\documentclass[article]{bioinf}

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\hypersetup{colorlinks=false,
   pdfborder=0 0 0,
   pdftitle={Rchemcpp -- Similarity measures for chemical compounds},
   pdfauthor={G\"unter Klambauer}}

\title{Rchemcpp -- Similarity measures for chemical compounds}
\author{Michael Mahr and G\"unter Klambauer}
\affiliation{Institute of Bioinformatics, Johannes Kepler University
Linz\\Altenberger Str. 69, 4040 Linz, Austria\\
\email{klambauer@bioinf.jku.at}}

%\VignetteIndexEntry{Rchemcpp}
%\VignetteKeywords{Kernel, kernels, graph, graphs, similarity, molecules, compounds}



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\begin{document}
<<echo=FALSE>>=
options(width=75)
set.seed(0)
library(Rchemcpp)
RchemcppVersion <- packageDescription("Rchemcpp")$Version
@
\newcommand{\RchemcppVersion}{\Sexpr{RchemcppVersion}}
\manualtitlepage[Version \RchemcppVersion, \today]


\vspace{1cm}

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\newcommand{\notebox}[1]{%
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\section{Introduction}

The \Rchemcpp\ package \citep{Klambauer2015} implements functions that compute
similarities between small drug-like molecules. Similarity measures for compounds are highly relevant
since they offer the possibility to find chemical clusters or to build models
that predict the biological activities and responses of a new drug. Further a 
database can be screened for structural analogs to given lead compound. This is
also crucial in drug-design.
This package extends the functionality of the {\tt ChemmineR} Bioconductor 
package using the functions and data structures that are introduced there, such
as the reading of molecule structures from SDF files and handling them as
S4 objects in \R.

The following similarity functions/kernels are implemented:
\begin{itemize}
\item the marginalized graph kernel between labeled graphs \citep{Kashima2004}.
\item extensions of the marginalized kernel \citep{Mahe2004}.
\item Tanimoto kernels \citep{Ralaivola2005}.
\item graph kernels based on tree patterns \citep{Mahe2009a}.
\item kernels based on pharmacophores for 3D structure of molecules \citep{Mahe2006}.
\end{itemize}

See \url{http://www.bioinf.jku.at/software/Rchemcpp/} for additional
information. A web-service for finding structural analogs using \Rchemcpp\ is
available at \url{http://www.bioinf.jku.at/services/analoging/}.

\section{Getting started and quick start}

To load the package, enter the following in your \R\ session:
<<>>=
library(Rchemcpp)
@

We enter the filename of and SDF file to the function {\tt sd2gram}. This function
computes the similarity of the molecules with the marginalized kernel 
\citep{Kashima2004} approach.\\

<<results=hide>>=
sdfolder <- system.file("extdata",package="Rchemcpp")
sdf <- list.files(sdfolder,full.names=TRUE,pattern="small")
K <- sd2gram(sdf)
@

The similarity values are now stored in {\tt K}. We visualize this matrix as
a heatmap.

<<eval=FALSE>>=
heatmap(K,Rowv=NA,Colv=NA,scale="none")
@


<<fig=FALSE,echo=FALSE,results=hide>>=
pdf("001.pdf")
heatmap(K,Rowv=NA,Colv=NA,scale="none",margins=c(12,12))
dev.off()
@

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\columnwidth]{001.pdf}
\end{center}
\end{figure} 



\newpage

\section{Molecular similarity for clustering}
Based on the similarity measure we can run clustering algorithms on the data
in order to find groups among the molecules. We use Affinity Propagation 
Clustering \citep{Frey2007} as implemented by \citet{Bodenhofer2011} 
for this task, because the cluster centers are real molecules. 

<<>>=
library(apcluster)
r <- apcluster(K)
@


<<eval=FALSE>>=
heatmap(r,K)
@

<<fig=FALSE,echo=FALSE,results=hide>>=
pdf("002.pdf")
heatmap(r,K,margins=c(12,12))
dev.off()
@

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{002.pdf}
\end{center}
\end{figure} 


\section{Molecular similarity for prediction of chromosome aberration}

The similarity measures can be used for building models to predict features of
the molecules, for example with the help of Support Vector Machines (SVMs). 
\citet{Mohr2010} used this approach to predict the Ames test
\citep{Mortelmans2000}, that tests whether a compound leads to chromosome 
aberration. The authors put together a data set of 940 compounds with known 
results of the Ames test. This data set is included in this package.

