# Combinatorial Therapy Discovery using Mixed Integer Linear Programming

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3 Running Time (seconds) 2.3 BTSC and MOTSC Problems are NP-hard The NP-hard property of the BTSC problem can be proved through reduction mapping. By setting α to 1, the maximum coverage problem becomes finding a maximum cover on the disease genes in T using at most k drugs from M. This reduction shows that BTSC is at least as hard as the maximum coverage problem, which is in the NP-hard problem set (Cohen and Katzir, 2008). MOTSC can be viewed as a modified version of the Red Blue Set Cover (RBSC), which is also a generalization of the standard set cover problem (Miettinen, 2009). RBSC is much harder than the standard set cover problem and there exists no polynomial approximation with a factor of 2 (4 log n)1 ϵ for any ϵ > 0 (Peleg, 2007). 2.4 Mathematical Programming Formulation Mathematically, BTSC can be formulated using the Mixed Integer Linear Programming (MILP) as follows: minimize α p i=1 (1 y i) + (1 α) p+q i=p+1 y i (1) subject to (Bx) i y i = 0 (2) by i y i 0 (3) y i y i 0 (4) m j=1 x j k (5) (Lx) 1 (6) y i Z+, y i, x j {0, 1} (7) The formulation has three variables, x, y, and y. The first two are binary, and the last one is nonnegative. The binary solution vector x indicates which drugs are selected. The non-negative cost variable y i counts the times that the i th gene is covered by the selected drugs. The binary cost vector y is derived from y, y i = 1 if y i 1; otherwise, y i = 0. The value of y i indicates whether the i th gene is covered. We note that in vectors y and y, the first p elements are associated with the on-target genes and the next q elements are associated with the off-target genes. The relation between a given disease D and the associated drugs M can be represented using a binary matrix B, where the rows are indexed by the on-targets (p) and off-targets (q), and the columns represent the drugs. B im = 1 if the i th gene in D is covered by the m th drug in M; otherwise, B im = 0. The non-zero elements in the product vector of B and x indicate the corresponding genes targeted by a selection of drugs. The drug-drug adverse interaction effects are encoded in L, a l m binary matrix where, for each row k, L ki = L kj = 1 if the drugs M i and M j have an adverse effect when used together. The intuition of the MILP formulation is the following. The equality constraint (2) counts the number of times that an on-target or an off-target is covered by the selected drug combination. However, y cannot be directly used in the objective cost function (1). Thus, the binary cost y is introduced and related to y by the inequality constraints (3) and (4). The inequality constraint (3) requires y i = 1 if y i 1. For that, the value of b needs to be at least the maximum value of all y i. The inequality constraint (4) guarantees that y i = 0 whenever y i = 0. The maximum number of drugs for any feasible solution is bounded by the inequality constraint (5). The inequality constraint (6) guarantees any feasible solution to avoid the drug-drug adverse effects encoded in L. In this paper, we solved BTSC problem using the GNU MILP solver GLPK which is based on a branch and cut algorithm. MOTSC can be formulated similarly (Supplementary Text 2). The dual problems of BTSC and MOTSC are further discussed in the Supplementary Texts 3 and Iterative Search and Online Tool To make our approach more accessible in practice, we developed a webbased interactive tool (http://www.drug.liuzlab.org/) that will allow users to iteratively refine the search results. Both BTSC and MOTSC are ES BTSC Number of Drugs Fig. 1. Running time difference between BTSC and ES. The number of associated drugs is simulated from 10 to 30 with a step size equal to 2. Each data point represents the average running time of 10 replicates. For the simulated data with 30 associated drugs, ES takes about 4.7 days, while BTSC only needs 9 seconds. implemented with the options to filter out drugs based on approval status, drug-drug adverse interaction, and drug action direction. In addition, our web tool can also generate the output result files for Cytoscape visualization (Shannon et al., 2003). 