#MCSS
Cancer is characterized by numerous somatic mutations, however, only a subset of mutations, called driver mutations, contribute to tumor growth and progression. Genes and pathways containing driver mutations are called driver genes and driver pathways respectively. Recent studies indicated that mutations in a driver pathway often appear in most samples while for any sample only a singe or few mutations appear because a single mutation is capable to perturb the whole pathway. Based on these findings, a new method called minimum cost subset selection (MCSS) is developed for de novo discovery of driver mutation pathways in cancer studies.
- We provide a novel algorithm MCSS based on a non-convex approximation to the original combinatorial optimization problem and with regularization for de novo discovery of mutated driver pathways.
- MCSS is designed to find multiple mutated driver pathways without the need to pre-specify a number of pathways and/or a number of genes.
- MCSS is flexible for integrative analysis of multiple types of genomic data. Currently, in addition to the standard analysis of a single mutation dataset, it can integrate a mutation dataset with a gene expression data (which may or may not be drawn on the same set of the subjects for the mutation data).
To allow the researchers to revise and extend MCSS more easily, we deliberately provide the source code instead of a toolbox. The researchers can either copy the source files into their working directory or use the following code to add a path. The source code is fully tested in Matlab 2012a and may fail in other versions of the Matlab. Please send us an email (wuxx0845@umn.edu) when you encounter any problems.
addpath(genpath(<path to your MCSS directory>))Based on the corresponding new concepts of coverage and mutual exclusivity, MCSS is designed for de novo discovery of mutated driver pathways in cancer. Since the computational problem is a combinatorial optimization with an objective function involving a discontinuous indicator function in high dimension, many existing optimization algorithms, such as a brute force enumeration, gradient descent and Newton’s methods, are practically infeasible or directly inapplicable. That is why we develop MCSS based on a novel formulation of the problem as non-convex programming with regularization. The method is computationally more efficient, effective and scalable than existing Monte Carlo searching and several other algorithms. For more details, see our paper.
- Binghui Liu, Chong Wu, Xiaotong Shen, and Wei Pan. "CA novel and efficient algorithm for de novo discovery of mutated driver pathways in cancer." Submitted to Annals of Applied Statistics.
We use the following simple simulated data example and corresponding code to help researchers understand and use our method.
- First, we generate a simple n by p mutation data set A with a 1 indicating a mutation and 0 otherwise, where n and p are the number of observations and the number of genes, respectively. For each patient, a gene in a driver pathway is randomly selected and it mutates with probability p1, and another gene in the driver pathway was randomly selected to have a mutation probability p2. Other genes outside the pathway mutated with probability p3.
n = 100; p = 500; p1 = 0.95; p2 = 0.01; p3 = 0.05
A=zeros(n,p); % A is n by p matrix
s=RandStream('mcg16807','Seed',1); % set the seed.
RandStream.setGlobalStream(s);
M1=1:4;
beta_M1=zeros(p,1);
beta_M1(M1,1)=ones(length(M1),1);
for i=1:n
g1=randperm(length(M1));
g1=M1(g1(1));
A(i,g1)=random('bino',1,p1,1,1);
if A(i,g1)==1
M2=setdiff(M1,g1);
g2=randperm(length(M1)-1);
g2=M2(g2(1));
A(i,g2)=random('bino',1,p2,1,1);
end
end
M3=setdiff(1:p,M1);
for j=1:(p-length(M1))
A(:,M3(j))=random('bino',1,p3,n,1);
end
A(1:5,1:5)- Second, we apply the main function MCSS to get the estimates. The non-zero estimates stand for the driver mutations/genes.
beta_new = MCSS(A,1,1,0.1);
% function beta_new = MCSS(A,lambda,tau1,tau2,alpha, beta_0, method,weight,max_iter_num, Err,max_iter_num_s,Err_s)
% Input:
% A--n*p mutation matrix
% lambda--a tuning parameter controlling the sparseness of the solutions
% tau1--a tuning parameter in truncated lasso penalty
% tau2--a tuning parameter relates to lambda
% alpha--a tuning parameter relates to ridge penalty
% The default value is 0.001.
% beta_0--starting value of beta.
% If we set beta_0=-1, a good starting value will be calculated.
% method--method for step2 in Algorithm 1. 2 (default) stands for CVX.
% weight--weight of exclusivity cost. The default is 1.
% max_iter_num--maximum number of iterations for difference of convex step.
% The default value is 20.
% Err--termination condition for difference of convex step.
% The default value is 0.001.
% max_iter_num_s--maximum number of iterations for sub-gradient algorithm.
% The default value is 20.
% Err_s--termination condition for sub-gradient algorithm.
% The default value is 0.001.
%
% Output:
% beta_new--a resulting estimate of beta.
