%% BioMed_Central_Tex_Template_v1.06 %% % % bmc_article.tex ver: 1.06 % % % %%IMPORTANT: do not delete the first line of this template %%It must be present to enable the BMC Submission system to %%recognise this template!! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% LaTeX template for BioMed Central %% %% journal article submissions %% %% %% %% <14 August 2007> %% %% %% %% %% %% Uses: %% %% cite.sty, url.sty, bmc_article.cls %% %% ifthen.sty. multicol.sty %% %% %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% For instructions on how to fill out this Tex template %% %% document please refer to Readme.pdf and the instructions for %% %% authors page on the biomed central website %% %% http://www.biomedcentral.com/info/authors/ %% %% %% %% Please do not use \input{...} to include other tex files. %% %% Submit your LaTeX manuscript as one .tex document. %% %% %% %% All additional figures and files should be attached %% %% separately and not embedded in the \TeX\ document itself. %% %% %% %% BioMed Central currently use the MikTex distribution of %% %% TeX for Windows) of TeX and LaTeX. This is available from %% %% http://www.miktex.org %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \NeedsTeXFormat{LaTeX2e}[1995/12/01] \documentclass[10pt,twocolumns]{bmc_article} % Load packages \usepackage{cite} % Make references as [1-4], not [1,2,3,4] \usepackage{url} % Formatting web addresses \usepackage{ifthen} % Conditional \usepackage{multicol} %Columns \usepackage[utf8]{inputenc} %unicode support \usepackage{epsfig} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \usepackage{easymat} \usepackage{lscape} \usepackage{array} \usepackage{multirow} \usepackage{arydshln} \usepackage{multirow} \usepackage{verbatim} \usepackage{algorithmic} \usepackage{algorithm} \usepackage{fancybox} \usepackage{graphicx} \usepackage{algorithmic} \usepackage{array} \usepackage{epstopdf} \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `basename #1 .tif`.png} %\usepackage{color,soul} %\definecolor{lightgreen}{rgb}{.93,1,.9} %\definecolor{lightyellow}{rgb}{1,1,.5} %\sethlcolor{lightyellow} \newtheorem{thm}{Theorem}[subsection] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{defn}[thm]{Definition} \newtheorem{remark}[thm]{Remark} %\usepackage[applemac]{inputenc} %applemac support if unicode package fails %\usepackage[latin1]{inputenc} %UNIX support if unicode package fails \urlstyle{rm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% If you wish to display your graphics for %% %% your own use using includegraphic or %% %% includegraphics, then comment out the %% %% following two lines of code. %% %% NB: These line *must* be included when %% %% submitting to BMC. %% %% All figure files must be submitted as %% %% separate graphics through the BMC %% %% submission process, not included in the %% %% submitted article. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\def\includegraphic{} %\def\includegraphics{} \setlength{\topmargin}{0.0cm} \setlength{\textheight}{21.5cm} \setlength{\oddsidemargin}{0cm} \setlength{\textwidth}{16.5cm} \setlength{\columnsep}{0.6cm} \newboolean{publ} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% You may change the following style settings %% %% Should you wish to format your article %% %% in a publication style for printing out and %% %% sharing with colleagues, but ensure that %% %% before submitting to BMC that the style is %% %% returned to the Review style setting. