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\title[]
{Functional Characterization and Topological Modularity of Molecular Interaction Networks}

\author[Grama et al.]
{Ananth~Grama\inst{1} 
}

\institute[Purdue]
{
  \inst{1}%
  Center for Science of Information and \\
  Department of Computer Science, Purdue University
}

\date[Aug. 2011] 
{\tiny Various parts of this talk involved collaborations with Jayesh Pandey, Mehmet Koyuturk,
Shahin Mohammadi, Giorgos Kollias, and Shankar Subramaniam. Acknowledgements to the
National Science Foundation.}

\subject{Bioinformatics}

%\AtBeginSection[]
%{
%  \begin{frame}<beamer>{Outline}
%    \tableofcontents[currentsection]
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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}{Outline}
%  \tableofcontents
%\end{frame}


\section{Modularity and Function: Background}

\begin{frame}{Molecular Interaction Networks}
\begin{columns}
\column{0.6\textwidth}
\begin{itemize}
  \item Provides a high level description of cellular organization
  \item Directed and undirected graph representation
  \bigitem{Nodes represent cellular components}
  {
  	\item Protein, gene, enzyme, metabolite
  }
  \bigitem{Edges represent reactions or interactions}
  {
    \item Binding, regulation, modification, complex membership, substrate-product relationship
  }
\end{itemize}
\column{0.45\textwidth}
\centerline{\includegraphics[height=\textwidth]{./images/pinsample.eps}}
\centerline{\scriptsize{\hspace{0.2in} {\em S.cerevisiae}}}
\centerline{\scriptsize{Protein-Protein Interaction (PPI) Network}}
\end{columns}
\end{frame}

\begin{frame}{Function : Gene Ontology}
\begin{columns}[c]
\column{0.6\textwidth}
\begin{itemize}
\item{\blue{Molecular annotation} provides a unified understanding of the underlying principles}
%\item{Significant progress on standardizing knowledge at the molecular level}
\item Gene Ontology: A controlled vocabulary of molecular functions, biological 
processes, and cellular components
\item Terms (concepts) related by {\em is-a, part-of} relationships
\item If a molecule is annotated by a term, then it is also annotated by terms on
the paths towards root.
\end{itemize}
\column{0.45\textwidth}
\centerline{\includegraphics[width=0.8\textwidth]{images/go1.eps}}
\end{columns}
\end{frame}

\begin{frame}{Function \& Topology in Molecular Networks}
How does function relate to network topology?
\centerline{\includegraphics[width=0.8\textwidth]{images/functionaltopology.eps}}
\end{frame}

\begin{frame}{Prior Work on Topology and Function}
Understanding functional composition of biochemical networks
\begin{itemize}
\item
Conservation [ISMB 04/Bioinf. 04]
\item
Alignment [RECOMB 05/JCB 06]
\item
Modularity [RECOMB 06/JCB 07]
\item
Inference [Bioinf. 06]
\item
Pathway Annotation [ISMB 07/Bioinf. 07, PSB 08]
\item
Network Abstractions/ Annotations [ECCB 08/ Bioinf. 08]
\item
Modularity and Domain Interactions [APBC 10/ BMC Bioinf. 10]
\item
Pathway Interaction Maps [PSB 12, Submitted]
\end{itemize}

\end{frame}

\begin{frame}{Functional Coherence in Networks}
\begin{itemize}
	\item Modularity manifests itself in terms of high connectivity in the network
	\item Functional association (similarity) is correlated with network proximity
%	\item Protein-protein interaction (PPI) networks are used extensively for functional inference
%	\item In PPI networks, functional coherence manifests itself in terms of network proximity
	\item A measure for annotation proximity of nodes (semantic similarity)
	\item A measure for network distance
\end{itemize}
\centerline{
	\includegraphics[width=0.35\textwidth]{images/funmod.eps}
	\hspace{0.1\textwidth}
	\includegraphics[width=0.35\textwidth]{images/funpro.eps}
}
\centerline{\small{Sharan \etal, {\em MSB}, 2007}}

