%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beamer %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % To run - pdflatex filename.tex % acroread filename.pdf %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\documentclass[handout,compress,red]{beamer} \documentclass[compress,red]{beamer} \mode \usetheme{Madrid}%Warsaw % other themes: AnnArbor, Antibes, Bergen, Berkeley, Berlin, Boadilla, boxes, CambridgeUS, Copenhagen, Darmstadt, default, Dresden, Frankfurt, Goettingen, % Hannover, Ilmenau, JuanLesPins, Luebeck, Madrid, Maloe, Marburg, Montpellier, PaloAlto, Pittsburg, Rochester, Singapore, Szeged, classic %\usecolortheme{lily} % color themes: albatross, beaver, beetle, crane, default, dolphin, dov, fly, lily, orchid, rose, seagull, seahorse, sidebartab, structure, whale, wolverine %\usefonttheme{serif} % font themes: default, professionalfonts, serif, structurebold, structureitalicserif, structuresmallcapsserif \hypersetup{pdfpagemode=FullScreen} % makes your presentation go automatically to full screen % define your own colors: \definecolor{Red}{rgb}{1,0,0} \definecolor{Blue}{rgb}{0,0,1} %\definecolor{Black}{rgb}{0,0,1} \definecolor{Green}{rgb}{0,1,0} \definecolor{magenta}{rgb}{1,0,.6} \definecolor{lightblue}{rgb}{0,.5,1} \definecolor{lightpurple}{rgb}{.6,.4,1} \definecolor{gold}{rgb}{.6,.5,0} \definecolor{orange}{rgb}{1,0.4,0} \definecolor{hotpink}{rgb}{1,0,0.5} \definecolor{newcolor2}{rgb}{.5,.3,.5} \definecolor{newcolor}{rgb}{0,.3,1} \definecolor{newcolor3}{rgb}{1,0,.35} \definecolor{darkgreen1}{rgb}{0, .35, 0} \definecolor{darkgreen}{rgb}{0, .6, 0} \definecolor{darkred}{rgb}{.75,0,0} \xdefinecolor{olive}{cmyk}{0.64,0,0.95,0.4} \useoutertheme[subsection=false]{smoothbars} \beamertemplateshadingbackground{red!9}{blue!4} \setbeamertemplate{footline}[text line]{} % makes the footer EMPTY % include packages \usepackage{subfigure} \usepackage{natbib} \usepackage{cite} \usepackage{multicol} \usepackage{amsmath} \usepackage{epsfig} \usepackage{graphicx} \usepackage[all,knot]{xy} \xyoption{arc} \usepackage{url} \usepackage{tikz} \usepackage{amsmath} \usepackage{verbatim} \usetikzlibrary{arrows,shapes} \usepackage{multimedia} \usepackage{hyperref} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Title Page Info %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vspace{-7 in} %\flushleft {{\upshape \tiny \em{\LaTeX\ seminar style \& Beamer}}} \\ \\ \title{Unifying theories of molecular, community and network evolution^{1}}\\ \tiny Carlos J. Meli\'an \\ {\em National Center for Ecological Analysis and Synthesis, University of California, Santa Barbara}\\ {\em Microsoft Research Ltd, Cambridge, UK.}} %\subtitle{ @ NCEAS} %\author{Carlos J. Meli\'an} \institute{Seminar @EEOB, ISU.\\ November 17, 2008} \vspace{0.25 in} \date{\tiny 1. Meli\'an, C. J., Alonso, D., Allesina, S., V\'azquez, D. P., and Regetz, J.\\ Unifying theories of molecular, community and network evolution, {\em Submitted}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Begin Document %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{ \titlepage } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section[Outline]{} % this puts the outline before EACH section automatically & will highlight the section you're about to talk about %\frame{\tableofcontents} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %16%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Thanks!} \begin{itemize} \item Computing-scientist staff at the National Center for Ecological Analysis and Synthesis \item Microsoft Research Ltd., Cambridge, UK. \item Drew Allen, Rick Condit, Jennifer Dunne, Rampal Etienne, Stanley Harpole, Pablo Marquet, Brad McRae, Mark Urban, and Tommaso Zillio, \end{itemize} } \section{Motivation \& Questions} \subsection{Motivation} %1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Paradox of Diversity \frame{\frametitle{Motivation} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} \begin{itemize} \item < 1-| alert@1 > {\Large Why are there so many species?}\\ {\small Ecological views have focused on the mechanisms that enable or constraint species coexistence. Yet what controls biodiversity remains a question still largely unanswered...