Reseach Projects- Optimal Perturbation Control of General Topology Molecular Networks


	Optimal Perturbation Control of General Topology Molecular Networks Abstract: We develop a comprehensive framework for optimal perturbation control of dynamic networks. The aim of the perturbation is to drive the network away from an undesirable steady-state distribution and to force it to converge towards a desired steady-state distribution. The proposed framework does not make any assumptions about the topology of the initial network, and is thus applicable to general-topology networks. We define the optimal perturbation control as the minimum-energy perturbation measured in terms of the Frobenius-norm between the initial and perturbed probability transition matrices of the dynamic network. We subsequently demonstrate that there exists at most one optimal perturbation that forces the network into the desirable steady-state distribution. In the event where the optimal perturbation does not exist, we construct a family of suboptimal perturbations, and show that the suboptimal perturbation can be used to approximate the optimal limiting distribution arbitrarily closely. Moreover, we investigate the robustness of the optimal perturbation control to errors in the probability transition matrix, and demonstrate that the proposed optimal perturbation control is robust to data and inference errors in the probability transition matrix of the initial network. Finally, we apply the proposed optimal perturbation control method to the Human melanoma gene regulatory network in order to force the network from an initial steady-state distribution associated with melanoma and into a desirable steady-state distribution corresponding to a benign cell. MATLAB & MATHEMATICA Code: DATA: Download the probability transition matrix of the Human melanoma network: P_matrix.xls CODE: The MATLAB code uses CVX, a package for specifying and solving convex programs. One needs to download and install the CVX package before running the MATLAB programs. For CVX installation instructions, please follow the guidelines in http://cvxr.com/cvx/download Download the MATLAB code: main.m with output the minimal-perturbation energy matrix.xls Download the MATLAB function: optimal_control.m The main code is also displayed below: 0001 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0002 %% The program "main.m" reads a square matrix, 0003 %% the probability transition matrix of the network, 0004 %% "P_matrix.xls" in excel format and returns as output 0005 %% the optimal perturbation matrix "C", which forces 0006 %% the network to transition from its initial steady-state to the desired steady-state pid. 0007 %% This code uses the CVX software. For installation instructions, please 0008 %% refer to: http://cvxr.com/cvx/download 0009 0010 %% Define the variables 0011 %%%%%%%%%%%%%%%%%%%%%%%% 0012 0013 [P0,Txt,Raw]=xlsread('P_matrix'); 0014 N = size(P0,1); 0015 OneVec = ones(N,1); 0016 0017 0018 %% Define the desired steady-state distribution 0019 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0020 0021 pid1 = 0.015525 * ones(64,1); 0022 pid2 = 10^(-4) * ones(64,1); 0023 pid = [pid1;pid2]; 0024 0025 0026 %% Call the optimal_control function 0027 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0028 0029 C = optimal_control(P0,pid); 0030 0031 0032 %% Writing and plotting 0033 %%%%%%%%%%%%%%%%%%%%%%%% 0034 0035 success = xlswrite('Perturbation-Matrix', C); 0036 Download the MATHEMATICA code (Figs. 2 and 3): Matematica_code.nb Download the matrices in Fig. 2 in excel format: P0.xls P1.xls P2.xls Pepsilon.xls Pa.xls Download the matrices in Fig. 4 in excel format: G0.xls G1.xls G2.xls Gepsilon.xls Ga.xls