%
% Author: Binghui Liu and Chong Wu (wuxx0845@umn.edu)
% Maintainer: Chong Wu (wuxx0845@umn.edu)
% Version: 1.0Selecting the tuning parameters is an important step for using MCSS. A better choice of tuning parameters will lead to a better result. Here, we provide a simple function that can select the tuning parameters via sample splitting procedure automatically. Specifically, we randomly partition the observations into training data and tuning data. We train the model based on the training data and use the tuning data to determine the tuning error (cost value) for each combination of tuning parameters. We repeat this process M time. Since the results only depend on the ratio of tau1/tau2, we only search tau2 and fix tau1. See more details in Model selection subsection (2.5) in our upcoming paper.
Lambda=(1:4:10)*1;
tau1=1;
tau2=0.1;
[beta_t,CF] = MCSS_CV1(A,Lambda,tau2,tau1);
% Help:
% function [beta_t,CF] = MCSS_CV1(A,Lambda,Tau2, tau1,alpha, beta_0, method,weight,max_iter_num, Err,max_iter_num_s,Err_s,M)
% Input:
% A--n*p mutation matrix
% Lambda--a set of tuning parameters for lambda.
% Tau2--a set of tuning parameters for tau2.
% tau1--a tuning parameter in truncated lasso penalty
% alpha--a tuning parameter relates to ridge penalty
% The default value is 0.001.
% beta_0--starting value of beta.
% If we set beta_0=-1, a good starting value will be calculated.
% method--method for step2 in Algorithm 1. 2 (default) stands for CVX.
% weight--weight of exclusivity cost. The default is 1.
% max_iter_num--maximum number of iterations for difference of convex step.
% The default value is 20.
% Err--termination condition for difference of convex step.
% The default value is 0.001.
% max_iter_num_s--maximum number of iterations for sub-gradient algorithm.
% The default value is 20.
% Err_s--termination condition for sub-gradient algorithm.
% The default value is 0.001.
% M--repeat CV for M times
%
% Output:
% beta_t--a resulting estimate of beta
% CF--cost values corresponding Lambda
%
% Author: Binghui Liu and Chong Wu (wuxx0845@umn.edu)
% Maintainer: Chong Wu (wuxx0845@umn.edu)
% Version: 1.0
MCSS is able to find multiple mutated driver pathways at the same time. Unlike existing methods, which usually need a pre-specified number of pathways and a pre-specified number of genes in each pathway, MCSS can automatically find multiple mutated driver pathways with low cost. The idea is that we can apply MCSS with different starting values and hopefully we can find multiple solutions that have low cost. We provide some code to generate a series of random starting values automatically and then apply MCSS_CV2 to select the "best" tuning parameters. Based on our empirical experience, using solutions from a subset of genes that have relatively large mutation rates may yield better solutions. The input of MCSS_CV2 is almost the same as MCSS_CV1 except beta_0 is a p by q matrix containing q different starting points.
Lambda=(1:4:10)*1;
tau1=1;
tau2=0.1;
alpha = 0.001;
T = 5;
%% Generate random starting values
beta_0_start = zeros(p,T);
b6=[];
for tt = 1:T
randIndex = tt;
s=RandStream('mcg16807','Seed',randIndex);
RandStream.setGlobalStream(s);
beta_0=random('bino',1,randi([2,8])/p,p,1);
b5=find(beta_0>0)';
while(ismember(sum(log([b5,0.5])),b6)==1)
beta_0=random('bino',1,randi([2,8])/p,p,1);
b5=find(beta_0>0)';
end
b6=[b6,sum(log([b5,0.5]))];
beta_0=zeros(p,1);beta_0(b5)=1;
beta_0_start(:,tt) = beta_0;
end
[beta_hat,M_list,CF_list] = MCSS_CV2(A,Lambda,tau2,tau1,alpha, beta_0_start);
% function [beta_hat,M_list,CF_list] = MCSS_CV2(A,Lambda,Tau2, tau1,alpha, beta_0, method,weight,max_iter_num, Err,max_iter_num_s,Err_s,M)
% Input:
% A--n*p mutation matrix
% Lambda--a set of tuning parameters for lambda.
% Tau2--a set of tuning parameters for tau2.
% tau1--a tuning parameter
% alpha--a tuning parameter with ridge penalty
% The default value is 0.001.
% beta_0--starting value of beta.
% If we set beta_0=-1, a good starting value will be calculated.
% method--method for step2 in Algorithm 1. 2 (default) stands for CVX.
% weight--weight of exclusivity cost. The default is 1.
% max_iter_num--maximum number of iterations for difference of convex step.
% The default value is 20.
% Err--termination condition for difference of convex step.
% The default value is 0.001.
% max_iter_num_s--maximum number of iterations for sub-gradient algorithm.
% The default value is 20.
% Err_s--termination condition for sub-gradient algorithm.
% The default value is 0.001.