\citep{Hutchinson:1959}} \item < 2-| alert@2 > {\Large Why are there so few kinds of animals?}\\ {\small Additional constraints on the process of speciation, constraints set by the genetics rather than ecology...\citep{Felsenstein:1981}} \item < 3-| alert@3 > \Large Lack of statistical test and hypothesis about the ecological and evolutionary processes underlying speciation and diversity...\citep{Gavrilets2:2004} \end{itemize} \end{beamerboxesrounded} }%make clear to a broad audience the work I'll present is at the interface ecol-evol and that is rare and why it is cutting-edge and important-what insights do we gain beyond those if i had only focused on the ecological or evolutionary aspects alone... %2%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Motivation} \begin{columns} \begin{column}{4cm} \vspace{0.05 in} \includegraphics[width=8cm]{speciation.pdf} \end{column} \begin{column}{9cm} \vspace{-2.5 in} \includegraphics[width=7cm,angle=90]{speciation1.pdf} \vspace{-1 in} \includegraphics[width=7cm,angle=90]{speciation2.pdf} \end{column} \end{columns} } %3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Challenges} \frame{\frametitle{Challenges} \setbeamercovered{transparent} \begin{itemize} \item < 1-| alert@1 > {\Large Huge amount of data on different biological levels are available.} % Huge intraspecific variability in empirical data (ref)/individual context dependence \item < 4-| alert@4 > Still not together $\rightarrow$ cross data sets! \citep{Jonesetal:2006} \item < 2-| alert@2 > {\Large It is possible to reconstruct multilevel systems and test their properties by integrating data, analytical work and simulations.} % Different theories explain similarly the same property in the same level (i.e., RSA) \item < 5-| alert@5 > It is challenging to do this combining analytical tools with novel computational methods! \item < 3-| alert@3 > {\Large The development of the NTME and NTB offer a framework to add the role of explicit mechanisms of speciation in the evolution of diversity and coexistence at multiple levels.} % Lack of theoretical framework to test simultaneously the importance of each level \item < 6-| alert@6 > Theory not yet developed to test simultaneously diversity at different biological levels! \end{itemize} } \subsection{Questions} %4%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Questions} \begin{itemize} \item < 1-| alert@1 > {\em This raises the important question of whether the type of selection (neutral or frequency-dependent) at molecular and ecological levels influence speciation and genetic--species diversity.} \item < 2-| alert@2> {\Large Can frequency-dependent selection at molecular and ecological levels maximize speciation rate and genetic and species diversity?} \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Background %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Background} \subsection{Introduction} %5%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Neutral Theory of Molecular Evolution ($NTME$)} \begin{enumerate} \item < 1-| alert@1 > $NTME$ was proposed when it was discovered that there is so much molecular diversity-polymorphism-in species \citep{Kimura:1968,King&Jukes:1969} \item < 2-| alert@2 > More than 30 years! \item < 3-| alert@3 > Common test of neutrality. \item < 4-| alert@4 > Increasing amount of DNA sequence and polymorphism data has stimulated re-examination of the NTME \citep{Wolfe&Wen-Hsiung:2003,Nei:2005}. \item < 5-| alert@5 > Better understanding of directional, stabilizing, balancing, and neutral selection and the structure-dynamics of DNA evolution \citep{Lynch:2007}. \end{enumerate} } %6%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Neutral Theory of Biodiversity ($NTB$)} \begin{enumerate} \item < 1-| alert@1 > $NTB$ proposed to test the effect of dispersal limitation and equivalence at individual level on diversity in ecological communities \citep{Hubbell2:2001} \item < 2-| alert@2 > Only a few years! $\rightarrow$ \citep{Leigh:1999} \item < 3-| alert@3 > Incipient test of neutrality. \item < 4-| alert@4 > Increasing amount of ecological data is stimulating the development of the $NTB$ at ecological level \citep{Alonso:2006}. \item < 5-| alert@5 > Better understanding of community assembly as a continuum between niche and neutral processes \citep{Tilman2:2004,Harpole&Suding:2007}. \end{enumerate} } %7%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{In common} \begin{enumerate} \item < 1-| alert@1 > First principles $\rightarrow$ birth-death process at individual level ($i = 1,2,...J$). \item < 2-| alert@2 > Analytical solutions in some cases $\rightarrow$ fast codes! \item < 3-| alert@3 > Non-interactive models \item < 4-| alert@4 > Equivalence and symmetric interactions at molecular and ecological levels \item < 5-| alert@5 > They have the same structure $\rightarrow$ starting point to add more complexity \end{enumerate} } %8%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{How to merge neutral molecular and ecological theories?} \begin{columns} \begin{column}{4cm} \begin{itemize} \item < 1-| alert@1 > Genome evolution\footnote{\hyperlink{GenomeEvolution}{\beamergotobutton{GenomeEvolution}}}%->whole genome and how to collapse the genome in the neutral case \item < 2-| alert@2 > Mutation rate ($\mu$)\footnote{\hyperlink{MutationRate}{\beamergotobutton{MutationRate}}} \item < 3-| alert@3 > Evolution of molecular constraints ($q^{min}$)\footnote{\hyperlink{MolecularConstraints}{\beamergotobutton{MolecularConstraints}}} \end{itemize} \end{column} \begin{column}{4cm} \begin{itemize} \item < 4-| alert@4 > Sexual behavior \item < 5-| alert@5 > Resource/trophic strategies/behavior \end{itemize} \end{column} \begin{column}{3cm} \begin{itemize} \item < 6-| alert@6 > Speciation\footnote{\hyperlink{Speciation}{\beamergotobutton{Speciation}}} \end{itemize} \end{column} \end{columns} } %9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Neutral and frequency-dependent speciation models} \begin{enumerate} \item < 1-| alert@1 > Kill one individual at random from a uniform distribution \item < 2-| alert@2 > Select one parent $G_1(k)$ from a uniform distribution \item < 3-| alert@3 > We chose at random a second parent $G_2(k)$ among the individuals compatible with the first parent (i.e., $q^{G_1(k) G_2(k)}$ $>$ $q^{min}$) \item < 4-| alert@4 > Given the similarity between the parent of the new individual $k$ and the individual $i$ already in the population we update the similarity matrix (Q) according to: \begin{equation} \begin{cases} q^{ki} = \frac{e^{-4\mu}}{2}\left(q^{G_1(k) i} + q^{G_2(k) i}\right)\label{A1}\\ q^{kk}=1. \end{cases} \end{equation} \item < 5-| alert@5 > The frequency--dependent model is the same but with individuals with rare alleles increasing the probability to mate.\footnote{\hyperlink{FrequencyDependent}{\beamergotobutton{FrequencyDependent}}} \end{enumerate} } %10%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Evolutionary graphs and the genetic species concept} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} %\begin{center} \includegraphics[width=10cm]{Fig1.pdf} %\end{center} \end{beamerboxesrounded} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Results and Summary %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results} \subsection{Results} \frame{\frametitle{Question} \begin{enumerate} \item < 1-| alert@1 > {\Large Can frequency-dependent selection at molecular and ecological levels maximize speciation rate and genetic and species diversity?} \end{enumerate} } %11%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Can we predict speciation rate in the neutral case?} {\headcol {What is the number of steps ($x_{n}$) at which $q^{min}$ $>$ $q^{A A_n}$?}} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} \vspace{-0.5 in} %\begin{center} \includegraphics[width=10cm]{mutation.pdf} %\end{center} \end{beamerboxesrounded} } %13%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{The expected speciation rate} \begin{itemize} \item < 1-| alert@1 > The number of steps ($x_{n}$) at which $q^{min}$ $>$ $q^{A A_n}$ that is proportional to the speciation rate is:\\ \begin{equation} \frac{1}{x_{n}} = \frac{- 4 \mu + log(\frac{1 + X}{2})}{log(q^{min})} \label{A3} \end{equation} \item < 2-| alert@2 > where we know the value of $X$ is in the range [1,$q^{min}$], thus the expected value of $X$ is = $\frac{1 + q^{min}}{2}$. \end{itemize} } \frame{\frametitle{We now can estimate the simulated speciation rate ($\nu$)} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} %\begin{center} \vspace{-0.5 in} \includegraphics[width=9cm]{Fig3aLog.pdf} %\end{center} \end{beamerboxesrounded} } %14%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Thus, the expected speciation rate is} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} \begin{equation} \nu = \alpha + \beta \left[\frac{- 4 \mu + log(\frac{1 + X}{2})}{log(q^{min})}\right],\label{A5} \end{equation} \end{beamerboxesrounded} \\ where $\alpha$ is equal to -0.34 and the slope $\beta$ is equal to 1.45. } \frame{\frametitle{This expression gives an accurate prediction of the speciation rate...} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} %\begin{center} \vspace{-0.5 in} \includegraphics[width=7.5cm]{Fig3c.pdf} %\end{center} \end{beamerboxesrounded} } \frame{\frametitle{and the predicted values are higher in the neutral case...} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} %\begin{center} \vspace{-0.25 in} \includegraphics[width=7cm]{SpeciationEvents.pdf} %\end{center} \end{beamerboxesrounded} } \frame{\frametitle{Does higher speciation rate in the neutral scenario imply higher genetic and species diversity?} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} %\begin{center} \vspace{-0.5 in} \includegraphics[width=7.5cm]{Fig3.pdf} %\end{center} \end{beamerboxesrounded} } %15%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{\frametitle{Summary} \begin{enumerate} \item < 1-| alert@1 > The expected speciation rate in the neutral case can be estimated using an approximation that takes into account the mutation rate, neutral genome evolution and a molecular filter to have fertile offspring. \item < 2-| alert@2 > The dynamics of speciation is dramatically different in both scenarios. While speciation events are predictable in the neutral case, rapid series of speciation events happen in the frequency-dependent scenario. Then the system reaches a plateau without further speciation events. Rare species in the system increases their abundance, decreasing the probability of extinction and increasing genetic--species diversity and coexistence. \item < 3-| alert@3 > Negative frequency--dependent scenario is a potent selection amplifier, suppress speciation rate and increases genetic--species diversity. \end{enumerate} } \frame{\frametitle{Conclusion} \begin{enumerate} \item < 1-| alert@1 > $NTME$ and $NTB$ offer a theoretical framework to add explicit mechanisms of speciation and link the ecology and evolution of coexistence and diversity at multiple levels. \item < 2-| alert@2 > Evolutionary graphs at multiple biological levels have fascinating applications. For example, from the approach presented here is it possible to develop a biodiversity number with explicit genome evolution and speciation. This approach can be tested assuming neutral genome evolution and ecological behavior using thousands of individuals simultaneously. \item < 3-| alert@3 > This approach represents a statistical test to explore simultaneously diversity at molecular and ecological levels. \end{enumerate} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Appendix} %17%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Reproductive Isolation} \frame[label=GenomeEvolution]{\frametitle{Genome evolution} \begin{itemize} \item < 1-| alert@1 > Let us assume that the genome of each individual is represented by a sequence of $N$ loci. Each locus can assume values for two alleles ($+1/-1$). The genome of each individual $i$ can be written as (${S_{1}^i,S_{2}^i,...,S_{N}^i}$), where $S_{u}^i$ is the $u^{th}$ locus for the individual $i$ \item < 2-| alert@2 > Genetic similarity between individual $i$ and $j$ is defined as $q^{ij}$ = $\frac{1}{N} \sum_{u=1}^N S_{u}^i S_{u}^j$. \item < 3-| alert@3 > The genetic similarity matrix is $Q$ = $[q_{ij}] = 1$ for all $q_{ij}$. \item < 4-| alert@4 > We know the evolution of this matrix in the limit $N$ $\rightarrow$ $\infty$ in the infinite genome limit \citep{Higgs&Derrida:1992}, because each pair of alleles contributing the similarity between each pair of individuals $k$ and $i$ comes with equal probability from one of the two possible combinations of the parents of $k$ and individual $i$. \item < 5-| alert@5 > The development of viable and fertile offspring is possible only between organisms having an overlap greater than $q^{min}$ loci responsible of reproductive isolation (postzygotic RI). \end{itemize} } %prezygotic RI \frame[label=MutationRate]{\frametitle{Mutation rate} \begin{itemize} \item < 1-| alert@1 > Evolving individuals $\rightarrow$ $\mu_{S_{u}^i}$ is neutral, equal and independent for all units of sequence N $\rightarrow$ $\mu_{S_{u}^i} = \mu$ \item < 2-| alert@2 > \end{itemize} } \frame[label=MolecularConstraints]{\frametitle{Molecular constraints} \headcol{from \citep{Coyne:1992}} \setbeamercolor{uppercol}{fg=black,bg=white} \setbeamercolor{lowercol}{fg=black,bg=white} \begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{} %\begin{center} \includegraphics[width=5cm]{SexualIsolation.pdf} %\end{center} \end{beamerboxesrounded} } \frame[label=Speciation]{\frametitle{Speciation} \begin{itemize} \item < 1-| alert@1 > If $\mu$ is $<<$ 1, then the mean similarity value of $Q$ has a solution $Q^{*}$ = $\frac{1}{\theta + 1}$ and if $q_{min}$ $<$ $Q^{*}$\footnote{\hyperlink{Similarity}{\beamergotobutton{Similarity}}} we will have always 1 species in the system \citep{Higgs&Derrida:1992}. However if $q^{min}$ $>$ $Q^{*}$ then we have speciation events. \item < 2-| alert@2 > {\tiny Two speciation modes can occur: fission and mutation--induced speciation (see Appendix). Fission happens because the death of an individual splits a previously connected component (i.e. species). This can originate one or more new species (Fig. 1). Mutation--induced speciation happens because a newly produced offspring cannot mate with any individual. What is the minimum mutation rate ($\mu_{min}$) for the mutation--induced speciation mode to happen? Because in this case we need $q^{ki} {\tiny Each individual $i$ of species $k$ is chosen for reproduction according to: \begin{equation} P_{i,k} = {\cal{N}} F_{i,k}, \end{equation} where individual fitness is defined as: \begin{equation} F_{i,k} = \frac{1}{\sum_{j=1}^{N_{k}} H(q^{ij}-q^{min})} \end{equation} Thus we write: \begin{equation} P_{i,k} = {\cal{N}} \frac{1}{\sum_{j=1}^{N_{k}} H(q^{ij}-q^{min})} \end{equation} where $\cal{N}$ is a normalization factor, $N_{k}$ is the abundance of species $k$, and $H(\alpha)$ is \begin{equation*} H(\alpha) = \begin{cases} 1 & \text{if $\alpha > 0$}\\ 0 & \text{otherwise} \end{cases} \end{equation*} We now calculate the normalization factor by using the normalization requirement, i.e., by summing $P_{i,k}$ across all individuals and species 1 must be obtained: \begin{equation} {\cal{N}} \sum_{i=1}^{S} \sum_{j=1}^{N_{k}} F_{i,k} = 1, \end{equation} where S is the number of species, then: \begin{equation} {\cal{N}} = \frac{1}{\sum_{i=1}^{S} \sum_{j=1}^{N_{k}} F_{i,k}} \end{equation} Therefore, the probability of birth for each $i$ individual is: \begin{equation} P_{i,k} = \frac{F_{i,k}}{\sum_{i=1}^{S} \sum_{j=1}^{N_{k}} F_{i,k}}} \end{equation} \end{itemize} } \subsection{Genetic Similarity} \frame[label=Similarity]{\frametitle{Genetic Similarity} \vspace{-1 in} \begin{center} \begin{figure} \includegraphics[width=10cm]{Nospeciesformation.pdf} \end{figure} \end{center} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% End Document %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame{ \bibliographystyle{ecology} {\tiny \bibliography{EvolutionMultilevel}} } \end{document} %\subsection{General Questions} %2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\frame{\frametitle{General Questions} %\begin{itemize} %\item < 1-| alert@1 > Is it really necessary to develop such an unification?%->pluralistic, first-principles, individual variability, raw data... %\item < 2-| alert@2 > {\Large If so, how to link the origin, evolution and persistence of diversity in a unified framework?}%parsimony, neutrality, main knowledge from different disciplines... %\item < 3-| alert@3 > Can we develop fast code/analytical approximations to test a unified theory with data at multiple biological levels?%methods->possible? %\end{itemize} %}