<<results=hide>>=
library(kernlab)
chromosomeAberrationDataSet <- list.files(sdfolder,
		full.names=TRUE,pattern="Mohr.sdf")
KCA <- sd2gramSpectrum(chromosomeAberrationDataSet,
		detectArom=FALSE,depthMax=4,silentMode=TRUE)
response <- getMoleculePropertyFromSDF(chromosomeAberrationDataSet,
		"chromosome_damage")
# C was set to 0.1 for computational speed - should be set to a higher value
model <- ksvm(KCA,y=as.factor(response),kernel="matrix",cross=10,type="C-svc",C=0.1)
@

An estimated accuracy of the prediction is
<<>>=
print(1-model@cross)
@

With this model we can predict whether a certain molecule will be positive 
in the Ames test
<<>>=
predict(model,as.kernelMatrix(KCA[3,SVindex(model),drop=FALSE]))
@


\section{Working with SDFset objects from the package ChemmineR}
In the {\tt ChemmineR} package {\tt SDFset} objects are defined, that make 
the handling of molecules in \R\ easy and provide a lot of utility functions,
like plotting.

<<fig=TRUE>>=
library(ChemmineR)
sdfFileName <- list.files(sdfolder,full.names=TRUE,pattern="small")
sdfSet <- read.SDFset(sdfFileName)
plot(sdfSet[2],print=FALSE)
@

It is possible to run the molecule kernels directly on {\tt SDFset} objects.
<<>>=
K1 <- sd2gramSubtree(sdfSet,silentMode=TRUE)
@

\section{Compound Screening: Comparing one compound against a data set}

Comparing the objects in one set (lead compound) to the objects in another set
(database of compounds) is possible.
<<>>=
leadCompound <- sdfSet[1]
compoundDataBase <- sdfSet[1:20]
K2 <- sd2gramSubtree(leadCompound,compoundDataBase,silentMode=TRUE)
@


\section{Graph kernels}
These kernels are based on sets of molecular fragments. Molecular fragments 
can be either {\em walks}, i.e. a sequence of atoms 
connected by bonds, or {\em subtrees}, i.e. directed tree-patterns. A molecule
is represented as graph. If we consider 
two molecules as graphs $X$ and $Y$, then the kernel $K$ is:
\begin{align}
K(X,Y)=\sum_{p\in \mathcal{P}} N(p,X) \cdot N(p,Y),
\end{align}

where $\mathcal{P}$ is the set of molecular fragments (all possible walks or
subtrees, and the function $N(p,X)$ usually (e.g. for the spectrum kernel \ref{s:spectrum})
counts, how often the pattern $p$ occurs in molecule graph $X$. The function
$N(p,X)$ will be defined for the different kernel types in the following.

Kernel similarity measures can be normalized to values between 0 and 1. 
Usually the following kernel normalization is used
 \begin{align}
    K_{\rm norm}(X,Y)=\frac{K(X,Y)}{\sqrt{K(X,X) \cdot K(Y,Y)}}.
  \end{align}
  

\subsection{Implementation}
Graph kernels are efficiently implemented in {\tt Chemcpp}\citep{Mahe2007b} or its {\tt R} interface 
\Rchemcpp\ \citep{Klambauer2015}.

\begin{itemize}
 \item {\bf sd2gram} Implements the marginalized kernel \ref{s:marginalized}.
 \item {\bf sd2gramSpectrum} Implements the spectrum kernel \ref{s:spectrum},
 the Tanimoto kernel \ref{s:tanimoto}, the MinMax kernel \ref{s:minmax}, 
 the lambda-k kernel \ref{s:lambdak} and the
 approximation of the marginalized kernel \ref{s:marginalizedapprox}.
 \item {\bf sd2gramSubtree} Implements the subtree kernel \ref{s:subtree}.
\end{itemize}

For all these kernels the set of molecular fragments are non-tottering walks of
or up to a certain length $n$ given by the {\tt depthMax} parameter. With the
option {\tt onlyDepthMax=FALSE} all molecular fragments up to a length of $n$
atoms/vertices are in the set $\mathcal{P}$, if {\tt onlyDepthMax=TRUE} only
molecular fragments with $n$ atoms/vertices are in the set $\mathcal{P}$. 
For all kernels there is a parameter {\tt returnNormalized} that scales the 
similarity measures to values between 0 and 1.