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Review style settings \newenvironment{bmcformat}{\begin{raggedright}\baselineskip20pt\sloppy\setboolean{publ}{false}}{\end{raggedright}\baselineskip20pt\sloppy} %Publication style settings %\newenvironment{bmcformat}{\fussy\setboolean{publ}{true}}{\fussy} % Begin ... \begin{document} \begin{bmcformat} \section*{\it {Supplementary materials}} %%%%%%%%%%%%% %Mutated TPM decomposition %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection*{Mutated TPM decomposition} Let $\mathbf{P}$ be the probability transition matrix of a BNp defined on $n$ genes with perturbation probability $p$ and suppose the network undergoes a single mutation, $(i,k)$. Then the transition probability matrix $\tilde{\mathbf{P}}$ of the mutated BNp possesses the decompostion \begin{equation} \tilde{\mathbf{P}}= \mathbf{P}% +(1-p)^{n}\mathbf{Q}(\mathbf{T}_{i,k}-\mathbf{I}) = \mathbf{P}+(1-p)^{n}.\sum_{j=1}^{u}{\mathbf{a}_{j}.% \mathbf{b}_{j}^{T}}, \label{decompositions} \end{equation}% for some vectors $\mathbf{a}_{j}$ and $\mathbf{b}_{j}$ satisfying $\mathbf{b}% _{j}^{T}.\mathbf{e}=0$, where $u\leq 2^{n-1}$ is a positive integer. Proof: Let $(x_1,...,x_n)$ denote the GAP of the network. We note that there is a natural bijection between GAP and its decimal representation, which takes values in $\{0,1,...,2^{n}-1\}$ which for the sake of simplicity is changed to $\{1,...,2^n\}$. Let $\mathcal S = \left\{{1,...,{2^n}}\right\} $ denote the state space. Based on the given mutation $(i,k)$, we can partition state space as follows: \[ \mathcal {P_S} = \left\{ s_1,...s_m\right\}\] where $m$ and $s_j$s are defined as \[ \begin{array}{c} m = \frac{2^n}{r}\\ s_j = \left\{{(j-1)r+1},...,{jr}\right\}, j=1,...,m\\ \end{array} \] where $r = 2^{n-i+1}.$ Now, suppose we define the transformation matrix as ${\bf T}_{i,k} = \left[{\bf t}_1,...,{\bf t}_{2^n}\right]$. In order to find ${\bf T}_{i,k}$, we split it in two cases: \begin{itemize} \item $k=0$: we define a bijection mapping from $s_{2j}$ to $s_{2j-1}$ for all $j=1,...,\frac{m}{2}$ as \[ {(2j-2)r+l} \leftarrow {(2j-1)r+l}, \forall l \in\left\{{1,...,r}\right\}, \] which means that the column corresponding to state ${(2j-1)r+l}$ in matrix $Q$ should be added to the column ${(2j-2)r+l}$ of this matrix, while the ${(2j-1)r+l}$th column should be changed to the all-zero vector. So, assuming ${\bf T}_{i,k}=[{\bf t}_1,{\bf t}_2,...,{\bf t}_{2^n}]$, we would have \begin{equation}\label{t-for-zero} {\bf t}_u = \begin{cases} {\bf 0} &; \forall u \in s_{2j}\\ \begin{cases} t_{u,u}=1& \\t_{v,u}=1;&\forall v \in s_{2j}\\t_{v,u}=0;& \text{otherwise} \end{cases} &; \forall u \in s_{2j-1}\end{cases} ; \forall j=1,...,\frac{m}{2}. \end{equation} \item $k=1$: in this case we have another bijection mapping from $s_{2j-1}$ to $s_{2j}$ for all $j=1,...,\frac{m}{2}$ as follows \[ {(2j-2)r+l} \rightarrow {(2j-1)r+l}, \forall l \in\left\{{1,...,r}\right\}. \] then we would have \begin{equation} \label{t-for-one}{\bf t}_u = \begin{cases} \bf{0}&; \forall u \in s_{2j-1}\\ \begin{cases} t_{u,u}=1& \\t_{v,u}=1;&\forall v \in s_{2j-1}\\t_{v,u}=0;& \text{otherwise} \end{cases} &; \forall u \in s_{2j}\end{cases} ; \forall j=1,...,\frac{m}{2}. \end{equation} \end{itemize} Hence, the overall altered transition matrix (i.e. under a given mutation $(i,k)$) of a given BNp with transition matrix ${\bf P}= (1-p)^n.