\end{frame}

\section{Modularity in PPIs and DDIs}

\begin{frame}{Assessing Functional Similarity}
\begin{columns}[c]
\column{0.5\textwidth}
\begin{itemize}
\item Gene Ontology (GO) provides a hierarchical taxonomy of biological process, molecular function and cellular component
\item Assessment of semantic similarity between concepts in a hierarchical taxonomy is well studied (Resnik, {\em IJCAI}, 1995)
\end{itemize}
\column{0.5\textwidth}
\centerline{\includegraphics[width=\textwidth]{images/go.eps}}
\end{columns}
\end{frame}

\begin{frame}{Semantic Similarity of GO Terms}
\begin{itemize}
\bigitem{Resnik's measure based on information content $I(c) = -\log_2(|G_c|/|G_r|)$
\begin{displaymath}
\delta_{I}(c_i, c_j) = \displaystyle\max_{c \in A_i \cap A_j} I(c)
\end{displaymath}
}
{\item$G_c$: Set of molecules that are associated with term $c$, $r$: Root term
\item $A_i$: Ancestors of term $c_i$ in the hierarchy
\item $\lambda(c_i, c_j)=\displaystyle\text{argmax}_{c \in A_i \cap A_j} I(c)$: Lowest common ancestor of $c_i$ and $c_j$
}
%\item {Normalize score:  
%$\delta_{JC}(c_i, c_j) = \frac{1}{1 - 2\delta_{I}(c_i, c_j) + I(c_i) + I(c_j)}$ (Jiang \& Conrath, {\em ICRCL}, 1997)}
\end{itemize}

\begin{columns}[c]
\column{0.22\textwidth}
\centerline{\includegraphics[height=\textwidth]{images/resnik.eps}}
\column{0.7\textwidth}
Resnik($c_3, c_4$) = Max(IC($c_1$), IC($c_2$)) \\
%JiangConrath($c_3, c_4$) = $\frac{1}{1 - 2*Resnik(c_3, c_4) + I(c_3) + I(c_4)}$ 
\end{columns}
\end{frame}

\begin{frame}{Functional Similarity of Molecules with Sets of Terms}
\begin{itemize}
%\item Each molecule has multiple GO terms
%\item Available annotations are incomplete
%\item Domain annotations are often derived from protein annotations
%\item Is it possible to compare functional similarity between domains and functional similarity between proteins at all?
\item{Average (Lord \etal, {\em Bioinformatics}, 2003)
\begin{displaymath}
\small
\rho_A(S_i, S_j) = \frac{1}{|S_i||S_j|}\displaystyle\sum_{c_k \in S_i} \displaystyle\sum_{c_l \in S_j} \delta(c_k, c_l)
\end{displaymath}}
\bigitem{Generalize the concept of lowest common ancestor to sets of terms (Pandey \etal, {\em ECCB}, 2008)
\begin{displaymath}
\small
\Lambda(S_i, S_j) = \displaystyle\bigsqcup_{c_k\in S_i, c_l\in S_j}{\lambda(c_k,c_l)}
\end{displaymath}
\begin{displaymath}
\small
\rho_I(S_i, S_j) = I(\Lambda(S_i, S_j)) = -\log_2\left(\frac{|G_{\Lambda(S_i, S_j)}|}{|G_r|}\right)
\end{displaymath}
}
{\item $G_{\Lambda(S_i, S_j)}=\displaystyle\bigcap_{c_k \in \Lambda(S_i, S_j)}{G_{c_k}}$
is the set of molecules that are associated with all terms in the MCA set}
\end{itemize}
\end{frame}

\begin{frame}{Functional Coherence of Module}
\begin{itemize}
\item A set of molecules that participates in the same biological processes or functions
\item sub-network with dense intra-connections and sparse interconnections
\item Each module is associated with set of molecular entities, and each molecule
associated with set of terms.
\end{itemize}
\begin{columns}[c]
\column{0.5\textwidth}
\begin{center}
  \includegraphics[width=0.4\linewidth]{images/sample_tree.eps}\\
  \vspace{0.02in}
  $S_1 = \{c_4\}, S_2 = \{c_4\}$, \\
  $S_3 = \{c_4, c_6\}$, $S_4 = \{c_1, c_6\}$, \\
  $S_5 = \{c_1\}$, $S_6 = \{c_6\}$
\end{center}
\column{0.5\textwidth}
Sets:
\begin{itemize}
\item $\mathcal{R}_1=\{S_1, S_2, S_3, S_4\}$
\item $\mathcal{R}_2=\{S_1, S_2, S_3\}$
\item $\mathcal{R}_3=\{S_3, S_4\}$
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}{Existing Measure}
\begin{itemize}
\item{Average (Pu \etal, {\em Proteomics}, 2007)
\begin{displaymath}
\sigma_A(\mathcal{R}) = \displaystyle\frac{1}{n(n-1)/2}\displaystyle\sum_{1\leq i < j\leq n} \rho(S_i, S_j).
\end{displaymath}}
\item{
%Example: $\sigma_A(\{ c_4\}, \{ c_4\}, \{ c_4, c_6\}, \{ c_1, c_6\}) = $
Example: $\sigma_A(S_1, S_2, S_3, S_4) = $
\begin{displaymath}
\displaystyle\frac{1}{6}(3*\sigma_A(S_1, S_2, S_3) + \rho(S_3, S_4) +  \red{\rho(S_1, S_4)+ \rho(S_2, S_4)})
% $\rho(S_1, S_4) = \rho(S_2, S_4) = 0$
\end{displaymath}}
% \item N^2 comparisons!
\end{itemize}
\end{frame}