\subsection{The Spectrum Kernel}
\label{s:spectrum}
Here the function $N(p,X)$ counts how often the
  walk $p$ occurs in the graph $X$. The kernel is
  \begin{align}
 K(X,Y)=\sum_{p\in \mathcal{P}} N(p,X) \cdot N(p,Y),
 \end{align}
 with 
  \begin{align}
    N(p,X)=\#\{p \in X \}.
  \end{align}
  For simplicity we denote a function $\#$ that the number of occurences of walk
  $p$ in graph $X$.
  The normalized version of this kernel is:
  \begin{align}
    K_{\rm norm}(X,Y)=\frac{K(X,Y)}{\sqrt{K(X,X) \cdot K(Y,Y)}}
  \end{align}
  
\subsection{The Tanimoto Kernel}
\label{s:tanimoto}

Here the function $N(p,X)$ indicates whether
  walk $p$ occurs in the graph $X$. 
  The kernel is
  \begin{align}
 K(X,Y)=\sum_{p\in \mathcal{P}} N(p,X) \cdot N(p,Y),
 \end{align}
 with 
  \begin{align}
    N(p,X)=\mathbf{1}\{p \in X \}.
  \end{align}
  For simplicity we denote an indicator function $\mathbf{1}$ that is one if the walk
  $p$ occurs in the graph $X$ and is zero otherwise.
   What is usually considered as the Tanimoto kernel \citep{Ralaivola2005} is the normalized version:
  \begin{align}
    K_{\rm Tanimoto}(X,Y)=\frac{K(X,Y)}{K(X,X)+K(Y,Y)-K(X,Y)}.
  \end{align}
  
  
\subsection{The MinMax Kernel}
\label{s:minmax}

This kernel \citep{Ralaivola2005} is a variation of the kernel
    presented above. Here the function $N(p,X)$ counts how often the
  walk $p$ occurs in the graph $X$.
  \begin{align}
      K_{\rm max}(X,Y)=\sum_{p\in \mathcal{P}} \max(N(p,X),N(p,Y)), \\
      K_{\rm min}(X,Y)=\sum_{p\in \mathcal{P}} \min(N(p,X),N(p,Y)) \\
  \end{align}
   with 
   \begin{align}
    N(p,X)=\#\{p \in X \}
   \end{align}
 the number of occurences of walk $p$ in the 
   graph $X$. The MinMax kernel is the already a normalized version:
  \begin{align}
    K_{\rm minmax}(X,Y)=\frac{K_{\rm min}(X,Y)}{K_{\rm max}(X,Y)}.
  \end{align}

\subsection{The Marginalized Kernel}
\label{s:marginalized}

  The marginalized kernel suggested by \citet{Kashima2003, Kashima2004} .
  Here the function $N(p,X)$ counts how often the walk $p$ occurs in the graph
  $X$ and weights it by the probability that it occurs. The kernel is
  \begin{align}
  \label{eq:marg}
 K(X,Y)=\sum_{p\in \mathcal{P}} N(p,X) \cdot N(p,Y),
 \end{align}
 with 
  \begin{align}
          N(p,X)=\sum_{h\in \mathcal{H(X)}}  w(h,X)\cdot  \mathbf{1}(h=p),
  \end{align}
   where $\mathcal{H(X)}$ is the set of walks of graph $X$, and $w(h,X)$ is
   the probability that the walk $h$ occurs in $X$. This probability is
   influenced by the stopping probability (parameter {\tt stopP}). The indicator
   function $\mathbf{1}(h=p)$ is one if the atoms and bonds of the walk $h$ 
   match the given walk $p$. The sum in Eq.~\ref{eq:marg} is a sum over an infinite
   number of walks. However, the probability of a walk decreases exponentially
   with its length, therefore the kernel converges.

