment{question}[1][]{\refstepcounter{question}\par\medskip + % \textbf{Question~\thequestion. #1} \rmfamily}{\medskip} + + +%\begin{document} +\section*{Equation Sheet} +\begin{table*} +% \begin{tabular}{llll} + % \csvreader[separator=semicolon]{math_background/Equation_sheet.txt} +%\end{tabular} + \csvautobooktabular[separator=semicolon]{math_background/Equation_sheet.txt} % can control this more https://mirror.hmc.edu/ctan/macros/latex/contrib/csvsimple/csvsimple.pdf + % https://tex.stackexchange.com/questions/324777/one-of-the-fields-in-the-csv-read-by-csvreader-has-a-lot-of-comma +\end{table*} %, table head=\hline, table foot=\hline + +%\newpage +%\begin{multicols}{2} + +%\end{multicols} + + +%\end{document} diff --git a/math_background/Equation_sheet.txt b/math_background/Equation_sheet.txt index 3879cc8..bc40477 100644 --- a/math_background/Equation_sheet.txt +++ b/math_background/Equation_sheet.txt @@ -3,7 +3,7 @@ Equation ; ref. ; Equation ; ref. $F_{ij}= 0 \times r_0 + (\nicefrac{1}{4}) r_1 + (\nicefrac{1}{2}) r_2$ ; \eqref{eqn:coeffkinship_step} ; $(1-F) p^2 + F p (1-F) 2pq (1-F) q^2 + F q$ ; \eqref{table:GeneralizedHWE} ; ; ; F statistics ; a ; b ; c -$\fst =1-\frac{H_S}{H_T}$ ; \eqref{eqn:FST}. ; $\fit =1-\frac{H_I}{H_T},~~\fis =1-\frac{H_I}{H_S}$ ; \eqref{eqn:FIT}, ~ \eqref{eqn:FIS} +$\fst =1-\frac{H_S}{H_T}$ ; \eqref{eqn:FST}. ; "$\fit =1-\frac{H_I}{H_T},~~\fis =1-\frac{H_I}{H_S}$" ; "\eqref{eqn:FIT}, ~ \eqref{eqn:FIS}" ; ; ; Relationship among F statistics ; ; Linkage disequilibrium (LD) ; ~ $(1-\fit) =(1-\fis)(1-\fst)$ ; \eqref{eqn:F_relationships} ; $D = p_{AB} - p_Ap_B $ ; \eqref{eqn:LD_def} @@ -12,16 +12,28 @@ Decay of LD ; ; Decay of Heterozygosity ; ~ $D_t= (1-r)^t D_0 \approx D_0 e^{-rt}$ ; \eqref{eqn_LD_decay} ; $H_t = \left(1-\frac{1}{2N_e} \right)^tH_0 \approx H_0 e^{-\nicefrac{t}{2N_e}}$ ; \eqref{eqn:loss_het_discrete} ; ; ; Equilibrium level of neutral polymorphism ; \eqref{eqn:hetero} ; Coalescent time and time to MRCA ; ~ - $H = \frac{4N_e\mu}{1+4N_e\mu} \approx 4N_e\mu $ ; ~ ; $\E[T_k] = \frac{2 N_e}{ {k \choose 2} },~~~~\E[T_{MRCA}] =4N_e(1-1/n) $ ; ~ + $H = \frac{4N_e\mu}{1+4N_e\mu} \approx 4N_e\mu $ ; ~ ; "$\E[T_k] = \frac{2 N_e}{ {k \choose 2} },~~~~\E[T_{MRCA}] =4N_e(1-1/n) $" ; ~ ; ; ; -Pairwise diversity \& number of segregating sites ; ~ ; Expectation of $\dNdS$ ; \eqref{eqn:dNDS_C_B} -$\E[\pi] = 4N_e\mu ,~~~~ \E[S] = 4N_e\mu \sum_{k=n}^2 \frac{1}{k-1} $ ; ; $\dNdS = (1-C-B) + 2 N B f_B $ ; ~ +Number pairwise diffs. \& segregating sites ; ~ ; Expectation of $\dNdS$ ; \eqref{eqn:dNDS_C_B} +"$\E[\pi] = 4N_e\mu ,~~~~ \E[S] = 4N_e\mu \sum_{k=n}^2 \frac{1}{k-1} $" ; ; $\dNdS = (1-C-B) + 2 N B f_B $ ; ~ ; ; ; -Model-based $\fst$ expectations. ; ~ ; Phenotypic covariance between relatives ($i$ \& $j$) ; ~ - $\fst = \frac{ T}{ T + 4N_e }, ~~~~F_{IM} = \frac{1}{1 + 4N_I m} $ ; \eqref{eqn:FST_split}, \eqref{eqn:FIM} ; $Cov(X_1,X_2) = 2F_{1,2} V_A + r_2 V_D~~~\textrm{if } V_D>0$ ; ~ +Model-based $\fst$ expectations. ; ~ ; Phenotypic covar. between relatives ($i$ \& $j$) ; ~ +" $\fst = \frac{ T}{ T + 4N_e }, ~~~~F_{IM} = \frac{1}{1 + 4N_I m} $" ; "\eqref{eqn:FST_split}, \eqref{eqn:FIM}" ; "$Cov(X_1,X_2) = 2F_{1,2} V_A + r_2 V_D~~~\textrm{if } V_D>0$" ; ~ ; ; ; -Cross trait ($1$ \& $2$) covariance between relatives ; ~ ; Breeder's equation ; ~ -$Cov(X_{1,i},X_{2,j}) = 2 F_{i,j} V_{A,1,2}$ ; ~ ; $R = h^2 S = V_A \beta = \frac{V_A}{\wbar} \frac{\partial \wbar}{\partial \bar{x}} $ ; \eqref{breeders_eqn}, \eqref{eqn:R_beta}, \eqref{eqn:pheno_fitness_landscape} +Cross trait ($1$ \& $2$) covar. between relatives ; ~ ; Breeder's equation ; ~ +"$Cov(X_{1,i},X_{2,j}) = 2 F_{i,j} V_{A,1,2}$" ; ~ ; $R = h^2 S = V_A \beta = \frac{V_A}{\wbar} \frac{\partial \wbar}{\partial \bar{x}} $ ; "\eqref{breeders_eqn}, \eqref{eqn:R_beta}, \eqref{eqn:pheno_fitness_landscape} " ; ; ; Multi-variate breeders equation ; ~ ; Hamilton's Rule ; ~ -$\bf{R} = \bf{G} \bf{V}^{-1} \bf{S} = \bf{G} \boldsymbol{ \beta} $ ; \ref{eqn:MV_breeders_eqn} ; $ 2 F_{i,j} B > C$ ; \eqref{eqn:Hamiltons_rule} +$\bf{R} = \bf{G} \bf{V}^{-1} \bf{S} = \bf{G} \boldsymbol{ \beta} $ ; \ref{eqn:MV_breeders_eqn} ; "$ 2 F_{i,j} B > C$" ; \eqref{eqn:Hamiltons_rule} + ; ; ; +Frequency next generation (haploid \& diploid). ; ; Frequency change ; +"$p_{t+1} = \frac{w_1 }{\wbar} p_t , ~p_{t+1} = \frac{w_{11} p_t^2 + w_{12} p_tq_t}{\wbar} $" ; "\eqref{eq:deltap_haploid}, \eqref{deltap_dip1}" ; $\Delta p_t=\frac{ (\wbar_1-\wbar_2)}{\wbar} p_t q_t =\frac{1}{2} \frac{p_tq_t}{\wbar} \frac{d \wbar}{dp}$ ; " \eqref{deltap_dip2}, \eqref{deltap_dip3}" + ; ; ; +Haploid cumulative change {\tiny (use $\nicefrac{s}{2}$ for diploid case)} ; ; Heterozygote advantage equilibrium ; +"$p_{\tau} \approx \frac{p_0}{p_0 + q_0 e^{-s\tau}},~~~\tau \approx \frac{1}{s} \log \left(\frac{p_{\tau} q_0}{q_{\tau} p_0}\right) $" ; \eqref{eq:haploid_logistic growth}~\eqref{eq:estTauExplSimpl} ; $p_e = \frac{s_2}{s_1+s_2}$ ; \eqref{eqn:het_ad_eq} + ; ; ; +Diploid mutation-selection equilibrium ; ; Migration-selection equil. \& cline width. ; +$q_e = \sqrt{\frac{\mu}{s}}~~(\textrm{if }h=0)$ ; "\eqref{eqn:mut_sel_bal}, \eqref{eqn:recess_mut_sel_bal} " ; "$q_{e,1} = \frac{m}{hs},~~~~0.6 \sigma/\sqrt{s} $" ; "\eqref{eqn:mig_sel_eq},~\eqref{eqn:cline_width}" + ; ; ; +Selected prob. fixation (haploid \& diploid) ; ; Prob. fixation for weakly selected alleles ($h=\nicefrac{1}{2}$) ; +"$p_F \left(\nicefrac{1}{2N} \right) = 2s, ~~~P_F \left(\nicefrac{1}{2N} \right) \approx 2 h s,\text{ \small , ~$Ns \gg 1$} $" ; \eqref{eqn:prob_fix_strong} \& \eqref{eqn:diploid_escape} ; "$P_F \left(\frac{1}{2N} \right) = \frac{1-e^{-s }}{1-e^{-2Ns}} \text{ {\small ,~ $s<0$ for deleterious allele.} } $" ; \eqref{eqn:new_mut_prob_fixed} \ No newline at end of file diff --git a/popgen_notes.pdf b/popgen_notes.pdf index 56080f9..4065c59 100644 Binary files a/popgen_notes.pdf and b/popgen_notes.pdf differ