% M--repeat CV for M times
%
% Output:
% beta_t--a resulting estimate of beta
% CF--cost values corresponding Lambda
%
% Author: Binghui Liu and Chong Wu (wuxx0845@umn.edu)
% Maintainer: Chong Wu (wuxx0845@umn.edu)
% Version: 1.0
MCSS can incorporate other types of genomic data; currently, it can integrate mutation data with gene expression data. Here, we show a simple example and provide the corresponding code（MCSS_ME, MCSS_ME_CV1, and MCSS_ME_CV2).
Since the genes in the same pathway collaborate with each other to perform the same or related biological function, their expression levels are usually more highly correlated than those from different pathways. Taking this prior knowledge into account, we can modify the objective function to include the correlations among the genes. For more details, see section 2.6 Integrative Analysis in our paper.
Along the way, we developed the MCSS_ME, MCSS_ME_CV1, and MCSS_ME_CV2 for implementing MCSS algorithm and selecting the tuning parameters with both mutation and gene expression data.
Unlike the MCSS, in MCSS_ME we need choose a suitable gamma to balance the contributions to the new cost function from mutation data and from gene expression data. After determining gamma, we can use MCSS_ME, MCSS_ME_CV1 and MCSS_ME_CV2 almost the same as before.
% generating mutation data
n = 100; p = 1000; p1 = 0.95; p2 = 0.01; p3 = 0.05
A=zeros(n,p); % A is n by p matrix
s=RandStream('mcg16807','Seed',1); % set the seed.
RandStream.setGlobalStream(s);
M1=1:4;
beta_M1=zeros(p,1);
beta_M1(M1,1)=ones(length(M1),1);
for i=1:n
g1=randperm(length(M1));
g1=M1(g1(1));
A(i,g1)=random('bino',1,p1,1,1);
if A(i,g1)==1
M2=setdiff(M1,g1);
g2=randperm(length(M1)-1);
g2=M2(g2(1));
A(i,g2)=random('bino',1,p2,1,1);
end
end
M3=setdiff(1:p,M1);
for j=1:(p-length(M1))
A(:,M3(j))=random('bino',1,p3,n,1);
end
A(1:5,1:5)
% Generating gene expression data, such that for the genes in the same set, their expression levels are highly correlated (0.9); for the genes in the different set, their expression levels are almost independent (0.1).
pG = p;
V=ones(pG,pG);
n_eig0=-1;
nnn=0;
while n_eig0 < 0
V=unifrnd(0.1,0.1,pG,pG); sum_random=4;
V(1:4,1:4)=unifrnd(0.9-0.0,0.9+0.0,4,4);
while pG-sum_random>20
a_random=randi([2,20]);
b_random=unifrnd(0.9,0.9); V((sum_random+1):(sum_random+a_random),(sum_random+1):(sum_random+a_random))=unifrnd(b_random-0.0,b_random+0.0,a_random,a_random);
sum_random=sum_random+a_random;
end
a_random=pG-sum_random;
b_random=unifrnd(0.9,0.9);
V((sum_random+1):(sum_random+a_random),(sum_random+1):(sum_random+a_random))=unifrnd(b_random-0,b_random+0,a_random,a_random);
V=tril(V); V=V+eye(pG); V=V+V'; V=min(V,1);
n_eig0=all(diag(eig(V)));
nnn=nnn+1;
end
V(1:10,1:10)
G=mvnrnd(zeros(pG,1),V,n); MCSS_ME combines the information from both mutation and gene expression data and returns the beta value.
beta_new = MCSS_ME(A,G,0.5,1,1,0.1);
% function beta_new =MCSS_ME(A,G,gamma,lambda,tau1,tau2,alpha, beta_0, method,weight,max_iter_num, Err,max_iter_num_s,Err_s)
%
% Input:
% A--n*p mutation matrix
% G--n*p gene expression matrix
% gamma--a tuning parameter controlling the contributions between mutation data and gene expression data
% lambda--a tuning parameter controlling the sparseness of the solutions
% tau1--a tuning parameter in truncated lasso penalty
% tau2--a tuning parameter relates to lambda
% alpha--a tuning parameter relates to ridge penalty
% The default value is 0.001.
% beta_0--starting value of beta.
% If we set beta_0=-1, a good starting value will be calculated.
% method--method for step2 in Algorithm 1. 1 (default) is sub-gradient and 2 stands for CVX.
% weight--weight of exclusivity cost. The default is 1.
% max_iter_num--maximum number of iterations for difference of convex step.
% The default value is 20.
% Err--termination condition for difference of convex step.
% The default value is 0.001.
% max_iter_num_s--maximum number of iterations for sub-gradient algorithm.
% The default value is 20.
% Err_s--termination condition for sub-gradient algorithm.
% The default value is 0.001.
%
% Output: beta_new--a resulting estimate of beta.