\subsection{The Marginalized Kernel Approximation}
\label{s:marginalizedapprox}

This is an approximation of 
  the marginalized kernel \citep{Kashima2003,Kashima2004} presented above, but
  the set of walks here is finite, since the length of the walks is bounded by
  $n$.
  Here the function $N(p,X)$ counts how often the walk $p$ occurs in the graph
  $X$ and weights it by the probability that it occurs. The kernel is
  \begin{align}
 K(X,Y)=\sum_{p\in \mathcal{P}} N(p,X) \cdot N(p,Y),
 \end{align}
 with 
  \begin{align}
          N(p,X)=\sum_{h\in \mathcal{H(X)}}  w(h,X)\cdot  \mathbf{1}(h=p),
  \end{align}
   where $\mathcal{H(X)}$ is the set of walks of graph $X$, and $w(h,X)$ is
   the probability that the walk $h$ occurs in $X$. This probability is
   influenced by the stopping probability (parameter {\tt stopP}). The indicator
   function $\mathbf{1}(h=p)$ is one if the atoms and bonds of the walk $h$ 
   match the given walk $p$.
   
   Because the length of the walks is limited by the parameter $n$, this is
   only an approximation of the marginalized kernel. Note that because of the 
   random walk process, the probabilities of the walkds exponentially decrease
   with their lengths. Long walks are barely taken into account in the marginalized
   kernel formulation. 
     The normalized version of this kernel is:
  \begin{align}
    K_{\rm norm}(X,Y)=\frac{K(X,Y)}{\sqrt{K(X,X) \cdot K(Y,Y)}}
  \end{align}
  
  
\subsection{The Lambda-k Kernel}
\label{s:lambdak}

  Here the function $N(p,X)$ counts how often the walk $p$ occurs in the graph
  $X$ and weights it by a function of the length of the walk. The kernel is
  \begin{align}
 K(X,Y)=\sum_{p\in \mathcal{P}} N(p,X) \cdot N(p,Y),
 \end{align}
 with 
  \begin{align}
          N(p,X)=\lambda^{|p|} \cdot \#\{p \in X \}.
  \end{align}
   For simplicity we denote a function $\#$ that the number of occurences of walk
  $p$ in graph $X$. For $\lambda=1$ this kernel corresponds exactly to the spectrum
   kernel. For $\lambda > 1$ the influence of longer walks is higher, and for $\lambda < 1$ 
   the similarity measure is more influenced by shorter walks.
     The normalized version of this kernel is:
  \begin{align}
    K_{\rm norm}(X,Y)=\frac{K(X,Y)}{\sqrt{K(X,X) \cdot K(Y,Y)}}
  \end{align}
  
\subsection{Efficiency of walk kernels}
A walk of a graph $X=(V,E)$ with vertices $V$ and edges $E$ is a sequence $v_1,\ldots,v_n \in V$
such that $(v_i,v_{i+1}) \in E$ for $i=1,\ldots,n-1$. This implementation mostly uses non-tottering
walks, that is walks with $v_i \neq v_{i+2}$ or provides an option to remove tottering walks.
We denote $\mathcal{H}_n(X)$ the set of walks of graph $X$ with length $n$.
The computation of these kernels 
is very efficient, since there is a bijection between the pairs of walks 
$p \in \mathcal{H}_n(X)$ and $q \in  \mathcal{H}_n(Y)$ with the same label sequences and
the walks on the product graph $r \in \mathcal{H}_n (X \times Y)$. Therefore we only have
to count the walks with length $n$ on the product graph of X and Y. 