{\bf Q}+{\bf H}$ can be written as \begin{equation} \label{additivepart} \tilde {\bf P} = {\bf P} + \underbrace{(1-p)^n.{\bf Q}.({\bf T}_{i,k}-{\bf I})}_{\text {additive perturbation}}.\end{equation} As we know, in general for any two matrices ${\bf A}$ and ${\bf B}$ we have \[Rank({\bf AB})\leq \min \left\{Rank({\bf A}),Rank({\bf B})\right\}.\] Now we prove the following: For every transformation matrix, ${\bf T}_{i,k}$, related to the mutation $(i,k)$ in an n-gene network, we have \begin{equation}\label{rank} Rank({\bf T}_{i,k}-{\bf I})= 2^{n-1}.\end{equation} proof: It is enough to show that ${\bf T}_{i,k}-{\bf I}$ has exactly $2^{n-1}$ nonzero eigenvalues. Every ${\bf T}_{i,k}$ can be considered as a transition matrix of a BN itself which exactly has $2^{n-1}$ singleton attractors. It shows that ${\bf T}_{i,k}$ has $2^{n-1}$ eigenvalues with value of $1$ and $2^{n-1}$ with value $0$, causing ${\bf T}_{i,k}-{\bf I}$ to have exactly $2^{n-1}$ zero eigenvalues and $2^{n-1}$ eigenvalues with value $-1$. From \eqref{additivepart} and \eqref{rank} we obtain that rank of perturbation is less than $2^{n-1}$, so \begin{equation*} Rank(\text{additive perturbation})\leq \min \left\{2^{n-1},Rank(\mathbf{Q}% )\right\}, \end{equation*} showing that the additive perturbation can be decomposed into at most $% 2^{n-1}$ rank-one matrices. Now, we want to show that these can be in the form of rank-one perturbations. Let ${\mathbf{b}_j}^T=\left[b_j^{(1)},...b_j^{(2^n)}% \right]$, and $\mathbf{a}_j$ make the $j$th rank-one perturbation. It can be shown that the corresponding components of these rank-one perturbation are as follow \begin{itemize} \item $k=0$ \begin{equation}\label{general-b-k-0} b_{(y-1)r+z}^{(w)}= \begin{cases} 1& \text{for $w= (2y-1)r+z-1$}\\ -1& \text{for $w=2yr+z-1$}\\ 0& \text{otherwise} \end{cases} ; \forall z \in \left\{1,...,r\right\} \forall y \in \{1,...,\frac{m}{2}\} \end{equation} and \begin{equation}\begin{array}{cccc} {\bf a}_{(y-1)r+z}={\bf q}_{{l_z}};& \forall l_z \in s_{2w},& \forall z \in \left\{1,...,r\right\},& \forall y \in \left\{1,...,\frac{m}{2}\right\}. \end{array} \end{equation} \item $k=1$ \begin{equation}\label{general-b-k-1} b_{(y-1)r+z}^{(w)}= \begin{cases} -1& \text{for $ w= (2y-1)r+z-1$,}\\ 1& \text{for $w=2yr+z-1$,}\\ 0& \text{otherwise} \end{cases} ; \forall z \in \{1,...,r\} \forall y \in \{1,...,\frac{m}{2}\} \end{equation} and \begin{equation} \begin{array}{cccc} {\bf a}_{(y-1)r+z} = {\bf q}_{{l_z}};& \forall l_z \in s_{2w-1},& \forall z \in \left\{1,...,r\right\}, &\forall y \in \left\{1,...,\frac{m}{2}\right\}. \end{array} \end{equation} As \eqref{general-b-k-0} and \eqref{general-b-k-1} show, ${\bf b}_j$s satisfy ${\bf b}_j^T.{\bf e}=0$. \end{itemize} \subsection*{Commutativity property of transformation matrices} For transformation matrices representing gene mutations we have \begin{equation}\label{commutative} {\bf T}_{i_1,k_1}.{\bf T}_{i_2,k_2} = {\bf T}_{i_2,k_2}.{\bf T}_{i_1,k_1} , \forall i_1 \neq i_2, \forall k_1,k_2 \in \{0,1\}. \end{equation} Proof: Let $\mathbf{X} = (x_1,...,x_n)$ determine the GAP. From \eqref{t-for-zero} and \eqref{t-for-one} one can see that each ${\bf T}_{i,k}$ corresponds to a BN itself which is shown here by $N_{i,k}$. We start our proof from the right hand side of \eqref{commutative}: ${\bf T}_{i_1,k_1}.