3 RESULTS 3.1 Time Complexity and Accuracy Analysis To evaluate the time complexity and accuracy of our algorithm, we compared it to exhaustive search (ES) on a simulated data set. According to the formulation in section 2.4, the simulated data set is controlled by four parameters: (1) the number of columns in B, which represents the number of drugs associated with a disease gene set; (2) the number of rows in B, which represents the total ontarget and off-target space; (3) the density of B, which represents the percentage of genes targeted by a drug; (4) the value of p, which represents the number of on-targets. The first parameter, the number of columns in B, is the most important factor affecting the running time of ES. Therefore, we generated B by varying the column size from 10 to 30 with a step size equal to 2. At the same time, the second parameter was set to 1000 and B was sampled from a Bernoulli distribution with hitting probability equal to 0.01 (the third parameter). In addition, we further removed those rows in B for which the sum across the columns is equal to 0. We sampled p indexes (the third parameter) of the rows in B, with a sampling rate equal to 10%. Then, we simulated the data ten times with each varying number of the columns in B, applied BTSC and ES to find the solution x and compared their differences in the cost and running time. Our results demonstrated that there is no difference in the cost function between the BTSC and ES solutions. The running time of ES however increases exponentially, while BTSC tends to generate the correct results significantly faster (Figure 1). From these results, we found that ES is impractical even for finding combinations among a small number of drugs. For example, a search for combinations among 30 drugs using ES will take about 4.7 days to get the optimal solution. However, BTSC can obtain the 3

4 N Formylmethionine LTF LCN2 DPP6 CTSD COX6C HLA B COX7B MBL2 GLA AMY2B MAN1A1 NPR3 COX6B1 GALNT1 IGHG2 UBC CLPP B2M ITGA4 CEACAM1 CXCL12 EFNB2 VEGFA GZMB BCHE IGF1R CPD IGF2R LDLR RNASE1 PTGS1 COX4I1 COX7A1 COX8A GP1BA SIGLEC7 IGHE COX6A2 ADAM33 CSF2RB CES1 NOS1 Tinzaparin Dalteparin S100B MT CO2 CHI3L1 CGA AZGP1 Alpha D Mannose CD2 COX7C SELP CTLA4 IFNB1 COX5B APOH DPP4 IL12B MT CO3 ACPP Heparin IL6R MT CO1 MT ND1 LYZ RABGGTA IGHG1 Ardeparin Sulodexide MAN1B1 LCT PLA2G1B SERPIND1 DNASE1 COX5A CES1P1 LGALS1 CD209 S100G RABGGTB C1R DB04673 DB CD46 RHO Nadroparin FOS EGF FGA Sucralfate PGA3 FGF2 LRP1 Alteplase Reteplase...3 Tenecteplase Fondaparinux_sodium Enoxaparin COL4A5 COL4A2 COL4A4 FGB FGG SERPINC1 TFPI COL4A6 F3 FCER1A Rivaroxaban DB07207 DB Coagulation_factor_VIIa DB04590 DB COL4A1 COL4A3 F7 F9 GGCX PF4 AHSP F8 HPN Gamma Carboxy Glutamic_Acid PROC AMI BGLAP THBD DB07211 DB PROS1 F5 Drotrecogin_alfa F2R DB07376 PLAT F2 ANXA2 CP DB04134 DB07684 Streptokinase CLEC3B SERPINB6 DB07718 CALR SERPINA5 SERPINB2 PLG Bicine KRT8 CANX PLAUR MIF Beta (2 Naphthyl) Alanine DB03865 PRSS1 Lepirudin Argatroban...72 Iloprost Urokinase DB06854 SERPINE1 PLAU PTGER1 YARS Suramin LRP2 PDE4A Aprotinin PIN1 PDE4D DB01767 DB DB02193 DB PDE4B NID1 ST14 DB00006 DB06695 PTGIR PDE4C FSHR SIRT5 PLA2G2A KLK1 RYR1 P2RY2 CTRB1 F10 RTN4R GUSB PROCR which is collected in the Molecular Signatures Database (version 3.1) (Subramanian et al., 2005; Liberzon et al., 2011). From 35 associated drugs, BTSC predicted that Streptokinase, Fondaparinux sodium Enoxaparin, DB04134, and DB07376 can be combined to target the AMI gene set (Figure 2). In the four-drug combination, Streptokinase is able to dissolve blood clots that have formed in the blood vessels. Fondaparinux, as well as Enoxaparin, is an anticoagulant that helps prevent the formation of blood clots. The co-administration of Streptokinase and Fondaparinux in a published cohort of ST-elevation myocardial infarction patients significantly reduced the risk of death, recurrence and severe bleeding (Peters et al., 2008). The combination of Streptokinase and Enoxaparin is effective in patients with AMI (Simoons et al., 2002; Giraldez et al., 2009). Two experimental compounds, DB04134 and DB07376, were also identified in the solution. DB04134 is an