\begin{frame}{Generalized Information Content}
Extend the notion of the minimum common ancestor of pairs of terms to tuples of terms 
$\lambda(c_{i_1},\ldots, c_{i_n})=\textrm{argmax}_{c \in \cap_{k=1}^{n} A_{i_k}}{I(c)}$

\begin{displaymath}
\sigma_I(\mathcal{R}) = I(\Lambda(S_1, \ldots ,S_n))
= -\log_2\left(\frac{|G_{\Lambda(S_i, \ldots, S_j)}|}{|G_r|}\right).
\end{displaymath}
where
\begin{displaymath}
\Lambda(S_1, S_2, \ldots, S_n) = 
\displaystyle\bigsqcup_{c_{i_j}\in S_j, 1\leq j\leq n} 
{\lambda(c_{i_1},c_{i_2},\ldots,c_{i_n})}
\end{displaymath}
Example: $\sigma_I(S_1, S_2, S_3, S_4) = I(r) = 0$, no common ancestor!
\end{frame}

\begin{frame}{Weighted Information Content}
Weigh the information content of shared functionality by the number of molecules 
that contribute to the shared functionality 
\begin{displaymath}
\sigma_W(\mathcal{R}) = 1 - \displaystyle\frac{\displaystyle\sum_{1\leq i\leq n}{\sum_{c \in {\mathcal{A}'}_i}{I(c)}}}{\displaystyle\sum_{1\leq i\leq n}{ \sum_{c \in \mathcal{A}_i}{I(c)} }}
\end{displaymath}

\begin{columns}[c]
\column{0.5\textwidth}
$\sigma_W(S_1, S_2, S_3, S_4) = 0.86$
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/tree2.eps}\\
\end{center}
\column{0.5\textwidth}
$\sigma_W(S_1, S_2, S_3) = 0.75$
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/tree1.eps}\\
\end{center}
\end{columns}

\end{frame}


\begin{frame}{Accounting for Multiple Paths}
\begin{itemize}
\bigitem{Is "shortest path" a good measure of network proximity?}
{
	\item Multiple alternate paths might indicate stronger functional association
	\item In well-studied pathways, redundancy is shown to play an important role in robustness \& adaptation (\eg, genetic buffering)
}
\end{itemize}
\vspace{0.2in}
\centerline{\includegraphics[width=0.9\linewidth]{images/proximity.eps}}
\end{frame}

\begin{frame}{Random walks with restarts}
\begin{itemize}
\item Consider a random walker that starts on a source node {\em s}. At every tick,
the walker chooses randomly among available edges or goes back to node {\em s} with
probability {\em c}.
\end{itemize}
\vspace{0.2in}
\centerline{\includegraphics[width=0.6\linewidth]{images/rwr.eps}}
\end{frame}

\begin{frame}{Proximity Based On Random Walks}
\begin{itemize}
\item Simulate an infinite random walk with random restarts at protein $i$
\bigitem{Proximity between proteins $i$ and $j$ is given by the relative amount of time spent
at protein $j$
	\begin{displaymath}
	\Phi(0)=I,~\Phi(t+1) = (1-c) A \Phi(t) + c I,~ \Phi = \lim_{t\rightarrow \infty}{\Phi(t)}
	\end{displaymath}
}
{
	\item $\Phi(i,j)$: Network proximity between protein $i$ and protein $j$
	\item $A$: Stochastic matrix derived from the adjacency matrix of the network
	\item $I$: Identity matrix
	\item $c$: Restart probability
}
\item Define proximity between proteins $i$ and $j$ as $\{ \Phi(i,j) + \Phi(j,i) \}/2$
\end{itemize}
\end{frame}

%\section{Results}

\begin{frame}{Network Proximity \& Functional Similarity}
\centerline{\includegraphics[width=0.75\linewidth]{images/proxcomp.eps}}
\centerline{\small{Correlation between functional similarity}}
\centerline{\small{and network proximity}}
\end{frame}


\begin{frame}{Topological Proximity and Functional Similarity}
\begin{columns}
\column{0.6\textwidth}
\centerline{\includegraphics[width=0.9\textwidth]{images/proxDistA.eps}}
\column{0.6\textwidth}
\centerline{\includegraphics[width=0.9\textwidth]{images/proxDistB.eps}}
\end{columns}
\vspace{0.5in}
Comparison of the DDI and PPI networks with respect to the relation 
between semantic similarity vs proximity and network distance
\end{frame}