{\bf T}_{i_2,k_2}$ is an altered copy of $N_{i_1,k_1}$ after the mutation $(i_2,k_2)$. It can be easily seen that, $N_{i_1,k_1}$ has ${2^{n-1}}$ basins in the general following form: \[\begin{array}{cc} & \mathbf{X} = (x_1,...x_{i_1-1},1-k_1,x_{i_1+1},...,x_n)\rightarrow {\hat{s}} = (x_1,...x_{i_1-1},k_1,x_{i_1+1},...,x_n)\rightarrow {\hat{\mathbf{X}}} ,\\ &\forall (x_1,...x_{i_1-1},x_{i_1+1},...,x_n)\in \{0,1\}^{n-1},\end{array}\] which contains both the following transitions \begin{equation} \label{transitions-in-the-first-matrix}\begin{array}{c}\mathbf{X} =(x_1,...x_{i_1-1},1-k_1,...,x_{i_2-1},1-k_2...,x_n)\rightarrow {\tilde{\mathbf{X}}}= (x_1,...x_{i_1-1},k_1,...,x_{i_2-1},1-k_2,...,x_n)\rightarrow {\tilde{\mathbf{X}}}\\ \mathbf{X} =(x_1,...x_{i_1-1},1-k_1,...,x_{i_2-1},k_2...,x_n)\rightarrow \tilde{\mathbf{X}} = (x_1,...x_{i_1-1},k_1,...,x_{i_2-1},k_2,...,x_n)\rightarrow {\tilde{\mathbf{X}}},\end{array}\end{equation} where without loss of generality we assumed $i_1 \leq {i_2 - 1}$. (we already know that $i_1\neq i_2.$) Now, we apply the matrix ${\bf T}_{i_2,k_2}$ to the transitions in \eqref{transitions-in-the-first-matrix} which produces the following transitions: \begin{equation} \label{transitions-final}\begin{array}{cc} & (x_1,...x_{i_1-1},1-k_1,...,x_{i_2-1},1-k_2...,x_n)\rightarrow {\hat{\mathbf{X}}}\rightarrow {\hat{\mathbf{X}}} \\ & (x_1,...x_{i_1-1},k_1,...,x_{i_2-1},1-k_2,...,x_n)\rightarrow {\hat{\mathbf{X}}}\rightarrow {\hat{\mathbf{X}}}\\ & (x_1,...x_{i_1-1},1-k_1,...,x_{i_2-1},k_2...,x_n)\rightarrow {\hat{\mathbf{X}}}\rightarrow {\hat{\mathbf{X}}} ,\end{array}\end{equation} where ${\hat{\mathbf{X}}}= (x_1,...x_{i_1-1},k_1,...,x_{i_2-1},k_2,...,x_n)$, and $(x_1,...x_{i_1-1},x_{i_1+1},...,x_{i_2-1},x_{i_2+1}...,x_n)\in \{0,1\}^{n-2}$. So, the final altered BN would have $2^{n-2}$ basins in the form of \eqref{transitions-final}. Now, suppose that one wants to compute ${\bf T}_{i_2,k_2}.{\bf T}_{i_1,k_1}$: in this case we should alter $N_{i_2,k_2}$ by the mutation $(i_1,k_1)$. In a similar way, $N_{i_2,k_2}$ has ${2^{n-1}}$ basins in the general following form: \[\begin{array}{cc} &\mathbf{X} = (x_1,...x_{i_2-1},1-k_2,x_{i_2+1},...,x_n)\rightarrow {\hat{\mathbf{X}}} = (x_1,...x_{i_2-1},k_2,x_{i_2+1},...,x_n)\rightarrow {\hat{\mathbf{X}}} ,\\ &\forall (x_1,...x_{i_2-1},x_{i_2+1},...,x_n)\in \{0,1\}^{n-1},\end{array}\] which contains both the following transitions \begin{equation} \label{transitions-in-the-first-matrix-2}\begin{array}{c}\mathbf{X} =(x_1,...x_{i_1-1},1-k_1,...,x_{i_2-1},1-k_2...,x_n)\rightarrow {\tilde{\mathbf{X}}}= (x_1,...x_{i_1-1},1-k_1,...,x_{i_2-1},k_2,...,x_n)\rightarrow {\tilde{\mathbf{X}}}\\ \mathbf{X} =(x_1,...x_{i_1-1},k_1,...,x_{i_2-1},1-k_2...,x_n)\rightarrow \tilde{\mathbf{X}} = (x_1,...x_{i_1-1},k_1,...,x_{i_2-1},k_2,...,x_n)\rightarrow {\tilde{\mathbf{X}}}.\end{array}\end{equation} Now, we apply the matrix ${\bf T}_{i_1,k_1}$ to the transitions in \eqref{transitions-in-the-first-matrix} which exactly produces the transitions in \eqref{transitions-final}. This shows that the produced BNs with the both multiplication orderings are the same and so are their transition matrices meaning that ${\bf T}_{i_1,k_1}.{\bf T}_{i_2,k_2} = {\bf T}_{i_2,k_2}.