The product graph between two labeled graphs $X=(V_X,E_X)$ and $Y=(V_Y,E_Y)$ is defined
as the graph $Z=(V_{X\times Y},E_{X\times Y})$ with 
\begin{align}
 V_{X\times Y}&=\{(v_1,v_2)\in V_X \times X_Y:  l(v_x)=l(v_y) \} \\
 E_{X\times Y}&=\{ \left( (v_1,v_2),(  v_1^\prime, v_2^\prime) \right) \in V_X \times X_Y 
 : (v_1, v_1^\prime) \in X\  \text{and\ } (v_2, v_2^\prime) \in Y \},
\end{align}

where $l(v)$ denotes the label of the vertex $v$. For a graphical explanation 
of the definition of the producte graph, see Fig.~\ref{fig:productgraph}.


\begin{figure}
  \centering
 \includegraphics[width=0.8\columnwidth]{Figures/productgraph.png}
 \label{fig:productgraph}
 \caption{An example for a product graph. Figure taken from \citet{Vert2007Presentation}.}
\end{figure}


\subsection{The Subtree Kernel}
\label{s:subtree}
This graph kernel is  based on the detection of common subtrees: the
so-called tree-pattern graph kernels, originally introduced in \citep{Ramon2003}, and revisited 
in \citep{Mahe2006}. For more details on the kernel definitions please refer to \citep{Mahe2006}. 
The size of the trees can either be bounded by the size of the tree (number of atoms) or 
the number of branches. Both trees of a given length and up to a given length can be used 
for the calculation of the kernel.

\section{The Pharmacophore Kernel}
The pharmacophore kernel is based on the 3D structure of molecules \citep{Mahe2006}.
The kernels are based on the comparison of the three-point pharmacophores 
present in the 3D structures of molecules, a set of molecular features known to 
be particularly relevant for virtual screening applications.  
There is a computationally demanding exact implementation of
these kernels, as well as fast approximations related to the classical fingerprint-based
approaches. 

\begin{itemize}
 \item {\bf sd2gram3Dpharma} Implements the exact pharmacophore kernel.
 \item {\bf sd2gram3Dspectrum} Implements some approximations of the pharmacophore
 kernel.
\end{itemize}





\section{How to cite this package}

If you use this package for research that is published later, you are kindly
asked to cite it as follows: \citep{Klambauer2015}.

To obtain Bib\TeX\ entries of the reference, you can enter the following
into your R session:
<<eval=FALSE>>=
toBibtex(citation("Rchemcpp"))
@ 

\tiny
\begin{verbatim}
@article{Klambauer2015,
author = {Klambauer, Guenter and Wischenbart, Martin and Mahr, Michael and Unterthiner, Thomas and Mayr, Andreas and Hochreiter, Sepp}, 
title = {Rchemcpp: a web service for structural analoging in ChEMBL, Drugbank and the Connectivity Map},
year = {2015}, 
doi = {10.1093/bioinformatics/btv373}, 
URL = {http://bioinformatics.oxfordjournals.org/content/early/2015/06/17/bioinformatics.btv373.abstract}, 
eprint = {http://bioinformatics.oxfordjournals.org/content/early/2015/06/17/bioinformatics.btv373.full.pdf+html}, 
journal = {Bioinformatics} 
}
\end{verbatim}
\normalsize


\begin{thebibliography}{}

\bibitem[Bodenhofer {\em et al.}(2011)Bodenhofer, Kothmeier, and
  Hochreiter]{Bodenhofer2011}
Bodenhofer, U., Kothmeier, A., and Hochreiter, S. (2011).
\newblock {APCluster: an R package for affinity propagation clustering}.
\newblock {\em Bioinformatics\/}, {\bf 27}, 2463--2464.

\bibitem[Frey and Dueck(2007)Frey and Dueck]{Frey2007}
Frey, B. J. and Dueck, D. (2007).
\newblock Clustering by passing messages between data points.
\newblock {\em Science\/}, {\bf 315}, 972--977.

\bibitem[Kashima {\em et al.}(2003)Kashima, Tsuda, and Inokuchi]{Kashima2003}
Kashima, H., Tsuda, K., and Inokuchi, A. (2003).
\newblock Marginalized kernels between labeled graphs.
\newblock In {\em Proceedings of the Twentieth International Conference on
  Machine Learning\/}, pages 321--328. AAAI Press.

\bibitem[Kashima {\em et al.}(2004)Kashima, Tsuda, and Inokuchi]{Kashima2004}
Kashima, H., Tsuda, K., and Inokuchi, A. (2004).
\newblock Kernels for graphs.
\newblock In {\em Kernel Methods in Computational Biology\/}, pages 155--170.
  MIT Press.

