\stitle{Genetic Interactome}

Double mutants exhibit unexpected phenotypes, as compared to joint single mutations.

\begin{definition}
\begin{itemize}
\item {\color{red} \textbf{Negative Interactions}}: more severe phenotype than expected
\begin{itemize}
\item Also known as \textit{aggravating} or \textit{synergistic}
\end{itemize}

\item {\color{green} \textbf{Positive Interactions}}: Less severe phenotype than expected
\begin{itemize}
\item Also known as \textit{alleviating} or \textit{epistatic}
\end{itemize}
\end{itemize}
\end{definition}

Most commonly used:
\begin{description}
\item [Phenotype]: Growth rate
\item[Model]: Multiplicative null model
\end{description}



\stitle{Organization of Genetic Interactions}

\begin{definition}
\begin{itemize}
\item \alert{Between-Pathway Model}
\begin{itemize}
\item Among genes participating in redundant functions
\end{itemize}
\item \alert{Within-Pathway Model}
\begin{itemize}
\item Among genes with additive effect
\end{itemize}
\item \alert{Indirect Effect}
\begin{itemize}
\item Among genes with distant functions that are not directly related
\end{itemize}
\end{itemize}
\end{definition}



\stitle{Between-Pathway Model (BPM)}

  \begin{columns}[c]
    \column{.4\textwidth}
	\includegraphics[scale=.25]{figures/BPM}	
	  
    \column{.6\textwidth}
		\begin{itemize}
		\item Bi-cliquish structure
		\item Have been used to:		
		\begin{enumerate}
			\item Predict co-pathway membership of gene pairs
			\item Extract redundant pathways
		\end{enumerate}
	\end{itemize}
  \end{columns}  




\stitle{The Genetic Landscape of a Cell}
  \begin{columns}[c]
    \column{.5\textwidth}
	\includegraphics[width=\columnwidth,height=.8\textheight]{figures/Costanzo}\\
	\tiny{Adopted from Costanzo et al., 2010}
	  
    \column{.5\textwidth}
		\begin{itemize}
		\item Baker's yeast, \textit{Saccharomyces cerevisiae}
		\item Synthetic Genetic Array (SGA)
		\item 1712 query genes
			\begin{enumerate}
				\item 1378 null alleles of non-essential genes
				\item 334 hypomorphic or conditional alleles of essential genes
			\end{enumerate}
		\item 3885 array strains
	\end{itemize}
  \end{columns}  



\stitle{Functional Annotations}
  \begin{columns}[c]
    \column{.5\textwidth}
	\includegraphics[width=\columnwidth]{figures/KEGG}
		  
    \column{.5\textwidth}
		\begin{itemize}
			\item KEGG Pathway Database
			\item Annotations for 1026 genes in the experiment
			\item 96 Pathways
			\begin{itemize}
				\item 80 pathways after filtering pathways with less than 10 genes.			
			\end{itemize}
		\end{itemize}
  \end{columns}  



\stitle{Local Neighborhood Similarity}{A Predictor of Co-Pathway Membership}
\begin{block}{Similarity prediction methods}
\begin{enumerate}
\item Number of Shared Neighbors
\item Congruence Score
\item Pearson Correlation of Interaction Profiles
\end{enumerate}
\end{block}


  \begin{columns}[t]
    \column{.45\textwidth}
	\includegraphics[width=\columnwidth]{figures/CS1}
	  
    \column{.35\textwidth}
	\includegraphics[width=\columnwidth]{figures/CS2}
  \end{columns}  
  

\tiny{Both $v_i$ and $v_j$ have three shared neighbors. However, in the first case their
congruence score is almost 0.6, while in the second case it is approximately 2
(assuming a graph of size 10).}





\stitle{Evaluating Ranking Methods}
Given a pathway \alert{$P_A$} and a cut size (target set) \alert{$l$}.

	\begin{definition}
		\begin{eqnarray*}
			P-value (X = k) &=& Prob(k\leq X) \nonumber \\
			&=&  HGT(k| N, N_A, l) \nonumber  \\
			&=& \sum_{x=k}^{min(N_A, l)} \frac{{C(l, x)} {C(N-l,N_A-x)}} {{C(N,N_A)}}
		\end{eqnarray*}
	\end{definition}

	\alert{$X$}: Random variable denoting the number of true positives in a random sample, \alert{$N$}: Total number of gene pairs, \alert{$N_A$}: Number of gene pairs in pathway $A$, \alert{$l$}: Size of target set