\begin{frame}{Comparison of Coherence Meaures}
\centerline{\includegraphics[width=0.7\linewidth]{images/coherence.eps}}
\centerline{Index of Dectectability vs. complex sizes}
\vspace{0.2in}
\small{\centerline{
$d(\sigma) = \frac{{\textrm{mean}_{t\in T}(\sigma(t)) - \textrm{mean}_{t\in C}(\sigma(t))}}
{\sqrt{((\textrm{std}_{t\in T}(\sigma(t)))^2+(\textrm{std}_{t\in C}(\sigma(t)))^2)/2}}$}}
\end{frame}

\begin{frame}{Lessons Learned}
  \begin{itemize}
%  \bigitem{Summary}
%  {
	\item Random walk based measures of topological proximity are better suited to existing interaction data
	\item Measures that quantify coherence among entire sets are superior to aggregares of known pair-wise measures
%  }
%  \item Future work : Using proximity measure to identify disease implicated genes in networks
%  \item
  \end{itemize}
\end{frame}

%\begin{frame}{Acknowledgments}
%
%\begin{itemize}
% \item \blue{Mehmet Koyut\"urk} (Case Western Reserve)
% \item \blue{Jayesh Pandey} (Purdue)
% \item \blue{Prof. Shankar Subramaniam} (UCSD)
%\end{itemize}
%
%\begin{itemize}
% \item NIH \& NSF
%\end{itemize}
%
%\end{frame}




\begin{frame}{Part II}

Building Pathway Maps Using Synthetic Lethality Networks

\end{frame}

\section{Pathway Maps: Background}
\subsection{Models and Definitions}
\begin{frame}{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}
\end{frame}



\begin{frame}{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}

\end{frame}


\begin{frame}{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}  

\end{frame}



\subsection{Datasets}
\begin{frame}{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}  
\end{frame}



\begin{frame}{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}  
\end{frame}


%\section{Motivation}
\subsection{Functional Similarity of Gene Pairs}

\begin{frame}{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).}

\end{frame}



\subsection{Performance Evaluation and Identifying Shortcomings}

\begin{frame}{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
\end{frame}


\begin{frame} {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}

\end{frame}

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

\end{frame}



\section{Constructing KEGG Crosstalk Map}
\subsection{Overview}
\begin{frame}{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}

\end{frame}

\subsection{Methods}
\begin{frame}{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.

\end{frame}

\begin{frame}{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}

\end{frame}


\begin{frame}{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}
\end{frame}

\begin{frame}{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}  
  
  
\end{frame}


\subsection{Results and Discussions}
\begin{frame}{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}  
\end{frame}



\begin{frame}{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{frame}


%\section*{Concluding Remarks}

\begin{frame}{Summary}

  % Keep the summary *very short*.
  \begin{itemize}
  \item
    The \alert{local neghborhood similarity} gives heterogeneous performance in predicting co-pathways membership of gene pairs.
  \item
    This phenomena is due to the specific structure around enriched pathways
  \item
	\alert{Decomposing the neighborhood} around each pathway sheds light on the cellular machinery.
  \end{itemize}
  
  % The following outlook is optional.
  \vskip0pt plus.5fill
  \begin{itemize}
  \item
    Future works:
    \begin{itemize}
    \item
      Analysing the hierarchy of ports instead of the most significant interaction port.
    \item
		Using our methodology to uncover dependencies among functional pathways.
    \end{itemize}
  \end{itemize}
\end{frame}



\appendix
\section<presentation>*{\appendixname}
\subsection<presentation>*{For Further Reading}

\begin{frame}[allowframebreaks]
  \frametitle<presentation>{For Further Reading}
    
  \begin{thebibliography}{10}
    
%  \beamertemplatebookbibitems
 
  \beamertemplatearticlebibitems
  \bibitem{Costanzo}
    M.~Costanzo et al.
    \newblock The Genetic Landscape of a Cell
    \newblock {\em Science}, 425
    2010.

  \bibitem{Dixon}
	SJ.~Dixon et al.
    \newblock Systematic Mapping of Genetic Interaction Networks
    \newblock {\em Annual review of genetics}, 43, 601
    2009.

  \bibitem{Segre}
	D.~Segre et al.
    \newblock Modular Epistasis in Yeast Metabolism
    \newblock {\em Nature Genetics}, 37, 77
    2005.

  \bibitem{Tucker}
	C.L.~Tucker and S.~Fields
    \newblock Lethal Combinations
    \newblock {\em Nature Genetics}, 35, 204
    2003.
        
%  \bibitem{CS}
%	P.~Ye et al.
%    \newblock Gene function prediction from congruent synthetic lethal interactions in yeast
%    \newblock {\em Molecular Systems Biology }, 1, msb4100034
%    2005.


  \end{thebibliography}
\end{frame}

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