\bibitem[Klambauer {\em et al.}(2015)Klambauer, Wischenbart, Mahr, Unterthiner,
  Mayr, and Hochreiter]{Klambauer2015}
Klambauer, G., Wischenbart, M., Mahr, M., Unterthiner, T., Mayr, A., and
  Hochreiter, S. (2015).
\newblock Rchemcpp: a web service for structural analoging in chembl, drugbank
  and the connectivity map.
\newblock {\em Bioinformatics\/}.

\bibitem[Mah{\'e} and Vert(2009)Mah{\'e} and Vert]{Mahe2009a}
Mah{\'e}, P. and Vert, J.-P. (2009).
\newblock Graph kernels based on tree patterns for molecules.
\newblock {\em Mach. Learn.}, {\bf 75}(1), 3--35.

\bibitem[Mah{\'e} {\em et al.}(2004)Mah{\'e}, Ueda, Akutsu, Perret, and
  Vert]{Mahe2004}
Mah{\'e}, P., Ueda, N., Akutsu, T., Perret, J.-L., and Vert, J.-P. (2004).
\newblock Extensions of marginalized graph kernels.
\newblock In R. Greiner and D. Schuurmans, editors, {\em Proc of the 21st
  {ICML}\/}, pages 552--559. ACM Press.

\bibitem[Mah{\'e} {\em et al.}(2006)Mah{\'e}, Ralaivola, Stoven, and
  Vert]{Mahe2006}
Mah{\'e}, P., Ralaivola, L., Stoven, V., and Vert, J.-P. (2006).
\newblock The pharmacophore kernel for virtual screening with support vector
  machines.
\newblock {\em J Chem Inf Model\/}, {\bf 46}(5), 2003--2014.

\bibitem[Mah{\'e} {\em et al.}(2007)Mah{\'e}, Perret, Akutsu, and
  Vert]{Mahe2007b}
Mah{\'e}, P., Perret, J.-L., Akutsu, T., and Vert, J.-P. (2007).
\newblock {Chemcpp: A C++ toolbox for chemoinformatics focusing on the
  computation of kernel functions between chemical compounds}.
\newblock {\em http://chemcpp.sourceforge.net/html/index.html\/}.

\bibitem[Mohr {\em et al.}(2010)Mohr, Jain, Sutter, Laak, Steger-Hartmann,
  Heinrich, and Obermayer]{Mohr2010}
Mohr, J., Jain, B., Sutter, A., Laak, A. T., Steger-Hartmann, T., Heinrich, N.,
  and Obermayer, K. (2010).
\newblock A maximum common subgraph kernel method for predicting the chromosome
  aberration test.
\newblock {\em J Chem Inf Model\/}, {\bf 50}(10), 1821--1838.

\bibitem[Mortelmans and Zeiger(2000)Mortelmans and Zeiger]{Mortelmans2000}
Mortelmans, K. and Zeiger, E. (2000).
\newblock The ames salmonella/microsome mutagenicity assay.
\newblock {\em Mutation Research/Fundamental and Molecular Mechanisms of
  Mutagenesis\/}, {\bf 455}(1), 29 -- 60.

\bibitem[Ralaivola {\em et al.}(2005)Ralaivola, Swamidass, Saigo, and
  Baldi]{Ralaivola2005}
Ralaivola, L., Swamidass, S. J., Saigo, H., and Baldi, P. (2005).
\newblock Graph kernels for chemical informatics.
\newblock {\em Neural Netw\/}, {\bf 18}(8), 1093--1110.

\bibitem[Ramon and G\"artner(2003)Ramon and G\"artner]{Ramon2003}
Ramon, J. and G\"artner, T. (2003).
\newblock Expressivity versus efficiency of graph kernels.
\newblock In {\em Proceedings of the First International Workshop on Mining
  Graphs, Trees and Sequences\/}, pages 65--74.

\bibitem[Vert(2007)Vert]{Vert2007Presentation}
Vert, J. P. (2007).
\newblock Graph kernels and applications in chemoinformatics.
\newblock In {\em {International Workshop on Graph-based Representations in
  Pattern Recognition (Gbr 2007), Alicante, Spain, June 13, 2007}\/}.

\end{thebibliography}

\end{document}