{\bf T}_{i_1,k_1}.$ %%%%%%%%%%%%%% Approximated probability of algorithm failure %%%%%%%%%%%%%%% \subsection*{Approximated probability of algorithm failure} Suppose that the probability of algorithm failure is defined as \begin{equation} \label{p-miss-definition appendix} P_{miss} = \Pr(\left[N_{normal};Path_c\right]\notin {\mathcal{BN}}^M). \end{equation} Define \begin{equation} \label{corresponding} p_i := {F_D}^{(M-i,{p_{cancer}})}(\beta_i),\forall i=1,...,M \newline \end{equation} where ${F_D}^{(M-i,{p_{cancer}})}(d)$ is the CDF of the distance $d$ between a BNp and its $(M-i)$-fold mutated counterpart. Defining \begin{align} \label{error approximate} & p_{miss}^{(i+1)} = (1-p_{i+1})^{M-i}(1-p_{miss}^{(i)})+p_{miss}^{(i)}; 1 \leq i\leq {M-1} \\ \nonumber &p_{miss}^{(1)} = (1-p_1)^M \end{align} we will have the following approximation \begin{equation} \label{missing error} P_{miss} \approx p_{miss}^{(M)}. \end{equation} Proof: We start proving \eqref{error approximate} by finding the probability of missing in the first step of the algorithm. As we mentioned before, $Path_c$ is a {\it set} of alterations meaning that the occurrence ordering does not matter. Hence the probability of missing in the first step can be found as follows \begin{equation}\label{appendix-p-miss-first}\begin{array}{cc} p^{(1)}_{miss} & = Pr(\nexists alt \in Path_c \ni [N_{normal};alt] \in \mathcal{BN}^1)\\ &\\ &= \Pr( \nexists alt \in Path_c \ni R([N_{normal};alt]) \leq \beta_1) \\ &\\ &\stackrel{A}{=} \Pr( \forall alt \in Path_c : \rho(SSD_{cancer},SSD([N_{normal};alt]))> \beta_1)\\ &\\ &\stackrel{B}{\approx} \left\{{\Pr}( \rho(SSD_{cancer},SSD([N_{normal};alt])) > \beta_1)\right\} ^{M}\\ &\\ &= {\{1- ({F_D}^{(M-1,p)}(\beta_1))\}}^M , \end{array}\end{equation} where in $B$, we assume independent which may not happen in general. \begin{comment}Generally, $p_{cancer}$ is unknown and has to be estimated from the data (e.g. gene expression); however, in our approximation we assume exact estimation. \end{comment} Since $Path$ does not have any order, there are many possibilities that can happen for the second step. We approximate this probability as \begin{align} p^{(2)}_{miss} = \sum_{i=0}^{M}{\left(\begin{array}{c}M\\i\end{array}\right)\Pr\left\{miss|E_i\right\}P(E_i)},\end{align} where $E_i$ is an {\it event} in which only $M-i$ mutations from $Path_c$ make the residual less than $\beta_1$ in the first iteration. We obtain \begin{equation}\begin{array}{cc} p^{(2)}_{miss}&=\sum_{i=0}^{M-1}{\left(\begin{array}{c}M\\i\end{array}\right)\Pr\left\{{miss}|E_i\right\}p(E_i)}+\Pr\left\{miss|E_M\right\}p(E_M)\\ &\\ &= \sum_{i=0}^{M-1}{\left(\begin{array}{c}M\\i\end{array}\right)\Pr\left\{miss|E_i\right\}p(E_i)} + p^{(1)}_{miss}\\ &\\ &\le \Pr\left\{miss|E_{M-1}\right\}\sum_{i=0}^{M-1}{\left(\begin{array}{c}M\\i\end{array}\right)p(E_i)} + p^{(1)}_{miss}\\ &\\ &= \Pr\left\{{miss}|E_{M-1}\right\}(1-{p_{miss}}^{(1)})+p^{(1)}_{miss}\end{array} \end{equation} which similar to \eqref{appendix-p-miss-first} can be written as \[\begin{array}{c} p^{(2)}_{miss} \le \Pr\left\{miss|E_{M-1}\right\}(1-{p_{miss}}^{(1)}) + p^{(1)}_{miss}\\ \\ \approx (1-{p_{miss}}^{(1)}){\{1- ({F_D}^{(M-2,p)}(\beta_2)\}}^{M-1} + p^{(1)}_{miss} \end{array}\] This procedure should be done for the next steps as before which completes the proof. \end{bmcformat} \end{document}