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Diffstat (limited to 'slides.tex')
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1 files changed, 78 insertions, 18 deletions
@@ -29,7 +29,7 @@ \addbibresource{slides.bib} \AtBeginBibliography{\small} -%% Tikz relative positioning https://tex.stackexchange.com/questions/89588/positioning-relative-to-page-in-tikz +% Tikz relative positioning https://tex.stackexchange.com/questions/89588/positioning-relative-to-page-in-tikz \makeatletter \def \parsecomma#1,#2 @@ -79,27 +79,87 @@ \newcommand{\dms}{\ac{dms} \xspace} +\definecolor{cb1}{HTML}{1b9e77} +\definecolor{cb2}{HTML}{d95f02} +\definecolor{cb3}{HTML}{7570b3} + \author{Justin Bed\H{o}} -\title{Exploration of deep mutational scanning data with unsupervised methods} +\title{Representation learning of compositional counts: exploration of deep mutational scanning data} \date{December 13, 2022} \begin{document} \maketitle - \section{Deep mutational scanning data} + \begin{frame}{Variants of Uncertain Significance + \footfullcite{Liu2020}} + \begin{center} + \input{clinvar.tikz} + \end{center} + \end{frame} - \begin{frame}{\dms data} Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. - Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. - Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. - Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. + \begin{frame}{\dms} + \begin{quote} Deep mutational scanning is a method for systematically introducing mutations into a gene and then analyzing the resulting protein products to see how the changes affect the protein's function. + \end{quote} + \begin{enumerate} + \item Growing resource of functional data + \item MaveDB + \footfullcite{Esposito2019} + \footnote{\url{https://www.mavedb.org}} catalogs a number of datasets and provides easy access + \end{enumerate} \end{frame} - \section{Compositional data} + \begin{frame}{Deep Mutational Scanning: Overview + \footfullcite{Fowler2014}} + \begin{tikzpicture} + \node at (page cs:0,0.75){\(t_0\)}; + \node at (page cs:0.53,0.75){\(t_1\)}; + \node(a) at (page cs:-0.75,0.5){\includegraphics[width=0.3 + \textwidth]{Protein-BRCA1.png}}; + \node(b) at (page cs:0,0.5){\begin{tikzpicture} + \node[circle,draw,fill=cb1] at (page cs:-0.06,0){}; + \node[circle,draw,fill=cb1] at (page cs:0,0){}; + \node[circle,draw,fill=cb1] at (page cs:0.06,0){}; + \node[circle,draw,fill=cb2] at (page cs:-0.06,0.1){}; + \node[circle,draw,fill=cb2] at (page cs:0.06,0.1){}; + \node[circle,draw,fill=cb2] at (page cs:0,0.1){}; + \node[circle,draw,fill=cb3] at (page cs:-0.06,-0.1){}; + \node[circle,draw,fill=cb3] at (page cs:0,-0.1){}; + \node[circle,draw,fill=cb3] at (page cs:0.06,-0.1){}; + \end{tikzpicture}}; + + \node(c) at (page cs:0.5,0.5){\begin{tikzpicture} + \node[circle,draw,fill=cb1] at (page cs:0.5,0){}; + \node[circle,draw,fill=cb1] at (page cs:0.56,0){}; + \node[circle,draw,fill=cb1] at (page cs:0.62,0){}; + \node[circle,draw,fill=cb1] at (page cs:0.68,0){}; + \node[circle,draw,fill=cb1] at (page cs:0.74,0){}; + \node[circle,draw,fill=cb2] at (page cs:0.5,0.1){}; + \node[circle,draw,fill=cb3] at (page cs:0.5,-0.1){}; + \node[circle,draw,fill=cb3] at (page cs:0.56,-0.1){}; + \end{tikzpicture}}; + + \node(d) at (page cs:0.2,-0.25){\includegraphics[width=0.3 + \textwidth]{nextseq500.jpg}}; + + \draw[->] (a) -- (b) node[midway,above]{mutagenesis}; + \draw[->] (b) -- (c) node[midway,above]{selection}; + \draw[->] (b) -- (d); + \draw[->] (c) -- (d); + \end{tikzpicture} + \end{frame} + + \begin{frame}{Deep Mutational Scanning: Integration issues} + \begin{enumerate} + \item Scores calculated a variety of ways, e.g., Rubin et al. + \footfullcite{Rubin2017}: + \[L_{v,t}=\log\left(\frac{(c_{v,t}+\frac12)(c_{wt,0}+\frac12)}{(c_{v,0}+\frac12)(c_{wt,t}+\frac12)}\right) \] + \end{enumerate} + \end{frame} \begin{frame}{Basics} \begin{definition}[Compositional