% Author: Binghui Liu and Chong Wu (wuxx0845@umn.edu)
% Maintainer: Chong Wu (wuxx0845@umn.edu)
% Version: 1.0MCSS_ME_CV1 works similar as MCSS_CV1, selecting the tuning parameters based on cross validation.
Lambda=(1:4:10)*1;
tau1=1;
tau2=0.1;
[beta_t,CF] = MCSS_ME_CV1(A,G,0.5,Lambda,tau2,tau1);
% function [beta_t,CF] = MCSS_ME_CV1(A,G, gamma, Lambda,Tau2, tau1,alpha, beta_0, method,weight,max_iter_num, Err,max_iter_num_s,Err_s,M)
% Input:
% A--n*p mutation matrix
% G--n*p gene expression matrix
% gamma--a tuning parameter controlling the contributions between mutation data and gene expression data
% Lambda--a set of tuning parameters for lambda.
% Tau2--a set of tuning parameters for tau2.
% tau1--tuning parameters
% alpha--tuning parameters with ridge penalty
% The default value is 0.001.
% beta_0--starting value of beta.
% If we set beta_0=-1, a good starting value will be calculated.
% method--method for step2 in Algorithm 1. 2 (default) stands for CVX.
% weight--weight of exclusivity cost. The default is 1.
% max_iter_num--maximum number of iterations for difference of convex step.
% The default value is 20.
% Err--termination condition for difference of convex step.
% The default value is 0.001.
% max_iter_num_s--maximum number of iterations for sub-gradient algorithm.
% The default value is 20.
% Err_s--termination condition for sub-gradient algorithm.
% The default value is 0.001.
% M--repeat CV for M times
%
% Output:
% beta_t--a resulting estimate of beta
% CF--cost values corresponding Lambda
%
% Author: Binghui Liu and Chong Wu (wuxx0845@umn.edu)
% Maintainer: Chong Wu (wuxx0845@umn.edu)
% Version: 1.0
MCSS_ME_CV2 works similar as MCSS_CV2. Based on multiple starting points, MCSS_ME_CV2 finds some gene pathways with low cost.
Lambda=(1:4:10)*1;
tau1=1;
tau2=0.1;
alpha = 0.001;
T = 5;
%% Generate random starting values
beta_0_start = zeros(p,T);
b6=[];
for tt = 1:T
randIndex = tt;
s=RandStream('mcg16807','Seed',randIndex);
RandStream.setGlobalStream(s);
beta_0=random('bino',1,randi([2,8])/p,p,1);
b5=find(beta_0>0)';
while(ismember(sum(log([b5,0.5])),b6)==1)
beta_0=random('bino',1,randi([2,8])/p,p,1);
b5=find(beta_0>0)';
end
b6=[b6,sum(log([b5,0.5]))];
beta_0=zeros(p,1);beta_0(b5)=1;
beta_0_start(:,tt) = beta_0;
end
[beta_hat,M_list,CF_list] = MCSS_ME_CV2(A,G,0.5,Lambda,tau2,tau1,alpha, beta_0_start);
% function [beta_hat,M_list,CF_list] = MCSS_ME_CV2(A,G,gamma,Lambda,Tau2, tau1,alpha, beta_0, method,weight,max_iter_num, Err,max_iter_num_s,Err_s,M)
% Input:
% A--n*p mutation matrix
% G--n*p gene expression matrix
% gamma--a tuning parameter controlling the contributions between mutation data and gene expression data
% Lambda--a set of tuning parameters for lambda.
% Tau2--a set of tuning parameters for tau2.
% tau1--tuning parameters
% alpha--tuning parameters with ridge penalty
% The default value is 0.001.
% beta_0--starting value of beta.
% If we set beta_0=-1, a good starting value will be calculated.
% method--method for step2 in Algorithm 1. 2 (default) stands for CVX.
% weight--weight of exclusivity cost. The default is 1.
% max_iter_num--maximum number of iterations for difference of convex step.
% The default value is 20.
% Err--termination condition for difference of convex step.
% The default value is 0.001.
% max_iter_num_s--maximum number of iterations for sub-gradient algorithm.
% The default value is 20.
% Err_s--termination condition for sub-gradient algorithm.
% The default value is 0.001.
% M--repeat CV for M times
%
% Output:
% beta_t--a resulting estimate of beta
% M_list--resulting list of gene sets corresponding to Lambda
% CF--cost values corresponding to Lambda
%
% Author: Binghui Liu and Chong Wu (wuxx0845@umn.edu)
% Maintainer: Chong Wu (wuxx0845@umn.edu)
% Version: 1.0
- For latest releases and announcements, follow on GitHub: @ChongWu-Biostat
Copyright (c) 2013-present, Chong Wu (wuxx0845@umn.edu) & Binghui Liu (liubh100@nenu.edu.cn)