antifibrinolytic agent that acts by inhibiting plasminogen activators which have fibrinolytic properties. DB07376 is an anticoagulant that can prevent blood clotting. Since these two compounds target different AMI genes, the use of these two compounds in combination with Streptokinase and/or Fondaparinux sodium Enoxaparin may exert additional benefits in treating AMI patients. This result indicated that BTSC is able to identify known drug combinations as well as novel compounds that are not covered by existing therapies. Fig. 2. The disease-gene-drug network of AMI. The disease gene set name AMI is represented using a cyan hexagon and there are 20 genes (red, orange or chocolate circles) in AMI gene set. Four (green squares) of 35 associated drugs (green or gray squares) selected by BTSC can cover 7 AMI genes (red circles) with no known off-target. Eight genes (orange circles) have no associated drugs and 5 genes (chocolate circles) associated with drugs are not covered by the selected drug combination. optimal solution within 9 seconds. We also varied the other three parameters in simulation. There is still no cost difference between BTSC and ES and the effects of these three parameters on the running time difference are much less important than that of the first parameter (data not shown). Similar results were obtained on MOTSC (Supplementary Figure 1). Thus, through this simulation, we demonstrated that our algorithms can achieve high accuracy with much fast running time. To demonstrate the power of our approach in searching for optimal drug combinations, we applied it on six disease gene sets: acute myocardial infarction, EGFR-PI3K-AKT-mTOR pathway, hypertension, type 2 diabetes mellitus, Parkinson s disease and schizophrenia. All the search parameters used for the following results are available in the Supplementary Text 5. All the following networks are visualized using Cytoscape (Shannon et al., 2003). Drugs are represented using DrugBank names, except that drugs with long names are represented using DrugBank IDs. Further details are provided in the online document. 3.2 Acute Myocardial Infarction Acute myocardial infarction (AMI) is the irreversible necrosis of heart muscle with the usual cause that a coronary artery is obstructed by an acute thrombus. The AMI related 20 genes were derived from BioCarta AMI pathway (http://www.biocarta.com) 3.3 EGFR-PI3K-AKT-mTOR Pathway EGFR-PI3K-AKT-mTOR (EPAM) pathway plays an important role in a lot of cellular processes and is abnormally activated in a variety of human diseases including cancer and neurological disorders. Eighteen genes in the EPAM pathway were obtained from Hennessy et al. (2005) and Morris et al. (2011). From 41 associated drugs, BTSC predicted that Lapatinib, Everolimus, DB08059, DB07812, and DB06831 can be combined to target the EPAM pathway (Figure 3). In the five-drug combination, Lapatinib is a selective inhibitor developed for both EGFR and ERBB2 (Opdam et al., 2012; Diaz et al., 2010). Everolimus, a derivative of Rapamycin, can act as an mtor inhibitor and block the signal transduction for cell proliferation. Through literature search, we discovered that the combination of Lapatinib and Everolimus shows efficacy in patients with advanced cancers in a phase I study (Gadgeel et al., 2013). This combination is also under a clinical trial (NCT ) in patients with advanced triple negative breast cancer, and under a phase 1b/2 clinical trial (NCT ) for treatment of HER-2 positive breast cancer with CNS metastasis. One consequence of mtor inhibition by Rapamycin or Rapamycin derivatives is the loss of inhibition on PI3K and AKT, resulting increased pathway activity. In addition, patients treated with EGFR inhibitors often develop drug resistance through secondary mutations (Kobayashi et al., 2005). To address these issues, our combinatorial therapy selected DB08059 (wortmannin), a known inhibitor of PI3K, and an experimental inhibitor (DB07812) of AKT and GSK3β. Therefore, simultaneously targeting multiple genes in the EPAM pathway will offer the hope to increase the efficacy of drug treatment and delay the development of drug resistance. 4

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