data] Data \(X \in \R^{n \times d}\) is compositional if rows \(\bx_i\) are in the simplex - \[S^d=\{\,\bx \in \R^d : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\} \] + \[S^d=\{\,\bx \in \R^d : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\} \] for constant \(\kappa > 0\). \end{definition} Information is therefore given only by the ratios of components and any composition can be normalised to the standard simplex where \(\kappa = 1\) (c.f., dividing by library size). \end{frame} @@ -107,10 +167,10 @@ \begin{frame}{Isomorphisms to Euclidean vector spaces} The simplex forms a \(d-1\) dimensional Euclidean vector space \footfullcite{Aitchison1982}: \begin{definition}[\ac{alr}] - \[\alr(\bx)_i = \log \frac{x_i}{x_0} \] + \[\alr(\bx)_i = \log \frac{x_i}{x_0} \] \end{definition} \begin{definition}[\ac{clr}] - \[\clr(\bx)_i = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}} \] + \[\clr(\bx)_i = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}} \] \end{definition} \end{frame} @@ -148,7 +208,7 @@ \begin{frame}{Traditional \ac{pca}} Given \(\X\in \R^{n\times d}\) minimise loss - \[\ell_{\textsc{pca}} \triangleq {\lVert \X - \V\A \rVert}^2_{\textrm{F}} \] + \[\ell_{\textsc{pca}} \triangleq {\lVert \X - \V\A \rVert}^2_{\textrm{F}} \] s.t. \(\V \in \R^{n \times k}\), \(\A \in \R^{k \times d}\), and \(\V^\intercal \V = \I\). @@ -161,24 +221,24 @@ \ac{pca}} \begin{definition}{Bregman Divergence} Let \(\varphi \colon \R^d \to \R\) be a smooth ($C^1$) convex function on convex set \(\Omega\). The Bregman divergence \(D_\varphi\) with generator \(\varphi\) is - \[ D_\varphi\left(\bu\,\Vert\,\bv\right) \triangleq \varphi(\bu)-\varphi(\bv)-\langle \nabla\varphi(\bv),\bu-\bv\rangle. \] + \[ D_\varphi\left(\bu\,\Vert\,\bv\right) \triangleq \varphi(\bu)-\varphi(\bv)-\langle \nabla\varphi(\bv),\bu-\bv\rangle. \] \end{definition} Denote the convex conjugate of \(\varphi\) as \(\varphi^*(\bu) \triangleq \sup_\bv\left\{\langle \bu,\bv\rangle-\varphi(\bv)\right\}\). The exponential family \ac{pca} is then given by minimising loss - \[\ell_{\varphi} \triangleq D_\varphi\left(\X\,\Vert\,\nabla\varphi^*\left(\V\A\right)\right) \] + \[\ell_{\varphi} \triangleq D_\varphi\left(\X\,\Vert\,\nabla\varphi^*\left(\V\A\right)\right) \] under the same constraints as previously, approximating \(\X \sim \nabla\varphi^*\left(\V\A\right)\). \end{frame} \begin{frame}{Aitchison's simplex and exponential \ac{pca}} Aitchison's log-transformation is a dual affine coordinate space made explicit with - \[\varphi(z) = z\log(z) - z \Leftrightarrow \varphi^*(z) = e^z, \] + \[\varphi(z) = z\log(z) - z \Leftrightarrow \varphi^*(z) = e^z, \] but what about normalisation? Consider \ac{alr}: - \[\alr(\bx) \triangleq x_0 \sum_{i=1}^d\varphi\left(\frac{x_i}{x_0}\right) \Leftrightarrow \alr^*(\bx) = x_0\sum_{i=1}^d e^{\frac{x_i}{x_0}} \] + \[\alr(\bx) \triangleq x_0 \sum_{i=1}^d\varphi\left(\frac{x_i}{x_0}\right) \Leftrightarrow \alr^*(\bx) = x_0\sum_{i=1}^d e^{\frac{x_i}{x_0}} \] \end{frame} @@ -186,9 +246,9 @@ \begin{theorem}{Scaled Bregman \footfullcite{nock2016scaled}} Let \(\varphi \colon \mathcal{X} \to \R\) be convex differentiable and \(g \colon \mathcal{X} \to \R\) be differentiable. Then - \[g(\bx)\cdot D_\varphi\left(\frac{\bx}{g(\bx)}\,\middle\Vert\,\frac{\by}{g(\by)}\right) = D_{\breve{\varphi}}\left(\bx\,\middle\Vert\,\by\right) \] + \[g(\bx)\cdot D_\varphi\left(\frac{\bx}{g(\bx)}\,\middle\Vert\,\frac{\by}{g(\by)}\right) = D_{\breve{\varphi}}\left(\bx\,\middle\Vert\,\by\right) \] where - \[\breve{\varphi} \triangleq g(\bx) \cdot \varphi\left(\frac{x}{g(\bx)}\right) \] + \[\breve{\varphi} \triangleq g(\bx) \cdot \varphi\left(\frac{x}{g(\bx)}\right) \] \end{theorem} Avalos et al. |