\stitle {Minimum HyperGeometric (mHG) Score}
Target size unknown:

\begin{definition}{	The \alert{Minimum HyperGeometric (mHG)} score}
\begin{equation*}
mHG(\lambda) = min_{1\leq l \leq N} HGT(b_l(\lambda); N, N_A, l),
\end{equation*}

where $b_l(\lambda) = \sum_{i=1}^l \lambda_i $

\end{definition}

 $\lambda_i$ is 1 if both of the genes in the $i^{th}$ ranked gene pair
are members of $P_A$, and 0 otherwise.

\begin{itemize}
\item mHG Adjusted for Multiple Comparison
\end{itemize}


\stitle{Predictions Are Not Equally Accurate in Different KEGG Pathways}
\includegraphics[width=.95\columnwidth] {figures/Complete_mHG_Bonferroni_01}




\stitle{Highlights}

\begin{block}{Basic Idea}
  \begin{columns}[c]
    \column{.4\textwidth}
	Heterogeneous performance of co-pathway membership predictions		

    \column{.05\textwidth}
	$\Longleftrightarrow$
			  
    \column{.4\textwidth}
	Existence of specific structure around enriched pathways

  \end{columns}  
\end{block}

\begin{itemize}
\item Decomposing neighborhood of each pathway
\item Inferring lethal crosstalk among pathways
\end{itemize}


\stitle{Modified Congruence Score (MCS)} {Evaluating Neighborhood Overlap of
Gene Pairs With Respect to a Given Pathway}

\begin{definition}
\begin{eqnarray*}
P-value (X = k_{ij}^B) &=& Prob(k_{ij}^B\leq X)  \\
&=&  HGT(k_{ij}^B| n_B, d_i^B, d_j^B) \\
&=& \sum_{x=k_{ij}^B}^{\min(d_i^B, d_j^B)} \frac{C(d_j^B, x) C(n_B-d_j^B, d_i^B-x)}{C(n_B, d_i^B)}
\end{eqnarray*}
\end{definition}
MCS is defined as $-log_{10}$ of the P-value.


\stitle{Modified Congruence Score (MCS)} {continued}
\begin{example}
  \begin{columns}[c]
    \column{.45\textwidth}
	\includegraphics[width=\columnwidth,height=.5\textheight]{figures/MCS}
	  
    \column{.45\textwidth}
	A sample neighborhood configuration for $v_i$ and $v_j$. Here $n = 15, D_i = 6, D_j = 5, n_B = 6, d_i = 3, d_j = 4$ and $k = 2$.
  \end{columns}  
\end{example}



\stitle{Constructing Neighborhood Overlap Graph \\ For a Given Pathway Pair}
\begin{definition}
The neighborhood overlap graph (NOG) of a given pathway $P_A$ with respect to
pathway $P_B$, denoted by $H_{A\rightarrow B} = (V_H, E_H)$, is an unweighted,
undirected graph defined over same vertices as $P_A$. In this graph, there is a
link between vertices $v_i$ and $v_j$ if the network structure around them
with respect to $P_B$ is statistically significant .
\end{definition}

\stitle{Pruning neighborhood overlap graph, finding cohesive subgraphs,
and identifying interaction ports}


  \begin{columns}[c]
    \column{.5\textwidth}
	\includegraphics[width=\columnwidth]{figures/KCore}\\
	\tiny{	Adopted from Batagelj and Zaversnik, 2002}		  
    \column{.5\textwidth}
    \begin{enumerate}
		\item Iterative peeling of K-shells\\Pruning hairy components
		\item Connected components in each core
		\item Evaluating the significance of components
			\begin{itemize}
				\item Evaluating significance using ER random graph model
			\end{itemize}		
	\end{enumerate}
  \end{columns}  
  
  


\stitle{KEGG Crosstalk Map}
  \begin{columns}[c]
    \column{.6\textwidth}
	\includegraphics[width=\columnwidth, height=.8\textheight]{figures/KEGG_final}
	  
    \column{.4\textwidth}
	\includegraphics[width=\columnwidth]{figures/Thresholding}
  \end{columns}  



\stitle{Interaction Port Case Study} {Crosstalk Between Protein Processing in ER and Proteasome}

 \begin{columns}[t]
    \column{.9\textwidth}
     \includegraphics[width=\textwidth]{figures/ProtER}
	  
  \end{columns}  
  


























\end{document}


